CONJECTURES OF ALPERIN AND BROUÉ FOR 2-BLOCKS WITH ELEMENTARY ABELIAN DEFECT GROUPS OF ORDER 8

RADHA KESSAR, SHIGEO KOSHITANI, AND MARKUS LINCKELMANN

Abstract. Using the classification of finite simple groups, we prove Alperin’s weight conjecture and the character theoretic version of Broué’s abelian defect conjecture for 2-blocks of finite groups with an elementary abelian defect group of order 8.

1. Introduction

Theorem 1.1. Let k be an algebraically closed field of characteristic 2, let G be a finite group and let b be a block of kG with an elementary abelian defect group P of order 8. Denote by c the block of kN_{G}(P) corresponding to b. Then b and c have eight ordinary irreducible characters and there is an isotypy between b and c; in particular, Alperin’s weight conjecture holds for all blocks of finite groups with an elementary abelian defect group P of order 8.

For principal blocks, Theorem 1.1 follows from work of Landrock [44] and Fong and Harris [30]. Theorem 1.1 implies Alperin’s weight conjecture for all blocks with a defect group of order at most 8. Indeed, the two remaining abelian cases C_{8} and C_{2} × C_{4} admit no automorphisms of odd order, hence arise as defect groups only of nilpotent blocks, and by work of Brauer [10] and Olsson [58], Alperin’s weight conjecture is known in the case of the two non-abelian groups D_{8} and Q_{8} of order 8.

The proof of Theorem 1.1 uses the classification of finite simple groups. Using a stable equivalence due to Rouquier we show in Theorem 4.1 that Alperin’s weight implies Broué’s isotypy conjecture. By the work of Landrock already mentioned, Alperin’s weight conjecture follows from the ‘if’ part of Brauer’s height zero conjecture (which predicts that all characters in a block have height zero if and only if the defect groups are abelian). The ‘if’ part of this conjecture has been reduced to quasi-simple finite groups by Berger and Knörr [5]; we verify in Theorem 3.1 that this reduction works within the realm of blocks with an elementary abelian defect group of order at most 8 and certain fusion patterns. We finally prove Alperin’s weight conjecture for blocks with an elementary abelian defect group of order 8 for quasi-simple groups in the remaining sections (making use of contributions of many authors, including the proof of Brauer’s height zero conjecture for blocks in non-defining characteristic of quasi-simple finite groups of type A and ^{2}A by Blau and Ellers [6]). Assembling these parts yields the proof of 1.1 in §19. It is noticeable how few blocks of quasi-simple groups have an elementary abelian defect group - and when they do, most of them are nilpotent; see Corollary 9.2, Theorems 11.1, 12.1, Lemma 12.4, Theorem 13.1 and Proposition 14.1 below.

Alperin announced the weight conjecture in [1]. At that time, the conjecture was known to hold for all blocks of finite groups with cyclic defect groups (by work of Brauer and Dade), dihedral, generalised quaternion, semidihedral defect groups (by work of Brauer and Olsson), and all defect groups admitting only the trivial fusion system (by the work of Broué and Puig on nilpotent blocks). Since then many authors have contributed to proving Alperin’s weight conjecture for various classes of finite groups - such as finite p-solvable groups (Okuyama), finite groups of Lie type in defining characteristic (Cabanes), symmetric and general linear groups (Alperin, Fong, An) and some sporadic simple groups (An). It appears that Theorem 1.1 is the first proof of a case of Alperin’s weight conjecture for all blocks with a fixed defect group since this conjecture was announced more than two decades ago.

2. Background material

For general background on block theory we refer to [69]. Let O be a complete discrete valuation ring with quotient field K and algebraically closed residue field k of positive characteristic p. We suppose that K is large enough for the finite groups under consideration. Given a finite group G and a block b of OG or of kG, we denote by Irr_{K}(G, b) the set of ordinary irreducible K-valued characters of G associated with b and by Irr_{k}(G, b) the set of irreducible Brauer characters of G associated with b. We set ℓ(b) = |Irr_{k}(G, b)|. We denote by ℤIrr_{K}(G, b) the group of class functions on G generated by Irr_{K}(G, b), and by ℤIrr_{k}(G, b) the corresponding group of class functions on the set of p-regular elements in G. By a classical result of Brauer, the decomposition map ℤIrr_{K}(G, b) → ℤIrr_{k}(G, b) induced by restriction of class functions to p-regular elements is surjective. The kernel of the decomposition map, denoted by L^{0}(G, b), consists of all class functions associated with b which vanish on the set of p-regular elements of G, or equivalently, all generalised characters in ℤIrr_{K}(G, b) which are perpendicular to the characters of the projective indecomposable OG-modules associated with b. We denote by C_{n} a cyclic group of order n. We review some of the standard reduction techniques due to Dade, Fong, Külshammer, Puig, Reynolds, and recall Landrock’s results on blocks with an elementary abelian defect group of order 8. The reduction techniques work irrespective of the characteristic, so for now, p is an arbitrary prime.

Proposition 2.1 (Puig’s version [60] of Fong-Reynolds reduction [29], [63]). Let G be a finite group, N a normal subgroup of G, c a G-stable block of kN and b a block of kG such that bc = b. Let P be a defect group of b. If P ∩ N = 1 then kNc is a block of defect zero and we have a canonical isomorphism

kGc ≅ kNc ⊗_{k} k_{α}(G/N)

for some α ∈ H^{2}(G/N; k^{×}) such that kGb is Morita equivalent to a block b̂ of a finite central p′-extension of G/N via a bimodule with diagonal vertex ΔP and endo-permutation source.

The isomorphism in the above statement can be described explicitly; see e.g. [37, 4.4]. This has been generalised by Külshammer as follows:

Proposition 2.2 (Külshammer [42, Proposition 5, Theorem 7]). Let G be a finite group, N a normal subgroup of G, c a G-stable block of kN and b a block of kG such that bc = b. Let P be a defect group of b. Suppose that for any x ∈ G conjugation by x induces an inner automorphism of kNc. Then there is a canonical isomorphism

kGc ≅ kNc ⊗_{Z(kNc)} Z(kNc)_{α}(G/N)

for some α ∈ H^{2}(G/N; (Z(kNc))^{×}). Moreover, if P ≤ N then the blocks kGb and kNc are source algebra equivalent.

As before, this isomorphism can be described explicitly; see e.g. [20, 2.1].

Proposition 2.3 (Dade [21, 3.5, 7.7]). Let G be a finite group, N a normal subgroup of G, c a G-stable block of kN and b a block of kG such that bc = b. Suppose that no element x ∈ G − N acts as inner automorphism on kNc. Then b = c and G/N has order prime to p.

We assume now that p = 2. Let P ≅ C_{2} × C_{2} × C_{2} be an elementary abelian group of order 8. The order of GL_{3}(2) is 8 · 21, from which one easily deduces that a non-trivial subgroup E of Aut(P) of odd order has either order 3 or 7, or is a Frobenius group of order 21. In all cases, E has a trivial Schur multiplier, and hence any block with a normal defect group P has as source algebra k(P ⋊ E). What is unusual in this case is that the number of characters at the local level does not depend on fusion (this is well-known; we include a sketch of a proof for the convenience of the reader):

Proposition 2.4. Let P be an elementary abelian group of order 8 and E a subgroup of Aut(P) of odd order. The group P ⋊ E has 8 ordinary irreducible characters.

Proof. This is trivial if E = 1. Suppose that |E| = 3. Then E fixes an involution in P, hence P ⋊ E ≅ C_{2} × (V_{4} ⋊ C_{3}), from which the statement follows. If |E| = 7 then P ⋊ E is a Frobenius group with E acting transitively on the involutions in P, which implies the result. The only remaining case is where E is a Frobenius group C_{7} ⋊ C_{3}. In that case, E has 5 characters (three with C_{7} in the kernel, and two more corresponding to the non-trivial C_{3}-orbits in the character group of C_{7}). Moreover, E acts again transitively on the involutions in P, and the centraliser of an involution in P ⋊ E is P ⋊ C_{3}, which has three modular simple modules, whence the result. □

One inequality of Alperin’s weight conjecture in this case, due to Landrock, can be proved without the classification of finite simple groups. Landrock’s results are more precise in that they also include information about defects and heights of characters; we briefly recall these notions. Given a block b of OG or of kG with a defect group P of order p^{d}, the defect of a character χ in the set Irr_{K}(G, b) of irreducible K-valued characters associated with b is the integer d(χ) such that p^{d(χ)} is the largest power of p dividing the rational integer \frac{|G|}{χ(1)}. It is well-known that d(χ) ≤ d; the integer h(χ) = d − d(χ) is called the height of χ. There is always at least one character in Irr_{K}(G, b) having height zero, and it has been conjectured by Brauer that P is abelian if and only if all characters in Irr_{K}(G, b) have height zero. The following summary of some of Landrock’s results in [44] implies in particular that Alperin’s weight conjecture holds for blocks with an elementary abelian defect group of order 8 if and only if Brauer’s height zero conjecture holds for these blocks:

Proposition 2.5 (Landrock, [44, 2.1, 2.2, 2.3]). Let G be a finite group and b a block of kG with an elementary abelian defect group P of order 8 and inertial quotient E ≤ Aut(P). Then 5 ≤ |Irr_{K}(G, b)| ≤ 8. If |Irr_{K}(G, b)| = 8 then all characters in Irr_{K}(G, b) have height zero. If |Irr_{K}(G, b)| < 8 then exactly four characters in Irr_{K}(G, b) have height zero, the remaining characters have height one, and ℓ(b) = 4. Moreover, the following hold.

(i) If E has order 1 then |Irr_{K}(G, b)| = 8 and ℓ(b) = 1. (ii) If E has order 3 then |Irr_{K}(G, b)| = 8 and ℓ(b) = 3. (iii) If E has order 7 then either |Irr_{K}(G, b)| = 5, ℓ(b) = 4 or |Irr_{K}(G, b)| = 8, ℓ(b) = 7. (iv) If E has order 21 then either |Irr_{K}(G, b)| = 7, ℓ(b) = 4 or |Irr_{K}(G, b)| = 8, ℓ(b) = 5.

Besides Landrock’s original proof it is also possible to prove this as a consequence of stronger results obtained later: the case |E| = 1 is a particular case of nilpotent blocks [15], the case |E| = 3 follows from Watanabe [72, Theorem 1]. In the case |E| = 7, the group E acts regularly on P −{1}, and by a result of Puig in [61] there is a stable equivalence of Morita type (cf. [13, §5]) between OGb and O(P ⋊ E). Any such stable equivalence induces an isometry L^{0}(G, b) ≅ L^{0}(P ⋊ E) between the generalised character groups which vanish on p-regular elements; in this case, these groups have rank one and are generated by an element of norm 8, whence the inequality. If |E| = 21 then there is again a stable equivalence of Morita type, by a result of Rouquier in [67]. Again by calculating a basis of L^{0}(P ⋊ E) - which in this case has rank 3 with a basis consisting of four elements of norm four, one also gets the inequality. See for instance [41] for an exposition of the well-known technique exploiting partial isometries induced by stable equivalences of Morita type; this will be used in the proof of Theorem 4.1. The following observation is a slight refinement of Proposition 2.5(iii) in the case where the inertial quotient has order 7 and |Irr_{K}(G, b)| < 8.

Proposition 2.6. Let G be a finite group, b a block of kG with an elementary abelian defect group P of order 8 and inertial quotient E of order 7. Suppose that b does not satisfy Alperin’s weight conjecture. Then there is a labelling Irr_{K}(G, b) = {χ_{i} | 1 ≤ i ≤ 5} with the following properties:

(i) χ_{1} has height one and χ_{i} has height zero, for 2 ≤ i ≤ 5. (ii) The group L^{0}(G, b) has rank 1 and a basis element of the form 2χ_{1} −∑_{i=2}^{5} δ_{i}χ_{i} for some signs δ_{i} ∈ {±1}, 2 ≤ i ≤ 5; moreover, at least one of the δ_{i} is positive. (iii) If i, j ∈ {2, 3, 4, 5} such that χ_{i}(1) = χ_{j}(1) then δ_{i} = δ_{j}.

Proof. Statement (i) is just a reformulation of 2.5 (iii). Using Puig’s stable equivalence of Morita type [61] we get that L^{0}(G, b) ≅ L^{0}(P ⋊ E), which is a free abelian group of rank one with a basis element of norm 8. The only way to write a norm 8 element in L^{0}(G, b) with less than 8 characters is with five characters, exactly one of which shows up with multiplicity 2, and then this character must have height one, as follows from comparing character degrees in conjunction with the fact that every generalised character in L^{0}(G, b) vanishes at 1. The signs δ_{i} cannot all be negative because the group L^{0}(G, b) does not contain any actual non-zero character (again because its elements vanish at 1). This proves (ii). If i, j ∈ {2, 3, 4, 5} then δ_{i}χ_{i} − δ_{j}χ_{j} is orthogonal to L^{0}(G, b), hence a generalised projective character. In particular, at 1, its value is divisible by the order of a Sylow 2-subgroup of G. But if χ_{i}(1) = χ_{j}(1) and δ_{i} ≠ δ_{j}, this value is ±2χ_{i}(1), which cannot be divisible by the order of a Sylow 2-subgroup of G as χ_{i} has height zero. The result follows. □

It is not known in general whether a Morita equivalence between two block algebras preserves their local structures, but some easy standard block theoretic arguments show that this is true, even for stable equivalences of Morita type, if one of the two blocks has an elementary abelian defect group of order 8.

Proposition 2.7. Let G, H be finite groups, b a block of kG with an elementary abelian defect group P of order 8 and c a block of kH with a defect group Q. If there is a stable equivalence of Morita type between kGb and kHc then Q ≅ P and the blocks b and c have isomorphic inertial quotients (or equivalently, isomorphic fusion systems).

Proof. A stable equivalence of Morita type preserves the largest elementary divisors of the Cartan matrices of the blocks, and these are equal to the orders of the defect groups, whence |Q| = |P |. A stable equivalence of Morita type preserves also the complexity (cf. [4, 5.3.4]) of modules; since the largest complexity of a module in a block is the rank of a defect group, we get that Q has rank 3, and thus Q ≅ P. Alternatively, a stable equivalence of Morita type preserves the Krull dimension of the Hochschild cohomology rings, which are also known to be equal to the ranks of the defect groups. (This part of the argument is well-known to remain valid for blocks with arbitrary elementary abelian defect groups, but we do not need this here.) Finally, a stable equivalence of Morita type preserves the rank of L^{0}(G, b), which is equal to |Irr_{K}(G, b)| − ℓ(b), or also equal to ∑_{(u,e_{u})} ℓ(e_{u}), where (u, e_{u}) runs over a set of representatives of the conjugacy classes of non-trivial (G, b)-Brauer elements. It happens so that this number determines the structure of the inertial quotient E of b. Indeed, by Proposition 2.5, this number is equal to 7 if and only if |E| = 1, equal to 5 if and only if |E| = 3, equal to 1 if and only if |E| = 7, and equal to 3 if and only if |E| = 21, whence the result. □

When dealing with the exceptional groups of type E_{7}(q) we will need a refinement of the preceding result because the finite group of Lie type E_{7}(q) is a central extension of the simple group of type E_{7}(q) by an involution, and so Bonnafé-Rouquier’s Jordan decomposition [8] will have to be applied to blocks with a defect group of order 2^{4}.

Proposition 2.8. Let G, H be finite groups, b a block of G with a defect group P and c a block of H with a defect group Q. Suppose that there are central involutions s ∈ Z(G) and t ∈ Z(H) such that P/⟨s⟩ is elementary abelian of order 8. Denote by b̄ and c̄ the images of b and c in kG/⟨s⟩ and kH/⟨t⟩, respectively. Suppose that the block algebras kGb and kHc are Morita equivalent. If b̄ does not satisfy Alperin’s weight conjecture then neither does c̄ and the defect groups of c̄ are elementary abelian of order 8.

Proof. Suppose that b̄ does not satisfy Alperin’s weight conjecture. Since the defect group P/⟨s⟩ of b̄ is elementary abelian of order 8 we have ℓ(b̄) = 4 by Proposition 2.5. Using that the number of isomorphism classes of simple modules is invariant under central 2-extensions and Morita equivalences we get that ℓ(c̄) = 4. Note that P and Q have the same order since b and c are Morita equivalent, and hence c̄ has a defect group R = Q/⟨t⟩ of order 8.

If R is cyclic or abelian of rank 2 then Aut(R) is a 2-group, hence c̄ is nilpotent, c̄ satisfies Alperin’s weight conjecture and ℓ(c̄) = 1, a contradiction. If R is one of the two non-abelian groups of order 8, then again c̄ satisfies Alperin’s weight conjecture and ℓ(c̄) ∈ {1, 3}, a contradiction. So, R is elementary abelian. Now it is immediate from Proposition 2.5 that c̄ does not satisfy the weight conjecture. □

Remark 2.9. Experts seem to agree that the Morita equivalences from Bonnafé-Rouquier’s Jordan decomposition [8] should preserve the local structure of the blocks, but at present there is no written reference for this fact. The two propositions 2.7 and 2.8 circumvent this issue by adhoc methods.

3. Reduction to quasi-simple groups

The ‘if’ part of Brauer’s height zero conjecture has been reduced to blocks of quasi-simple finite groups in work of Berger and Knörr [5]. In order to avoid having to check the ‘if’ part of Brauer’s height zero conjecture for all abelian 2-defect groups of quasi-simple finite groups we need to make sure that in the reduction we can indeed restrict the problem to checking only defect groups of order at most 8; this is not entirely obvious since in Step 6 of the proof of [5, Theorem] the order of the defect group may possibly go up, an issue which arises also in the alternative proof given by Murai in [54, §6]. As a consequence of Landrock’s results quoted above, if Alperin’s weight conjecture fails for a block b with an elementary abelian defect group of order 8 then there is a quasi-simple finite group H and a block c of kH with an elementary abelian defect group of order 8 such that Alperin’s weight conjecture fails for c.

Theorem 3.1. Let G be a finite group and b a block of kG with an elementary abelian defect group P of order 8. Suppose that |G/Z(G)| is minimal such that |Irr_{K}(G, b)| < 8. Then Z(G) has odd order and G/Z(G) is simple. If moreover we also choose |Z(G)| minimal then G is quasi-simple. In addition, the inertial quotient of b is either cyclic of order 7 or a Frobenius group of order 21.

Proof. If |Z(G)| is even then P ∩ Z(G) is non-trivial, hence contains a subgroup Z of order 2. The image b̄ of b in kG/Z is then a block of kG/Z with a Klein four defect group P/Z. Thus, since b̄ satisfies Alperin’s weight conjecture, so does b, a contradiction to the assumption. Thus Z(G) has odd order. Let (P, e) be a maximal (G, b)-Brauer pair and set E = N_{G}(P, e)/C_{G}(P). By Proposition 2.5, the order of E is either 7 or 21. In both cases, E acts transitively on P − {1}. Thus, if N is a normal subgroup of G then either N ∩ P = {1} or P ≤ N. Moreover, the minimality of |G/Z(G)| implies that if N is a normal subgroup of G containing Z(G) then there is a unique block c of kN covered by b; that is, bc = b. Let now N be a maximal normal subgroup of G containing Z(G) and let c be the block of kN satisfying bc = b. Consider first the case P ∩ N = {1}. In that case, by Proposition 2.1, we have an isomorphism

kGc ≅ kNc ⊗_{k} k_{α}(G/N)

for some α ∈ H^{2}(G/N; k^{×}) such that kGb is Morita equivalent to a block b̂ of a finite central p′- extension H of the simple group G/N. Consider next the case P ≤ N. Let G[c] be the subgroup of G consisting of all x ∈ G such that conjugation by x induces an inner automorphism of kNc. Since N is maximal normal in G we have either N = G[c] or G[c] = G. Suppose first that G[c] = G. Then, by Proposition 2.2, we have an isomorphism

kGc ≅ kNc ⊗_{Z(kNc)} Z(kNc)_{α}(G/N)

for some α ∈ H^{2}(G/N; (Z(kNc))^{×}), and the blocks kGb and kNc are source algebra equivalent - but this contradicts the minimality of |G| since source algebra equivalent blocks have in particular the same number of ordinary irreducible characters. Thus we have N = G[c]. Then, by Proposition 2.3, we have b = c and G/N has odd order. Since also G/N is simple, this implies that G/N is cyclic of odd prime order ℓ, by Feit-Thompson’s Odd Order theorem. By standard results in Clifford theory, any η ∈ Irr_{K}(H, c) is either G/N-stable, in which case it extends to exactly ℓ different characters in Irr_{K}(G, b), or Ind_{N}^{G}(η) ∈ Irr_{K}(G, b). It follows that

|Irr_{K}(G, b)| = ℓ · m + r < 8

where m is the number of characters in Irr_{K}(H, c) fixed by G/N and where r is the number of non-trivial G/N-orbits in Irr_{K}(H, c); in particular, using induction, we have

8 = |Irr_{K}(H, c)| = ℓ · r + m

If ℓ > 7 this forces r = 0, hence m = 8, and hence ℓ · m + r > 8, a contradiction. If ℓ = 7 either m = 1 or m = 7; in both cases we get ℓ · m + r ≥ 8, a contradiction. If ℓ = 5 then either m = 3 or m = 8, and if ℓ = 3 then either m = 2 or m = 5 or m = 8; in all cases one easily checks that ℓ · m + r ≥ 8, a contradiction. Thus the assumption P ≤ N is not possible, and therefore the above implies that G/Z(G) is simple. Finally, since G/Z(G) is simple we have G = Z(G)[G, G], so G/[G, G] acts trivially on all characters of [G, G], and so |Irr_{K}(G, b)| = |Irr_{K}([G, G], d)|, where d is a block of [G, G] satisfying bd = b. After repeating this, if necessary, we also may assume that G is perfect, hence quasi-simple. □

4. Perfect isometries

For a block b of a finite group G with an abelian defect group P, Alperin’s weight conjecture is equivalent to stating that the block algebra of b and that of its Brauer correspondent c of N_{G}(P) have the same number of ordinary irreducible characters (which is also equivalent to having the same number of isomorphism classes of simple modules). Using Rouquier’s stable equivalence for blocks with an elementary abelian defect group of order 8 we can show that Alperin’s weight conjecture implies the character theoretic version of Broué’s abelian defect conjecture in this case:

Theorem 4.1. Let k be an algebraically closed field of characteristic 2, let O be a complete discrete valuation ring with residue field k and quotient field K of characteristic zero. Let G be a finite group and let b be a block of OG with an elementary abelian defect group P of order 8. Set H = N_{G}(P) and denote by c the block of OH with defect group P corresponding to b via the Brauer correspondence. Suppose that K is large enough for b and c. If |Irr_{K}(G, b)| = |Irr_{K}(H, c)| then the blocks b and c are isotypic; in particular, there is a perfect isometry ℤIrr_{K}(G, b) ≅ ℤIrr_{K}(H, c).

See [11, 6.1, 6.2], [12], [13] for more precise versions of Broué’s abelian defect conjecture, as well as background material on perfect isometries and isotypies.

Proof of Theorem 4.1. We refer to [41, §§2, 3] for notation and an expository account of the standard techniques on extending partial isometries induced by stable equivalences of Morita type. As before, we denote by L^{0}(G, b) the subgroup of ℤIrr_{K}(G, b) consisting of all generalised characters which are perpendicular to all characters of finitely generated projective OGb-modules with respect to the canonical scalar product on ℤIrr_{K}(G, b), and we write L^{0}(G) instead of L^{0}(G, b) if OG has a unique block. By a result of Rouquier in [67] (extending Puig’s results in [61] in that case), there is a stable equivalence of Morita type between the block algebras of b and of c over O (a proof of this result is given in Theorem 20.1 below) given by a bounded complex of bimodules whose indecomposable summands all have diagonal vertices and trivial source. Denote by E the inertial quotient of b. Since the block algebra OHc is Morita equivalent to O(P ⋊E) via a bimodule with diagonal vertex and trivial source this implies that there is a stable equivalence of Morita type between OGb and O(P ⋊ E), induced by a bounded complex of bimodules with diagonal vertices and trivial source. It is well-known (see e.g. [41, 3.1]) that any such stable equivalence induces an isometry L^{0}(P ⋊ E) ≅ L^{0}(G, b). It suffices to show that this partial isometry extends to an isometry ℤIrr_{K}(P ⋊ E) ≅ ℤIrr_{K}(G, b) because any such extension is then a p-permutation equivalence by [41, 3.3], hence induces an isotypie by [46, Theorem 1.4]. We do this by running through all possible inertial quotients E. If E = {1} the block b is nilpotent, hence Morita equivalent to OP, and so the result holds trivially in this case. Assume that |E| = 3. Then P ⋊ E ≅ C_{2} × A_{4}, and hence we can list the eight ordinary irreducible characters of P ⋊ E in such a way that the three characters of the projective indecomposable O(P ⋊ E)-modules are of the form χ_{i} + χ_{i+3} + χ_{7} + χ_{8} where 1 ≤ i ≤ 3. Thus a basis of L^{0}(P ⋊ E) is of the form

{χ_{1} − χ_{4}, χ_{2} − χ_{5}, χ_{3} − χ_{6}, χ_{7} − χ_{8}, χ_{1} + χ_{2} + χ_{3} − χ_{7}}

The four elements of norm 2 in this basis must be sent to norm 2 elements under the isometry L^{0}(P ⋊ E) ≅ L^{0}(G, b) no two of which involve a common irreducible character in Irr_{K}(G, b), and hence are mapped to elements of the form δ_{1}(η_{1} − η_{4}), δ_{2}(η_{2} − η_{5}), δ_{3}(η_{3} − η_{6}), δ_{7}(η_{7} − η_{8}), for some labelling Irr_{K}(G, b) = {η_{i} |1 ≤ i ≤ 8} and some signs δ_{i}. We may then choose notation (after possibly exchanging η_{1} and η_{4} etc.) in such a way that the image in L^{0}(G, b) of the norm four element χ_{1} +χ_{2} +χ_{3} −χ_{7} is equal to δ_{1}η_{1} +δ_{2}η_{2} +δ_{3}η_{3} −δ_{7}η_{7}. Setting δ_{i+3} = δ_{i} for 1 ≤ i ≤ 3, and δ_{8} = δ_{7} it follows that the map sending χ_{i} to δ_{i}η_{i} induces an isometry ℤIrr_{K}(P ⋊E) ≅ ℤIrr_{K}(G, b) extending the isometry L^{0}(P ⋊P) ≅ L^{0}(G, b) as required. Assume next that |E| = 7. Then P ⋊E is a Frobenius group, whose seven characters of the projective indecomposable modules are of the form χ_{i} + χ_{8} with 1 ≤ i ≤ 7, for some labelling Irr_{K}(P ⋊ E) = {χ_{i} | 1 ≤ i ≤ 8}; the characters χ_{i}, 1 ≤ i ≤ 7 have P in their kernel, and χ_{8} is induced from a nontrivial character of P to P ⋊ E. The group L^{0}(P ⋊ E) has rank 1, with a basis element ∑_{i=1}^{7} χ_{i} − χ_{8}. This element has norm 8, hence its image in L^{0}(G, b) has norm 8 as well. Moreover, all irreducible characters in Irr_{K}(G, b) have to be involved in this element. Since we assume that Alperin’s weight conjecture holds for b, we have |Irr_{K}(G, b)| = 8, and so the image of this element in L^{0}(G, b) is of the form ∑_{i=1}^{7} δ_{i}η_{i} − δ_{8}η_{8} for some labelling Irr_{K}(G, b) = {η_{i} | 1 ≤ i ≤ 8} and some signs δ_{i}. Again, the map sending χ_{i} to δ_{i}η_{i} induces the required isometry ℤIrr_{K}(P ⋊ E) ≅ ℤIrr_{K}(G, b). Finally, assume that |E| = 21. Then E is itself a Frobenius group, isomorphic to C_{7} ⋊ C_{3} with the obvious nontrivial action of C_{3} on C_{7}. The group E has 5 ordinary irreducible characters, hence O(P ⋊E) has five isomorphism classes of projective indecomposable modules, and thus L^{0}(G, b) has rank 3. We can label Irr_{K}(P ⋊ E) = {χ_{i} | 1 ≤ i ≤ 8} in such a way that χ_{1}, χ_{2}, χ_{3} have degree 1, the characters χ_{4}, χ_{5} have degree 3 and χ_{6}, χ_{7}, χ_{8} have degree 7. An easy calculation shows that L^{0}(P ⋊ E) has a basis of the form

{χ_{6} − χ_{4} − χ_{5} − χ_{1}, χ_{7} − χ_{4} − χ_{5} − χ_{2}, χ_{8} − χ_{4} − χ_{5} − χ_{3}}

consisting of three elements of norm 4, such that any two different of these basis elements involve two common irreducible characters. Thus the same is true for L^{0}(G, b). Using again the hypothesis |Irr_{K}(G, b)| = 8 one deduces that L^{0}(G, b) has a basis of the form

{δ_{6}η_{6} − δ_{4}η_{4} − δ_{5}η_{5} − δ_{1}η_{1}, δ_{7}η_{7} − δ_{4}η_{4} − δ_{5}η_{5} − δ_{2}η_{2}, δ_{8}η_{8} − δ_{4}η_{4} − δ_{5}η_{5} − δ_{3}η_{3}}

for some labelling Irr_{K}(G, b) = {η_{i} | 1 ≤ i ≤ 8} and some signs δ_{i}. As before, the map sending χ_{i} to δ_{i}η_{i} induces the required isometry ℤIrr_{K}(P ⋊ E) ≅ ℤIrr_{K}(G, b), which concludes the proof. □

5. Sporadic finite simple groups

Let O be a complete discrete valuation ring having an algebraically closed residue field k of prime characteristic p and a quotient field K of characteristic zero, which is always assumed to be large enough for the finite groups arising in this section. Let G be a finite group and b a block of OG. The kernel Ker_{G}(b) of b is defined by

Ker_{G}(b) = ⋂_{χ∈Irr_{K}(G,b)}Ker(χ),

see [9, §3] and [55, Chap.5, §8.1]. By a result of Brauer [9, Proposition (3B)] we have Ker_{G}(b) = O_{p′}(G) ∩ Ker(χ) for any χ ∈ Irr_{K}(G, b). Hence, Ker_{G}(b) is a normal p′-subgroup of G, see [55, Chap.5, Theorem 8.1]. We say that b is faithful if Ker_{G}(b) = 1.

Proposition 5.1. Let G be a quasi-simple finite group such that p divides |G|.

(i) We have O_{p′}(Z(G)) = O_{p′}(G) and Ker_{G}(b) = O_{p′}(Z(G)) ∩ Ker(χ) for any χ ∈ Irr_{K}(G, b). (ii) Set Ḡ = G/Ker_{G}(b) and denote by b̄ the image of b in OḠ. Then b̄ is a faithful block of OḠ and the canonical map G → Ḡ induces an O-algebra isomorphism OGb ≅ OḠb̄.

Proof. Statement (i) follows from Brauer’s result mentioned above, and (ii) is an easy consequence of [55, Chap.5, Theorem 8.8]. □

The following table is due to Noeske [56]. By Proposition 5.1(ii) it is enough to consider faithful p-blocks.

Proposition 5.2 ([56]). The following is a list of all faithful non-principal 2-blocks with noncyclic abelian defect groups of sporadic simple groups and their covers. Each number in the 2nd column corresponds to the number attached to each block in the Modular Atlas [52].

group blocks b defect groups  k(b) ℓ(b)
M_{12} 2 C_{2} × C_{2} 4 3
12.M_{22} 4, 5  C_{2} × C_{2}, C_{2} × C_{2} 4, 4 1, 1
J_{2}  2  C_{2} × C_{2}  4  3
HS  2  C_{2} × C_{2}  4  3
Ru  2  C_{2} × C_{2}  4  3
Co_{3}  2  C_{2} × C_{2} × C_{2}  8  5
2.F i_{22} 3  C_{2} × C_{2}  4  1
F i_{24}′ 2  C_{2} × C_{2}  4  3

Proposition 5.3. Let G be a quasi-simple finite group such that G/Z(G) is a sporadic simple group, and let b be a 2-block of G with elementary abelian defect group of order 8. Then, |Irr_{K}(G, b)| = 8; in particular, Alperin’s weight conjecture holds for b.

Proof. If b is the principal 2-block the result follows from [44, Theorem 3.8]. Suppose that b is a non-principal 2-block; by Proposition 5.1 we may assume that b is faithful. Proposition 5.2 implies that G = Co_{3} and |Irr_{K}(G, b)| = 8. □

.

6. Finite simple groups with exceptional Schur multipliers

Proposition 6.1 ([36, Table 6.1.3, p.313], [52]). For a simple group G ∈ Chev(3), we get the following table;

\begin{matrix}G & A_{1}(9) ≅ A_{6} & ^{2}A_{3}(3) = U_{4}(3) & B_{3}(3) = O_{7}(3) & G_{2}(3) \\ Me & 3 & 3, 3 & 3 & 3 \\ M(G) & 6 & 12 & 6 & 3 \end{matrix}

where Me denotes the elementary divisors of the exceptional parts of the Schur multipliers M(G) of G.

Proposition 6.2. If G is a quasi-simple finite group such that G/Z(G) is of type either A_{1}(9), 2A_{3}(3), B_{3}(3) or G_{2}(3), then G has no 2-blocks b with elementary abelian defect group of order 8.

Proof. Set Ḡ = G/Z(G). Recall that A_{1}(9) ≅ A_{6} and^{2}A_{3}(3) = U_{4}(3) and B_{3}(3) = Ω_{7}(3) = O_{7}(3), see [35, p.8, Table I]. First, assume that Ḡ = A_{6}. Then, G ∈ {A_{6}, 2.A_{6}, 3.A_{6}, 6.A_{6}}. In any case, G has no 2-blocks with elementary abelian defect group of order 8 by [52]. For the other three cases Ḡ ∈ {U_{4}(3), O_{7}(3), G_{2}(3)}, similarly we know that G has no 2-blocks with elementary abelian defect group of order 8 by [52]. □

7. Alternating groups

We denote in this section - and only in this section - by A_{n} the alternating group of degree n, where n is a positive integer.

Proposition 7.1. If G ≅ A_{n} for some n ≥ 5 then G has no 2-blocks with an elementary abelian defect group of order 8.

Proof. By results in [40], a 2-block of an alternating group A_{n} with defect group P is source algebra equivalent to a block of an alternating group A_{m} for some m ≤ n having P as Sylow 2-subgroup. But there is no alternating group with an elementary abelian Sylow 2-subgroup of order 8. □

Proposition 7.2 (See [36, Table 6.1.3]). If G is 3.A_{6} or 3.A_{7}, then G has no 2-block with an elementary abelian defect group of order 8.

Proof. This is clear since Sylow 2-subgroups of G are dihedral of order 8, see [19, p.4, p.10]. □

8. Finite groups of Lie type in characteristic 2

Proposition 8.1. Let G be a quasi-simple group such that G/Z(G) is a finite group of Lie type in characteristic 2. Suppose that Z(G) has odd order. Let b be a 2-block of G having an elementary abelian defect group of order 8. Then G ≅ PSL_{2}(8), the block b is the principal 2-block of G, and |Irr_{K}(G, b)| = 8; in particular, Alperin’s weight conjecture holds for b.

Proof. Consider first the case where G/Z(G) is not isomorphic to the Tits simple group ^{2}F_{4}(2)′, PSp_{4}(2)′ ≅ A_{6}, or G_{2}(2)′ ≅ PSU_{3}(3). Then, by [20, Proposition 8.7], the defect groups of any 2-block of G are either trivial or the Sylow 2-subgroups of G. It is well-known that the only finite groups of Lie type in characteristic 2 having abelian Sylow 2-subgroups are the groups PSL_{2}(2^{n}), and hence G ≅ PSL_{2}(8), where we use that the Schur multiplier of PSL_{2}(8) is trivial. By the pages on decomposition numbers in the Modular Atlas [52], the principal 2-block of PSL_{2}(8) contains 8 irreducible characters and that the remaining irreducible character belongs to a block of defect zero. The group ^{2}F_{4}(2)′ has trivial Schur multiplier and by the pages on decomposition numbers in the Modular Atlas [52], ^{2}F_{4}(2)′ has three 2-blocks: the principal block (of defect 11) and two defect zero blocks; ^{2}F_{4}(2)′ has no block with an elementary abelian defect group of order 8. The case PSp_{4}(2) ≅ A_{6} is already checked in §6 (and can also be read off the Modular Atlas). Finally, PSU_{3}(3) has trivial Schur multiplier and again by the pages on decomposition numbers in the Modular Atlas [52], PSU_{3}(3) has besides the principal block (of defect 5) two blocks of defect zero; in particular, PSU_{3}(3) has no blocks with an elementary abelian defect group of order 8. □

Remark 8.2. One could have excluded the Tits simple group ^{2}F_{4}(2)′ also by observing that its order is not divisible by 7, and hence the inertial quotient of a hypothetical block with elementary abelian defect group of order 8 can only be trivial or cyclic of order 3, in which case Alperin’s weight conjecture holds by Proposition 2.5 above.

9. Further background results on finite reductive groups

The following notation will be in effect for this section. Let r and ℓ be disctinct primes and let q be a power of r. Let G a connected reductive group over 𝔽̄_{q} and F : G → G a Frobenius morphism with respect to an 𝔽_{q}-structure on G. Let (G^{∗}, F^{∗}) be a pair in duality with (G, F) with respect to some choice of an F -stable (respectively F^{∗}-stable) maximal torus of G (respectively G^{∗}) and with respect to a fixed isomorphism 𝔽̄_{q}^{×} ≅ (ℚ/ℤ)_{r′} and a fixed embedding 𝔽̄_{q}^{×} ↪ ℚ̄_{ℓ}^{×}. For an F^{∗}- stable semi-simple element s of G^{∗}, we denote by E(G^{F} , (s)) ⊆ Irrℚ̄_{ℓ}(G) the subset of characters corresponding to the geometric conjugacy class (s) and by E(G^{F} , [s]) the subset of characters corresponding to the rational conjugacy class [s]. We set C(s) = C_{G^{∗}}(s), C^{∘}(s) = C_{G^{∗}}(s)^{∘}, the connected component of C(s), C̄^{∘}(s) = C^{∘}(s)/Z(C^{∘}(s)), the quotient of the connected centraliser by its centre, and C^{∘}(s)′ = [C^{∘}(s), C^{∘}(s)] the derived subgroup of the connected centraliser. Set a_{s} = \frac{|C(s)^{F^{∗}} |}{|C^{∘}(s)^{F^{∗}} |}, Z^{∘}(s) = Z(C^{∘}(s)) and z(s) = |Z^{∘}(s)^{F^{∗}}|. For any positive integer m we denote by m_{+} the highest power of 2 dividing m.

9.1. Jordan decomposition of characters. By the work of Lusztig ([47], [48], see also [22, 13.24]), there is a bijection between E(G^{F} , (s)) and the set E(C(s)^{F^{∗}}, 1) of unipotent characters of C(s)^{F^{∗}} such that if χ ∈ E(G^{F} , (s)) corresponds to τ ∈ E(C(s)^{F^{∗}}, 1), then

(1)  χ(1) = \frac{|G^{F} |_{r′}}{|C(s)^{F^{∗}}|_{r′}} τ(1).

Here we note that if C(s) is not connected, then E(C(s)^{F^{∗}}, 1) is defined to be the set of irreducible characters of C(s)^{F^{∗}} covering the set E(C^{∘}(s)^{F^{∗}}, 1) of unipotent characters of C^{∘}(s)^{F^{∗}}. By standard Clifford theory, if τ is an irreducible character of C(s)^{F^{∗}} covering an irreducible character λ of C^{∘}(s)^{F^{∗}}, then τ(1) = aλ(1), for an integer a dividing a_{s}. Thus, to each element of E(G^{F} , (s)) is associated a C(s)^{F^{∗}}-orbit of E(C^{∘}(s)^{F^{∗}}, 1) of unipotent characters of C(s)^{F^{∗}} such that if χ ∈ E(G, (s)) corresponds to the orbit of λ ∈ E(C^{∘}(s)^{F^{∗}}, 1), then

(2)  χ(1) = \frac{|G^{F} |_{r′}}{a_{χ}|C^{∘}(s)^{F^{∗}}|_{r′}} λ(1)

for some integer a_{χ} dividing a_{s}. Restriction induces a degree preserving bijection λ ↦ λ′ between the sets E(C^{∘}(s)^{F^{∗}}, 1) and E(C^{∘}(s)^{′F^{∗}}, 1) (cf. [48, 5.1]). There is also a degree preserving bijection λ → λ̄ between E(C^{∘}(s)^{F^{∗}}, 1) and E(C̄^{∘}(s)^{F^{∗}}, 1) (cf. [48, 5.1]). Further, the group C̄^{∘}(s)^{F^{∗}} = ∏_{ω} C̄^{∘}_{ω}^{F^{∗}}, where ω runs through the F -orbits of the Dynkin diagram Δ_{s} of C̄^{∘}(s), and for each ω, C̄_{ω}^{∘}(s) is the direct product of subgroups of C̄^{∘}(s) corresponding to the elements of ω. The elements of E(C̄^{∘}(s)^{F^{∗}}, 1) are products ∏_{ω} ϕ_{ω} where ϕ_{ω} is a unipotent character of C̄^{∘}(s)_{ω}^{F^{∗}} for each ω. Tracing through the above bijections, and noting that

|C^{∘}(s)^{F^{∗}}| = z(s) + |C′^{∘}(s)^{F^{∗}}| = z(s) + |C̄^{∘}(s)^{F^{∗}}|

it follows that if χ corresponds to λ and λ corresponds to the unipotent character λ̄ := ∏_{ω} ϕ_{ω} of C̄^{∘}(s)^{F^{∗}}, then

χ(1) = \frac{|G^{F} |_{r′}}{z(s)a_{χ}|C^{∘}(s)^{′F^{∗}}|_{r′}} λ′(1)
(3)  = \frac{|G^{F} |_{r′}}{z(s)a_{χ}} ∏_{ω} \frac{ϕ_{ω}(1)}{|C̄_{ω}^{∘F^{∗}}(s)|_{r′}} .

The above correspondences have the following consequence, which we record for use in later sections. If r is odd, then

(4)  2-defect of χ = α_{χ} + ζ_{s} + 2-defect of λ′
(5)  = α_{χ} + ζ_{s} + ∑_{ω}(2-defect of ϕ_{ω})

where 2^{ζ_{s}} = z(s)_{+} and 2^{α_{χ}} = |a_{χ}|_{+}. We note that we get an analogous formula for ℓ-defects, for any prime ℓ different from r.

9.2. Jordan decomposition of blocks. We denote by O a complete discrete valuation ring with an algebraically closed residue field k of characteristic ℓ, different from r and a quotient field K of characteristic zero which is assumed to be big enough for all finite groups considered and we identify without further comment the sets Irrℚ̄_{ℓ}(G^{F}) and Irr_{K}(G^{F}). Let t be a semi-simple element of G^{∗F^{∗}} of order prime to ℓ and let E_{ℓ}(G^{F} , [t]) = ∪_{u}E(G^{F} , [tu]), where u runs over the ℓ-elements of C_{G^{∗}}^{∘}(t)^{F^{∗}}. By [14], E_{ℓ}(G^{F} , [t]) = ∪_{u}E(G^{F} , [tu]), is a union of ℓ-blocks of G; a block b of G^{F} is in this union if and only if E(G, [t]) ∩ Irr_{K}(G^{F} , b) ≠ ∅ (cf. [33]). In this case, we say that b is in the series [t] or that [t] is the semi-simple label of b. Blocks with semi-simple label [1] are called unipotent. Let L^{∗}(t) be the (necessarily F^{∗}-stable) minimal Levi subgroup of G^{∗}-containing C(t) (if C(t) is not contained in any proper Levi subgroup of G^{∗}, we take for L^{∗}(t) the group G itself) and let L be an F -stable Levi subgroup of G dual to L^{∗}(t). Let e_{t}^{G^{F}} and e_{t}^{L^{F}} be the sum of block idempotents of OG^{F} and OL^{F} in the series [t]. Then by Theorem B’ of [8], the algebras OG^{F} e_{t}^{G^{F}} and OL^{F} e_{t}^{L^{F}} are Morita equivalent. Further, if C(t) is itself a Levi subgroup of G^{∗}, ie. if C(t) = L^{∗}(t), then by Theorem 11.8 of [8], OG^{F} e_{t}^{G^{F}} is Morita equivalent to the sum of unipotent blocks OL^{F} e_{1}^{L^{F}} of OL^{F}. We record a consequence of these results for classical groups for ℓ = 2.

Theorem 9.1. Suppose that ℓ = 2 and either G = GL_{n}(𝔽̄_{q}) or that G is simple of classical type B, C or D. Let t be an odd order semi-simple element of G^{∗F^{∗}}. Then, C(t) is a a Levi subgroup of G^{∗}. Let L be an F -stable Levi subgroup of G in duality with C(t) as above. Then e_{t}^{G^{F}} is a block of OG^{F}, and e_{1}^{L^{F}} is the principal block of OL^{F}. The block algebras OG^{F} e_{t}^{G^{F}} and OL^{F} e_{1}^{L^{F}} are Morita equivalent and a Sylow 2-subgroup of L^{F} is a defect group of OG^{F} e_{t}^{G^{F}}.

Proof. The element t has odd order, whereas Z(G)/Z^{∘}(G) is a 2-group, hence C(t) is connected. The prime 2 is the only bad prime for G (if G is a general linear group, then all primes are good for G). Thus, the order of t is not divisble by any bad prime, which means that C^{∘}(t) is a Levi subgroup of G^{∗}. Thus by the Bonnafé-Rouquier theorem, OG^{F} e_{t}^{G^{F}} and OL^{F} e_{1}^{L^{F}} are Morita equivalent. Now the components of L are all of classical types A, B, C or D. Hence by [16, Theorem 13], the principal block of OL^{F} is the unique unipotent 2-block of OL^{F}, and the Morita equivalence implies that e_{t}^{G^{F}} is a block of OG^{F}. The assertion on defect groups is in [27, Proposition 1.5(ii), (iii)]. □

Corollary 9.2. Suppose that ℓ = 2 and either G = GL_{n}(𝔽̄_{q}) or that G is simple of classical type B, C or D. Let b be a block of OG^{F}. If b has abelian defect groups, then OG^{F} b is nilpotent.

Proof. Let t be the semi-simple label of b and suppose that b has abelian defect groups. By the theorem, b = e_{t}^{G^{F}} and the Sylow 2-subgroups of L^{F} are abelian. Suppose that [L, L] ≠ 1. If L has a component different from A_{1}, then [L, L]^{F} contains a subquotient isomorphic to SL_{2}(q′) for some power q′ of q (see Theorem 3.2.8 of [36]). If all components of L are of type A_{1}, then [L, L]^{F} is a commuting product of finite special linear and projective general linear groups of degree 2. But the Sylow 2-subgroups of SL_{2}(q′) are quaternion and those of PGL_{2}(q′) are dihedral of order at least 8 for any odd prime power q′ (see 10.1 below), a contradiction. Hence, L and therefore L^{F} is an abelian group. In particular, any block of OL^{F} is nilpotent. Since nilpotent blocks with abelian defect groups are precisely the blocks with a symmetric centre (cf. [57, Theorems 3 and 5] and [53]), any block Morita equivalent to a block of OL^{F} is nilpotent with an abelian defect group, whence the result. Alternatively, Morita equivalences of blocks preserve nilpotence by [62, Theorem 8.2]. □

10. On the 2-local structure of finite classical groups

Let n be a natural number, q an odd prime power and let L denote one of the groups GL_{n}(q), GU_{n}(q), O_{2n+1}(q), Sp_{2n}(q), O_{2n}^{+}(q), or O_{2n}^{−}(q). Let Z be a central subgroup of L contained in [L, L] and set G = [L, L]/Z. We gather together a few well-known facts on the Sylow 2-structure of the groups L and G.

Lemma 10.1. With the notation above,

(i) If n ≥ 3, then the Sylow 2-subgroups of G (and hence of [L, L] and L) are non-abelian. (ii) If n = 2, the Sylow 2-subgroups of [L, L] are non-abelian. Further, if n = 2 and L is not one of GL_{2}(q), q ≡ ±3 (mod 8), GU_{2}(q), q ≡ ±3 (mod 8), or O_{4}^{+}(q), q ≡ ±3 (mod 8), then the Sylow 2-subgroups of G are non-abelian. (iii) If L is one of GL_{2}(q), GU_{2}(q) or Sp_{2}(q) then the Sylow 2-subgroups of [L, L] are generalised quaternion groups. They have order at least 16 if q ≡ ±1 (mod 8) and in this case the Sylow 2-subgroups of G are non-abelian. If q ≡ ±3 (mod 8), then the Sylow 2-subgroups of [L, L] have order 8 and the Sylow 2-subgroups of G are Klein 4-groups. The Sylow 2-subgroups of PGL_{2}(q) are dihedral of order at least 8. (iv) If L is one of O_{2}^{+}(q), respectively then O_{2}^{−}(q), then L is a dihedral group of order 2(q − 1), respectively 2(q + 1).

(v) If L = O_{3}(q) and if q ≡ 1 (mod 4) then the Sylow 2-subgroups of L are isomorphic to the direct product of a cyclic group of order 2 with a Sylow 2-subgroup of O_{2}^{+}(q) and the Sylow 2-subgroups of SO_{3}(q) are isomorphic to the Sylow 2-subgroups of O_{2}^{+}(q). If L = O_{3}(q) and q ≡ 3 (mod 4) then the Sylow 2-subgroups of L are isomorphic to the direct product of a cyclic group of order 2 with a Sylow 2-subgroup of O_{2}^{−}(q) and the Sylow 2-subgroups of SO_{3}(q) are isomorphic to the Sylow 2-subgroups of O_{2}^{−}(q).

Proof. Statements (iii), (iv), (v) can be found in [18]. The only simple groups with abelian Sylow 2-subgroups are P SL_{2}(q′), q′ ≅ ±3 (mod 8), P SL_{2}(2^{a}), ^{2}G_{2}(q′), q′ = 3^{2u+1}, u ≥ 1, ^{2}G_{2}(3)′ ≅ PSL_{2}(8) or J_{1} (cf. [70], [3]). Also, if n ≥ 2, then unless L is one of O_{4}^{+}(q), GL_{2}(2), GL_{2}(3) or GU_{2}(2), the groups G are all quasi-simple. Statement (i) and the second assertion of (ii) are immediate from this. If L = GL_{2}(q), or L = GU_{2}(q), the second assertion of (ii) is immediate from (iii). Now consider L = O_{4}^{+}(q). Then L contains a subgroup isomorphic to GL_{2}(q) or to GU_{2}(q) (as a centraliser of a semi-simple element), hence [L, L] = Ω_{4}^{+}(q) contains a subgroup isomorphic to SL_{2}(q) or to SU_{2}(q) and SL_{2}(q) and SU_{2}(q) have non-abelian Sylow 2-subgroups. □

In the next sections, we will analyse closely the structure of centralisers of semi-simple elements in classical groups. The following elementary lemma will be useful in this context. In what follows, O_{0}^{±}(q) are to be interpreted as the trivial group, and O_{1}(q) as a cyclic group of order 2.

Lemma 10.2. Let t ≥ 0, let d_{i}, m_{i}, 1 ≤ i ≤ t, be positive integers and let m_{0} be a non-negative integer. Let

H = ∏_{0≤i≤t} H_{i}

be a subgroup of L such that H_{0} is one of the groups Sp_{2m_{0}}(q), O_{2m_{0}+1}(q) or O_{2m_{0}}^{±}(q) and H_{i} is isomorphic to GL_{m_{i}}(q^{d_{i}}) or GU_{m_{i}}(q^{d_{i}}) for 1 ≤ i ≤ t. Let Z be a central subgroup of L contained in H such that if the above decomposition of H has more than one non-trivial factor, then Z ∩H_{i} = 1 for all i, 0 ≤ i ≤ t. Let T be a Sylow 2-subgroup of H and set P = (T ∩ [L, L]Z)/Z. Suppose that P is abelian. Then,

(i) m_{i} ≤ 2, 0 ≤ i ≤ t. (ii) If m_{0} = 2, then H = H_{0} = O_{4}^{+}(q). (iii) If m_{i} = 2 for some i ≥ 1, then t = 1 and m = 0, that is H = H_{1}. Further, d_{1} is odd and q ≡ ±3 (mod 8). (iv) If H_{0} = Sp_{2n}(q) and if m_{0} ≠ 0, then t = 0, that is H = H_{0}.

Proof. For 0 ≤ i ≤ t, [H_{i}, H_{i}] ≤ H ∩ [L, L], and hence

[H_{i}, H_{i}]/[H_{i}, H_{i}] ∩ Z ≅ [H_{i}, H_{i}]Z/Z ≤ (C_{L}(s) ∩ [L, L])/Z.

Suppose first that two of the factors of H are non-trivial, say H_{i} andH_{j}, i ≠ j. Then, by assumption [H_{i}, H_{i}]∩Z = 1. It follows from the above that [H_{i}, H_{i}] is a subgroup of (H ∩[L, L])/Z. In particular, the Sylow 2-subgroups of [H_{i}, H_{i}] are abelian. But by Lemma 10.1, [H_{i}, H_{i}] has non-abelian Sylow 2-subgroups if m_{i} ≥ 2. This proves the first assertions of (ii) and (iii).

Now supppose that some m_{i} ≥ 2. By what has just been proved, either t = 0 or t = 1 and m_{0} = 0. If t = 0, then H = H_{0} is an orthogonal or symplectic group of dimension m_{0}. By Lemma 10.1, it follows that H_{0} = O_{4}^{+}(q), proving (ii). If t = 1 and m_{0} = 0, then [H, H]/Z([H, H]) is a projective special linear or unitary group, and by Lemma 10.1 (i), m_{1} = 2, H ≡ GL_{2}(q^{d_{1}}), and q^{d_{1}} ≡ ±3 (mod 8). The last can only hold if d_{1} is odd and q ≡ ±3 (mod 8). This proves (iii). The proof of (iv) is similar, using the fact that symplectic groups Sp_{2n}(q) are perfect for n ≥ 2. □

11. Type A in odd characteristic

By results of Blau and Ellers in [6], Brauer’s height zero conjecture holds for all blocks in non-defining characteristic of quasi-simple groups of type A and ^{2}A, and hence so does Alperin’s weight conjecture for all blocks of these groups in odd characteristic with an elementary abelian defect group of order 8, by Landrock’s results quoted in Proposition 2.5. This proves in particular Theorem 1.1 for these groups. For the sake of completeness, and using methods similar to those in [6], we prove in this and the following section that elementary abelian 2-defect groups of odd rank at least three do not occur in type A and type ^{2}A. Throughout this section k is an algebraically closed field of characteristic 2.

Theorem 11.1. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) ≅ PSL_{n}(q) for some positive integer n and some odd prime power q. Let r ≥ 3 be an odd integer. Then kG has no block with an elementary abelian defect group of order 2^{r}.

Lemma 11.2. Let n, m, d be positive integers such that n = md. Consider GL_{m}(q^{d}) as subgroup of GL_{n}(q) through some 𝔽_{q}-decomposition (𝔽_{q})^{n} ≅ (𝔽_{q^{d}})^{m}. Denote by det the determinant function on GL_{n}(q). Then there is an element x ∈ GL_{m}(q^{d}) such that det(x) has order q − 1.

Proof. Let λ be a generator of 𝔽_{q^{d}}^{×} and f ∈ 𝔽_{q}[X] the minimal polynomial of λ over 𝔽_{q}. Then f

has degree d, and the roots of f are λ, λ^{q}, λ^{q2},..., λ^{q^{d−1}}. Let y ∈ GL_{d}(q) with minimal polyomial f. Define x ∈ GL_{n}(q) via d × d-block diagonal matrices where the first block is y and the remaining m blocks are the identity matrices Id_{d}. Then, in GL_{n}(𝔽̄_{q}), the element x is conjugate to a diagonal matrix whose diagonal entries are λ, λ^{q}, λ^{q2},...,λ^{q^{d−1}}, 1, 1, ...,1. Thus the determinant of x is

det(x) = λ·λ^{q} ⋯ λ^{q^{d−1}} = λ^{\frac{q^{d}−1}{q−1}} . Since λ has order q^{d} −1 it follows that det(x) has order q−1. □

Lemma 11.3. Let n, m, d be positive integers such that n = md. Consider GL_{m}(q^{d}) as subgroup of GL_{n}(q) through some 𝔽_{q}-decomposition (𝔽_{q})^{n} ≅ (𝔽_{q^{d}})^{m}. If m ≥ 2 and q^{d} ≡ 1 (mod 4) then there is an element y ∈ GL_{m}(q^{d}) of order 4 such that the image of y in PGL_{n}(q) has order 4. If moreover m ≥ 3 we can choose such an element y in SL_{n}(q).

Proof. Since m ≥ 2, the group GL_{m}(q^{d}) contains a subgroup isomorphic to 𝔽_{q^{d}}^{×} × 𝔽_{q^{d}}^{×}; choose y = (y_{1}, 1) in this subgroup, where y_{1} ∈ 𝔽_{q^{d}}^{×} has order 4. Then y has an eigenvalue 1, hence if some power y^{r} is a scalar multiple of Id_{n} then y^{r} = Id_{n}, which shows that the order of y remains unchanged upon taking its image in PGL_{n}(q). If m ≥ 3 then GL_{m}(q^{d}) contains a subgroup isomorphic to (𝔽_{q^{d}}^{×})^{3}; choose y = (y_{1}, 1, y_{3}) with y_{3} such that det(y_{3}) = det(y_{1})^{−1}, which is possible thanks to Lemma 11.2, and as before, y has the required properties. □

Since the case PSL_{2}(9) ≅ A_{6} is dealt with in §6 in order to prove Theorem 11.1 we may assume that G = SL_{n}(q)/Z_{+}, where Z_{+} is the Sylow 2-subgroup of Z(SL_{n}(q)). Note that |Z_{+}| is equal to the 2-part (n, q − 1)_{+} of (n, q − 1).

Let b be a block of kG and denote by P a defect group of b. Since PSL_{n}(q) ≅ G/Z_{−}, where Z_{−} is the complement of C_{+} in Z(SL_{n}(q)) identified to its image in G, the image of P in PSL_{n}(q) is isomorphic to P. Since Z_{+} is a central 2-subgroup, b is the image of a unique block b̃ of k SL_{n}(q), and the inverse image P̃ of P in SL_{n}(q) is a defect group of b̃. Let d be a block of k GL_{n}(q) covering b̃ with a defect group T such that T ∩ SL_{n}(q) = P̃. By [20, Proposition 6.3] the block b̃ is stable under the 2-part of GL_{n}(q)/ SL_{n}(q), and hence T/P̃ is cyclic of order the 2-part (q − 1)_{+} of q − 1. By results of Fong-Srinivasan [31], there exists a semi-simple element s of odd order in GL_{n}(q) such that T is a Sylow 2-subgroup of C_{GL_{n}(q)}(s) (this can also be seen as a consequence of the Jordan decomposition of Theorem 9.1). Further there exist positive integers m_{i}, d_{i}, 1 ≤ i ≤ t such that

n = ∑_{1≤i≤t} m_{i}d_{i},

and setting H_{i} = GL_{m_{i}}(q^{d_{i}}), there is a decomposition

C_{GL_{n}(q)}(s) ≡ ∏_{1≤i≤t} H_{i}

corresponding to a subspace decomposition of the underlying 𝔽_{q}-vector vector space as isotypic 𝔽_{q}[s]-modules. In particular, H_{i} = GL_{m_{i}}(q^{d_{i}}) is a subgroup of GL_{m_{i}d_{i}}(q) through some 𝔽_{q}- decomposition (𝔽_{q})^{m_{i}d_{i}} ≅ (𝔽_{q^{d_{i}}})^{m_{i}}.

Lemma 11.4. With the notation above, suppose that P is elementary abelian and that t ≥ 3. Then the following holds.

(i) q ≡ 3 (mod 4). (ii) m_{i} = 1 for 1 ≤ i ≤ t. (iii) d_{i} is odd for 1 ≤ i ≤ t. (iv) If t is even then |P | = 2^{t−2}, and if t is odd then |P | = 2^{t−1}; in particular, the 2-rank of P is even.

(v) The block d of GL_{n}(q) is nilpotent with an elementary abelian defect group T of order 2^{s}.

Proof. Clearly, P = (T ∩ SL_{n}(q))/Z_{+} and Z_{+} ∩ H_{i} = 1 for any i such that H_{i} ≠ C_{L}(s). Thus Lemma 10.2 applies with L = GL_{n}(q), H = C_{L}(s) and Z = Z_{+}, and assertion (ii) is immediate. If q ≡ 1 (mod 4) or if d_{1} is even, then the group H_{1} = GL_{1}(q^{d_{1}}) contains an element y_{1} of order 4. By Lemma 11.2 the group H_{2} = GL_{1}(q^{d_{2}}) contains a 2-element y_{2} such that det(y_{2}) = det(y_{1})^{−1}. Thus x = y_{1}y_{2} ∈ SL_{n}(q) = [L, L]. Since t ≥ 3, we have H_{1} × H_{2} ∩ Z_{+} = 1, and it follows that the image xZ_{+} of x in P has order 4, a contradiction. Thus (i) and (iii) hold. Hence T is a Sylow 2-subgroup of ∏_{i=1}^{t} 𝔽_{q^{d_{i}}}^{×}. Since the d_{i} are odd and q ≡ 3 (mod 4) this implies that T is elementary abelian of rank t, and hence P̃ is elementary abelian of rank t − 1. If n is odd then P ≅ P̃ and since n = ∑_{i=1}^{t} d_{i} and the d_{i} are odd it follows that t is odd, hence |P | = 2^{t−1}. If n is even then |P | = \frac{|P̃|}{2} and since n = ∑_{i=1}^{t} d_{i} and the d_{i} are odd it follows that t is even and |P | = 2^{t−2}, which proves (iv). Since T is clearly abelian, (v) is immediate from Corollary 9.2. □

Lemma 11.5. Suppose that P is elementary abelian and that t = 2. Then m_{1} = m_{2} = 1, |P | ≤ 4 and P̃ contains an element of order max((q^{d_{1}} − 1)_{+}, (q^{d_{2}} − 1)_{+}).

Proof. The fact that m_{1} = m_{2} = 1 is a consequence of Lemma 10.2. So H_{i} ≅ GL_{1}(q^{d_{i}}). By Lemma 11.2, it follows that P̃ = T ∩ SL_{n}(q) contains elements of order max((q^{d_{1}} − 1)_{+}, (q^{d_{2}} − 1)_{+}). □

Lemma 11.6. Suppose that P is elementary abelian and that t = 1. Then m_{1} ≤ 2. If m_{1} = 2, then d_{1} is odd, n = 2d_{1}, |P | = 4 and P̃ is a quaternion group of order 8. If m_{1} = 1, then T, P̃ and P are cyclic, |P | = 2 if and only if n is even, and either q ≡ 3 (mod 4)

and n_{+} ≤ (q − 1)_{+} or q ≡ 1 (mod 4) and n_{+} = 2(q − 1)_{+}.

Proof. By Lemma 10.2, m_{1} ≤ 2. Suppose that m_{1} = 1, so T is a Sylow 2-subgroup of GL_{1}(q^{n}) and in particular, T is a cyclic group of order (q^{n} − 1)_{+}. It follows from Lemma 11.2 that P̃ = P ∩ SL_{n}(q) is cyclic of order \frac{(q^{n}−1)_{+}}{(q−1)_{+}} , and hence P is cyclic of order \frac{(q^{n}−1)_{+}}{(q−1)_{+}|Z_{+}|}. Now, |Z_{+}| = min(n_{+}, (q − 1)_{+}). If n is odd then (q^{n} − 1)_{+} = (q − 1)_{+} and if n is even then (q^{n} − 1)_{+} = \frac{(q^{2}−1)_{+}n_{+}}{2} . The statement of the lemma for the case m_{1} = 1 follows by an easy calculation. Now suppose m_{1} = 2. Then by Lemma 10.2, (ii) we have C_{GL_{n}(q)}(s) = GL_{2}(q^{d_{i}}), q ≡ ±3 (mod 8), d_{i} is odd and n = 2d_{i}. With this arithmetic, one sees easily that |P | = 4 and |P̃| = 8. Finally, P̃ is quaternion since it contains a subgroup isomorphic to the Sylow 2-subgroups of SL_{2}(q^{d_{i}}), which by Lemma 10.1 are quaternion. □

Proof of Theorem 11.1. If t ≥ 3, Lemma 11.4 shows that the 2-rank of S is at least 4 and even, while if t ≤ 2, Lemma 11.5 and Lemma 11.6 show that the rank of P is at most 2; in particular G has no block with an elementary abelian defect group of odd rank at least 3. □

We also note the following.

Lemma 11.7. Suppose that n = 2m, m ≥ 1, and P is elementary abelian of order 2 or 4. Then the inverse image of P in a non-split central extension 2.G has an element of order 4 unless t = 4, q ≡ 3(mod 4), and d_{i} is odd for 1 ≤ i ≤ 4. In particular, if P has order 2, then the inverse image has order 4.

Proof. The central extension 2.G may be assumed to be a central quotient of SL_{n}(q) and the inverse image, P_{0} of P in 2.G is a quotient of P̃ by a cyclic (central) group of order \frac{1}{2}|Z_{+}|. We will show that unless we are in the exceptional case above, that either P̃ is cyclic or that P̃_{0} contains an element of order 2|Z|_{+}. If t = 1 and m_{1} = 1, then C_{L}(s), and hence P̃ is cyclic. So certainly P_{0} is cyclic. If t = 1 and m_{1} = 2, then [H_{2}, H_{2}] ≤ C_{L}(s) ∩ [L, L] is a special linear group of dimension 2 and hence P̃ ∩[H_{2}, H_{2}] is a quaternion group of order q^{2d} −1. In particular, P̃ contains an element of order \frac{1}{2}(q^{2d} − 1). Since |Z|_{+} ≤ (q − 1)_{+}, if d is even, then P̃ contains an element of order at least 2|Z|_{+}. So we may assume that d is odd. Then |Z|_{+} = 2, and \frac{1}{2}(q^{2d} − 1) ≥ 4 = 2|Z|_{+}. Now suppose t = 2. So, m_{1} = m_{2} = 1 and n = d_{1} + d_{2}. Thus, T = T_{1} × T_{2}, with H_{i} a cyclic group of order (q^{d_{i}} − 1)_{+}, i = 1, 2. We assume without loss of generality that |T_{1}| ≥ |T_{2}|. By Lemma 11.2, it is easy to see that P̃ = T ∩ SL_{n}(q) is a direct product Q_{1} × Q_{2} such that Q_{1} is cyclic of order |T_{1}| and Q_{2} is cyclic of order \frac{|T_{2}|}{(q−1)_{+}} . If d_{1} is even, then |T_{1}| ≥ 2(q − 1)_{+} ≥ 2|Z|_{+}. If d_{1} is odd, then d_{2} is also odd (as n is even), hence T_{2} = 1, that is T̃ is cyclic. Finally, suppose that t ≥ 3. By Lemma 11.4, P is not of order 2, and P is of order 4 if and only if t = 4, q ≡ 3(mod 4), and d_{i} is odd for 1 ≤ i ≤ 4. Note that in this case, P̃ is elementary abelian of order 8. □

12. Type ^{2}A in odd characteristic

Throughout this section k is an algebraically closed field of characteristic 2. We show that type 2A yields no blocks with elementary abelian defect groups of order 8; in fact, more generally, we have the following result:

Theorem 12.1. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) ≅ PSU_{n}(q) for some positive integer n and some odd prime power q. Let r ≥ 3 be an odd integer. Then kG has no block with an elementary abelian defect group of order 2^{r}.

The proof of this follows the same lines as the untwisted case. We give details for the convenience of the reader. We single out two elementary observations which we will use in the proof below:

Lemma 12.2. Let n, m, d be positive integers and denote by det the determinant function on GL_{n}(𝔽̄_{q}).

(i) Suppose that n ≥ 2d. Consider the inclusions GL_{1}(q^{2d}) ≤ GL_{d}(q^{2}) ≤ GU_{2d}(q) ≤ GU_{n}(q), where GL_{1}(q^{2d}) is a subgroup of GL_{d}(q^{2}) through some 𝔽_{q^{2}}-vector space isomorphism (𝔽_{q^{2}})^{d} ≅ 𝔽_{q^{2d}}, GL_{d}(q^{2}) is a subgroup of GU_{2d}(q) through some 𝔽_{q^{2}}-vector space embedding (𝔽_{q^{2}})^{d} ↪ (𝔽_{q^{2}})^{d} ⊕ (𝔽_{q^{2}})^{d} of the form λ → λ + λ^{−q} and GU_{2d}(q) is a subgroup of GU_{n}(q) through some decomposition (𝔽_{q^{2}})^{n} ≅ (𝔽_{q^{2}})^{2d} ⊕ (𝔽_{q^{2}})^{n−2d}. There is an element x ∈ GL_{1}(q^{2d}) such that det(x) has order q + 1.

(ii) Suppose that d is odd and that n ≥ d. Consider the inclusions GU_{1}(q^{d}) ≤ GU_{d}(q) ≤ GU_{n}(q), where GU_{1}(q^{d}) is a subgroup of GU_{d}(q) through an irreducible unitary representation of GU_{1}(q^{d}) on a d-dimensional 𝔽_{q^{2}}-space, and where GU_{d}(q) is a subgroup of GU_{n}(q) through some decomposition (𝔽_{q^{2}})^{n} ≅ (𝔽_{q^{2}})^{2d} ⊕ (𝔽_{q^{2}})^{n−2d}. There is an element x ∈ GL_{1}(q^{2d}) such that det(x) has order q + 1.

Proof. (i) Let λ be a generator of 𝔽_{q^{2d}}^{×} and f ∈ 𝔽_{q^{2}}[X] the minimal polynomial of λ over 𝔽_{q^{2}}.

Then f has degree d, and the roots of f are λ, λ^{q^{2}}, λ^{q^{4}},..., λ^{q^{2(d−1)}}. Let x ∈ GL_{d}(q^{2}) with minimal polyomial f. Then, in GL_{n}(𝔽̄_{q}), the element x is conjugate to a diagonal matrix whose diagonal entries are λ, λ^{q^{2}}, λ^{q^{4}},...,λ^{q^{2(d−1)}}, λ^{−q}, (λ^{q^{2}})^{−q}, (λ^{q^{4}})^{−q},...,(λ^{q^{2(d−1})})^{−q}, 1 ⋯ 1. Thus the

determinant of x is det(x) = aa^{−q}, where a = λ · λ^{q^{2}} ⋯ λ^{q^{2(d−1)}} = λ^{\frac{q^{2d}−1}{q^{2}−1}} . Since λ has order q^{2d} − 1 it follows that det(x) has order q + 1.

(ii) Let λ be an element of order q^{d+1} in 𝔽_{q^{2d}}^{×} and f ∈ 𝔽_{q^{2}}[X] the minimal polynomial of λ over 𝔽_{q^{2}}. Since d is odd, f has degree d. Let x ∈ GU_{d}(q) with minimal polyomial f. In GL_{n}(𝔽̄), the element x is conjugate to a diagonal matrix whose diagonal entries are λ, λ^{q^{2}}, λ^{q^{4}},...,λ^{q^{2(d−1)}}. Thus the determinant of x is

det(x) = λ · λ^{q^{2}} ⋯ λ^{q^{2(d−1)}} = λ^{\frac{q^{2d}−1}{q^{2}−1}} .

Since λ has order q^{d} + 1 and since d being odd, the integers, \frac{q^{d}−1}{q−1} and q^{d} + 1 are relatively prime, it follows that det(x) has order q + 1. □

We turn now towards the proof of Theorem 12.1. Since the case PSU_{4}(3) is dealt with in §6 in order to prove Theorem 11.1 we may assume that G = SU_{n}(q)/Z_{+}, where Z_{+} is the Sylow 2-subgroup of Z(SU_{n}(q)). Note that |Z_{+}| is equal to the 2-part (n, q + 1)_{+} of (n, q + 1). Let b be a block of kG and denote by P a defect group of b. Since PSU_{n}(q) ≅ G/Z_{−}, where Z_{−} is the complement of C_{+} in Z(SL_{n}(q)) identified to its image in G, the image of P in PSU_{n}(q) is isomorphic to P. Since Z_{+} is a central 2-subgroup, b is the image of a unique block b̃ of k SU_{n}(q), and the inverse image P̃ of P in SU_{n}(q) is a defect group of b̃. Let d be a block of k GU_{n}(q) covering b̃ with a defect group T such that T ∩ SU_{n}(q) = P̃. By [20, Proposition 6.3] the block b̃ is stable under the 2-part of GU_{n}(q)/ SU_{n}(q), and hence T/P̃ is cyclic of order the 2-part (q + 1)_{+} of q + 1. By Fong-Srinivasan [31], T is isomorphic to a Sylow 2-subgroup of

H = ∏_{i=1}^{s} GL_{m_{i}}(q^{2d_{i}}) × ∏_{j=1}^{t} GU_{n_{j}}(q^{e_{j}})

for some non-negative integers s, t, and some positive integers m_{i}, n_{i} d_{i}, e_{j} such that all e_{j} are odd and satisfy

n = ∑_{i=1}^{s} 2m_{i}d_{i} + ∑_{j=1}^{t} n_{j}e_{j}.

We keep this notation for the remainder of this section.

Lemma 12.3. (i) Suppose that ∑_{1≤i≤s} m_{i} + ∑_{1≤j≤t} n_{j} ≥ 3. If for some i, j, either GL_{1}(q^{2d_{i}}) or GU_{1}(q^{e_{j}}) contains an element of order 2^{a}, then so does P. (ii) Suppose that ∑_{1≤i≤s} m_{i} + ∑_{1≤j≤t} n_{j} ≥ 4 and either m_{i} ≥ 2 for some i, 1 ≤ i ≤ s or n_{j} ≥ 2 for some j, 1 ≤ j ≤ t. Then P contains an element of order 8. (iii) Suppose that n_{j} ≥ 2 for some j, 1 ≤ j ≤ t and ∑_{1≤i≤s} m_{i} + ∑_{1≤j≤t} n_{j} ≥ 3. Then, P̃ contains an element of order 8.

Proof. (i) This follows easily from Proposition 12.2 using arguments similar to Lemma 11.3.

(ii) Similar argument to (i) using the fact that GL_{2}(q′) and GU_{2}(q′) contain elements of order 8 for any odd prime power q′.

(iii ) Again, we use the fact that GU_{2}(q^{e_{i}}) has an element of order 8. □

Lemma 12.4. Suppose that P is elementary abelian and that s + t ≥ 3. Then the following hold.

(i) s = 0. (ii) n_{i} = 1 for all i, 1 ≤ i ≤ t. (iii) q ≡ 1 (mod 4).

(iv) If s + t is even then |P | = 2^{s−2}, and if s + t is odd then |P | = 2^{s−1}; in particular, the 2-rank of P is even.

(v) The block d of GU_{n}(q) is nilpotent with an elementary abelian defect group T of order 2^{s}.

Proof. Since GL_{1}(q^{2d_{1}}) contains an element of order 8, conclusion (i) is immediate from Lemma 12.3(i). Assume from now on that s = 0. If n_{1} ≥ 2, then the fact that s + t ≥ 3 implies that ∑_{1≤j≤s} n_{j} ≥ 4. So, by Lemma 12.3(ii), P has an element of order 8, a contradiction. So, (ii) holds. If q ≡ 3 (mod 4), then GU_{1}(q^{e_{i}}) contains an element of order 4, and again by Lemma 12.3(i), P has an element of order 4. This proves (iii). Hence T is a Sylow 2-subgroup of ∏_{j=1}^{t} 𝔽_{q^{e_{j}}}^{×}. Since the e_{j} are odd and q ≡ 1 (mod 4) this implies that T is elementary abelian of rank t, and hence P̃ is elementary abelian of rank t − 1. If n is odd then P ≅ P̃ and since n = ∑_{j=1}^{t} e_{j} and the e_{j} are

odd it follows that t is odd, hence |P | = 2^{t−1}. If n is even then |P | = \frac{|P̃|}{2} and since n = ∑_{j=1}^{t} e_{j} and the e_{j} are odd it follows that t is even and |P | = 2^{t−2}, which proves (iv). Finally, the block d of GU_{n}(q) is nilpotent in that case because (ii) implies that the centraliser of the semi-simple element labelling d is a torus, whence (v). □

Lemma 12.5. Suppose that P is elementary abelian and that s + t ≤ 2 . Then |P | ≤ 4.

Proof. Suppose s + t = 1; that is, T is a Sylow 2-subgroup of GL_{m}(q^{2d}), where m = m_{1} and d = d_{1}, and we have n = 2md or T is a Sylow 2-subgroup of GU_{m}(q^{e}), where m = n_{1} and e = e_{1}, and we have n = me. In the former case, if m = 1 then T is cyclic, hence |P | ≤ 2, so we may assume m ≥ 2. If m ≥ 3, then since GL_{1}(q^{2d}) contains an element of order 8, by Lemma 12.3 (i), P has an element of order 8, a contradiction. This contradiction shows m = 2. Thus n = 4d, and so T is a Sylow 2-subgroup of GL_{2}(q^{2d}), and one easily checks that any elementary abelian subquotient of T has rank at most 2. In the latter case, again if m = 1, then T is cyclic. If m ≥ 4, then by Lemma 12.3 (ii), P contains an element of order 8, a contradiction. If m = 3, then by Lemma 12.3 (iii), P̃ has an element of order 8. But n = 3e is odd, and hence P ≡ P̃. This contradiction shows that m = 2, and T is a Sylow 2-subgroup of GU_{2}(q^{2}). But any elementary abelian subquotient of T has rank at most 2. Now suppose that s + t = 2 and ∑_{i} m_{i} + ∑_{j} n_{j} ≥ 3. By Lemma 12.3 (i), s = 0, whence t = 2. By Lemma 12.3 (ii), at least one of n_{1} or n_{2} equals 1, say n_{2} = 1. If n_{1} ≥ 3, then by Lemma 12.3 (ii), P contains an element of order 8. I If n_{2} = 2, then by Lemma 12.3 (iii), P̃ contains an element of order 8. But in this case, n = 2e_{1} + e_{2} is odd, which means that P̃ ≡ P, and hence P has an element of order 8. If n_{1} = 1, then T is abelian of rank 2, hence P has order at most 4. Finally suppose that s + t = 2 and ∑_{i} m_{i} + ∑_{j} n_{j} = 2. If s = 0, t = 2, then n_{1} = n_{2} = 1; if s = 1, t = 1, then m_{1} = n_{1} = 1 and if s = 2, t = 0, then m_{1} = m_{2} = 1. In all cases T is abelian of rank 2 whence P has order at most 4. □

Proof of Theorem 12.1. This is immediate from the preceding lemmas. □

13. Orthogonal and symplectic groups in odd characteristic

Throughout this section k is an algebraically closed field of characteristic 2. We show that blocks of orthogonal and symplectic groups with elementary abelian defect groups of order at least 8 are all nilpotent.

Theorem 13.1. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is isomorphic to one of PSp_{2n}(q), n ≥ 2, PΩ_{2n+1}(q) , n ≥ 3 or PΩ_{2n}^{±}(q), n ≥ 4 for some odd prime power q. Let b be a block of kG with elementary abelian defect groups of order at least 8. Then b is nilpotent.

Notation. The group G will denote one of the groups in the above theorem. Further, we define L, L_{0} and G̃ as follows.

If G/Z(G) = PSp_{2n}(q), then L = L_{0} = Sp_{2n}(q) and G̃ = Sp_{2n}(q).

If G/Z(G) = PΩ_{2n+1}(q), then L = O_{2n+1}(q), L_{0} = SO_{2n+1}(q) and G̃ = Ω_{2n+1}(q).

If G/Z(G) = PΩ_{2n}^{+}(q), then L = O_{2n}^{+}(q), L_{0} = SO_{2n}^{+}(q) and G̃ = Ω_{2n}^{+}(q).

If G/Z(G) = PΩ_{2n}^{−}(q), then L = O_{2n}^{−}(q), L_{0} = SO_{2n}^{−}(q) and G̃ = Ω_{2n}^{−}(q).

So, G̃◁L_{0} ◁L with the indices of the inclusions being 1 or 2 and G = G̃/Z where Z is a central subgroup of G̃ of order 1 or 2. Let b be a block of kG and denote by P a defect group of b. Since Z is a central 2-subgroup, b is the image of a unique block b̃ of kG̃, and the inverse image P̃ of P in kG̃ is a defect group of b̃. Let d_{0} be a block of kL_{0} covering b̃ with a defect group T_{0} such that T_{0} ∩ G̃ = P̃. We note that the block b̃ is L_{0}-stable (see [20, Corollary 6.4]) whence, d_{0} = b and T_{0}/P̃ is cyclic of order [L_{0} : G̃]. Let d be a block of kL covering d_{0} = b̃ and T a defect group of b containing T_{0}. As a byproduct of the Jordan decomposition of blocks (see Theorem 9.1), there exists a semi-simple element s ∈ L_{0} of odd order such that T_{0} is a Sylow 2-subgroup of C_{L_{0}}(s) and T is a Sylow 2-subgroup of C_{L}(s) (see [2, (5A)]).

Lemma 13.2. If the defect groups of kL_{0}d_{0} are abelian, then kL_{0}d_{0}, kLd, kG̃b̃ and kGb are all nilpotent blocks.

Proof. L_{0} = G_{0}^{F}, where L_{0} is a simple algebraic group of type B, C or D over 𝔽̄_{q} and F : L_{0} → L_{0} is a Frobenius endomorphism with respect to an 𝔽_{q}-structure on L_{0}. Thus, by Corollary 9.2, kL_{o}d_{0} is nilpotent. Since G̃ is a normal subgroup of L_{0} of index a power of 2 and d_{0} covers b̃, kG̃b̃ is nilpotent. Since G is a quotient of G̃ by a central 2-subgroup, and b̃ lifts b, kGb is nilpotent. Finally L_{0} is of index 1 or 2 in L, and d covers d_{0}, hence kLd is nilpotent. □

For the rest of this section s will denote a semi-simple element of odd order in L_{0} such that T is a Sylow 2-subgroup of C_{L_{0}}(s) and T is a Sylow 2-subgroup of C_{L}(s).

Lemma 13.3. Suppose that G = PSp_{2n}(q), n ≥ 2. If P is abelian, then b is nilpotent.

Proof. Note that G̃ = [L, L] = L_{0} = L, G = L/Z, |Z| = 2, and d = d_{0} = b̃. Further, by [32, §1], and noting that s has odd order, the group H = C_{L}(s) is a direct product of groups H_{i} as in Lemma 10.2 with H_{0} ≅ Sp_{2m_{0}}(q) and

n = m_{0} + ∑_{1≤i≤t} m_{i}d_{i}.

The above decomposition of C_{L}(s) corresponds to the orthogonal decomposition of the underlying symplectic space as a direct sum of isotypic 𝔽_{q}[s]-modules (for instance, H_{0} corresponds to the 1-eigen space of s). In particular, if the decomposition has more than one non-trivial factor, then Z ∩ H_{i} = 1 for all i, 0 ≤ i ≤ t. We have T = T_{0} = P̃ and P = T ∩ [L, L]/Z = T/Z. Suppose that P is abelian. If m_{0} ≠ 0, then by Lemma 10.2(ii) and (iv), m_{0} = 1, H = H_{0} = Sp_{2}(q). In particular, n = m_{0} = 1, a contradiction. Thus m_{0} = 0. Suppose that m_{i} = 2 for some i ≥ 1. Then by Lemma 10.2, t = 1 and H = GL_{2}(q^{d_{1}}) or H_{1} = GU_{2}(q^{d_{1}}). But as observed above, P is the quotient of a Sylow 2-subgroup of H by Z(H), and PGL_{2}(q′) and PGU_{2}(q′) have non-abelian Sylow 2-subgroups for any odd prime power q′, a contradiction. Thus m_{0} = 0 and m_{i} ≤ 1 for all i, 1 ≤ i ≤ t. In particular, H and therefore T_{0} is abelian. The result follows by Lemma 13.2. □

Before we proceed, we recall the structure of C_{L}(s) when L is an orthogonal group as described in [32, §1]. Let V be an underlying 𝔽_{q}-vector space for L, and let τ : V → 𝔽_{q} be a non-degenerate quadratic form underlying L. So L = I(V), L_{0} = I_{0}(V) and G̃ = Ω(V) = [I(V), I(V)]. Let V_{0} denote the 1-eigen space of s. The space V decomposes as an orthogonal direct sum

V = V_{0} ⊕ (⊕_{1≤i≤t}V_{i})

where for i ≥ 1, V_{i}, is an isotypic 𝔽_{q}[s]-module, such that V_{i} and V_{j} have no common irreducible 𝔽_{q}[s]-summands for 0 ≤ i ≠ j ≤ t and such that

C_{L}(s) = ∏_{i} C_{L}(s) ∩ GL(V_{i}).

Here ∏_{i} GL(V_{i}) is considered as a subgroup of GL(V) in the standard way. Further, setting H_{i} = C_{L}(s) ∩ GL(V_{i}), we have that

H_{i} = \{ \begin{matrix}L ∩ GL(V_{i}) & if & i = 0 \\ GL_{m_{i}}(ϵ_{i}q^{d_{i}}) & if & i ≥ 1. \end{matrix}

Here for each i ≥ 1, 2d_{i}m_{i} is the 𝔽_{q}-dimension of V_{i} and H_{0} = I(V_{0}). Also, ϵ_{i} ∈ {±1} and GL_{m_{i}}(ϵ_{i}q^{d_{i}}) denotes GU_{n_{i}}(q^{d_{i}}) if ϵ_{i} = −1. If i ≥ 1, then H_{i} ≤ L_{0}. Thus,

C_{L_{0}}(s) = (H_{0} ∩ L_{0}) × ∏_{1≤i≤s} H_{i}.

For each i, 0 ≤ i ≤ t, the restriction of τ to V_{i} is non-degenerate. This form has maximal Witt index if ϵ_{i}^{m_{i}} = 1 and is of non-maximal Witt index if ϵ_{i}^{m_{i}} = −1. For 1 ≤ i ≤ t, let t_{i} be the unique involution in the centre of H_{i}. As an element of GL(V), t_{i} acts as −1 on V_{i} and as 1 on all V_{j}, j ≠ i. Now G̃ = [I(V), I(V)] is the kernel of the spinorial norm from I_{0}(V) to 𝔽_{q}^{×}/𝔽_{q}^{×2} (see [36, §2.7]). From this it follows that t_{i} ∈ Ω(V) if and only if q^{d_{i}m_{i}} ≡ ϵ_{i}^{m_{i}} (mod 4). We will use this fact in the sequel.

Lemma 13.4. Suppose that G = PΩ_{2n+1}(q), n ≥ 3 and that P is elementary abelian with |P | ≥ 8. Then b is nilpotent.

Proof. Note that since the dimension of the underlying vector space is odd, G = G̃ = [L, L], G̃ is of index 2 in L_{0} and L_{0} is of index 2 in L. In particular, P̃ = P. By Lemma 10.2, either m_{1} = 2 and m_{i} = 0 for all i different from 1, or all m_{i} ≤ 1. In the former case, again by Lemma 10.2, T is isomorphic to a Sylow 2-subgroup of a 2-dimensional general linear group. In particular, T has an element of order 8. Since T ≤ L in this case, P is of index 2 in of T, hence P has an element of order 4, a contradiction. We assume from now on that all m_{i} ≤ 1. If m_{0} = 0, then T and therefore T_{0} is abelian and we are done by Corollary 13.2. So, m_{0} = 1, i.e. H_{0} = O_{3}(q). Since n ≥ 2, i ≥ 1, i.e., m_{1} ≠ 0. Let T^{i} be the i-th component of T. We claim that T^{i} ≰ P for any i ≥ 1. Indeed, suppose the contrary. Then T^{i} = ⟨t_{i}⟩ has order 2. In particular, this means that q^{d_{i}} ≢ ϵ_{i} (mod 4). But since m_{i} = 1, this means that t_{i} ∉ Ω(V) and hence t_{i} ∉ P. This proves the claim. By Lemma 10.2, T^{0} ∩ H_{0} is a dihedral group of order at least eight. Also, clearly T^{0} ∩ H_{0} ≤ T_{0}. Since P is of index 2 in T_{0}, and since as just shown T^{1} ≰ P, it follows that P contains a subgroup isomorphic to T^{0} ∩ H_{0}, an impossibility as P is abelian and T^{0} ∩ SO_{3}(q) is not. □

Lemma 13.5. Suppose that G = PΩ_{2n}^{+}(q), or PΩ_{2n}^{−}(q), n ≥ 4 and that P is elementary abelian with |P | ≥ 8. Then b is nilpotent.

Proof. We first consider G = PΩ_{2n}^{+}(q). If m_{1} = 2, then by Lemma 10.2(i) and (iii), m_{j} = 0 for j ≠ i, C_{L}(s) = H_{1} ≅ GL(ϵ_{1}q^{d_{1}}), where q ≡ ±3 (mod 8) and d_{1} is odd. But q^{d_{1}m_{1}} ≡ 1 ≡ ϵ_{1}^{m_{1}} (mod 4), hence the central involution t_{1} of H_{1} is in [L, L] and P is a subgroup of T/⟨t_{i}⟩. Since T/⟨t_{i}⟩ is a dihedral group (see [18]), P has rank at most 2, a contradiction. Now suppose m_{0} = 2. Then, m_{i} = 0 for all i ≥ 1, and n = 2, a contradiction, as n is assumed to be at least 3. Thus, m_{i} ≤ 1 for all i. If m_{0} = 0, then T_{0} is abelian and we are done by Theorem 13.2. So, suppose that m_{1} = 1. Then H_{0} = O_{2}^{±}(q) is a dihedral group and H_{0} ∩ L_{0} = SO_{2}^{±}(q) is a cyclic group (see [36, §2.7]). Since H_{i} is also cyclic for all i ≥ 1, T_{0} is abelian and we are done by Theorem 13.2. The proof for G = PΩ_{2n}^{−}(q) is similar. □

Proof of Theorem 13.1. This is immediate from the preceding lemmas. □

14. Type G_{2}, ^{2}G_{2} and ^{3}D_{4} in odd characteristic

Let k be an algebraically closed field of characteristic 2 and q an odd prime power. The 2-rank of G_{2}(q) and ^{2}G_{2}(q) is 3; see e.g. [34, §1].

Proposition 14.1. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is simple of type G_{2}(q). Then kG has no block with an elementary abelian defect group of order 8.

Proof. This follows from [38], where the 2-blocks of G are determined. Alternatively, one can use the arguments in [20, 12.2]: there is a unique conjugacy class of involutions u in G and by [36, 4.5.1], C_{G}(u) ≅ Z(G)× (2.(PSL_{2}(q) × PSL_{2}(q)).2). The acting 2-automorphism in the second factor is inner-diagonal, hence stabilises any block of 2.(PSL_{2}(q)×PSL_{2}(q)). Thus a block of C_{G}(u) with elementary abelian defect group of order 8 would cover a block of 2.(PSL_{2}(q) × PSL_{2}(q)) with a Klein four defect group, whose image modulo the central involution would yield a block of PSL_{2}(q) × PSL_{2}(q) with a defect group of order 2. Any block of this direct product is of the form c_{0} ⊗ c_{1}, where c_{0}, c_{1} are blocks of PSL_{2}(q), so exactly one of c_{0}, c_{1} would have defect zero and the other defect one. But since any inverse image of an involution in PSL_{2}(q) in SL_{2}(q) has order 4 this is impossible. □

Proposition 14.2. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is simple of type ^{2}G_{2}(q). The principal block b_{0} is the unique block of kG having an elementary abelian defect P group of order 8, and we have |Irr_{K}(G, b_{0})| = 8.

Proof. The Sylow 2-subgroups of ^{2}G_{2}(q) are elementary abelian of order 8. The simple group of type ^{2}G_{2}(q) has trivial Schur multiplier, hence Z(G) = 1. In that case we have N_{G}(P) ≅ P ⋊ E, with E a Frobenius group of order 21 acting faithfully on P, and hence, by Brauer’s First Main Theorem, the principal block b_{0} of kG is the unique block having P as defect group. By Ward’s explicit calculations in [71] or Landrock’s general results in [44, §3] we have |Irr_{K}(G, b_{0})| = 8. □

Finally for the triality D_{4}-groups we have the following proposition due to Deriziotis and Michler [25, Proposition 5.3]

Proposition 14.3. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is simple of type ^{3}D_{4}(q). Let P be a defect group of some block of kG. Then either P is non-abelian or P has rank at most 2. In particular, no block of kG has defect groups which are elementary abelian of order 8.

15. Unipotent characters with small 2-defects

We denote in this section by O a complete discrete valuation ring with an algebraically closed residue field k of characteristic 2 and a quotient field K of characteristic zero which is assumed to be large enough for all finite groups under consideration. Recall that the 2-defect of an irreducible character χ of a finite group G is the largest integer d(χ) such that 2^{d(χ)} divides the rational integer \frac{|G|}{χ(1)}. The notation of the finite groups of Lie type and the labelling of their unipotent characters due to Lusztig in the following two propositions are from the tables page 75/76 and pages 465–488 in Carter’s book [17]. In particular, if X is some simple type we denote by X(q) the finite group of Lie type (that is, the group of fixed points in the algebraic group under some Frobenius endomorphism). We will deviate from the notation of [17] in one respect, i.e., we will denote by ^{2}A_{l}(q), ^{2}D_{l}(q), and ^{3}D_{4}(q) the twisted groups denoted by ^{2}A_{l}(q^{2}), ^{2}D_{l}(q^{2}), and 3D_{4}(q^{3}) respectively in [17]. For classical groups, we will also draw upon Olsson’s treatment of the combinatorics of symbols [59]. Given a positive integer n, let n_{+} denote as before the 2-part of n and n_{−} the 2′-part of n. So, n = n_{+}n_{−}, no odd prime divides n_{+} and 2 does not divide n_{−}.

Proposition 15.1. Let q be a power of an odd prime r. Let (q − 1)_{+} = 2^{d} and (q + 1)_{+} = 2^{e}.

(i) Every unipotent character of A_{l}(q) has 2-defect greater than or equal to dl. If l + 1 is not a triangular number, then the 2-defect of any unipotent character of A_{l}(q) is at least dl + e ≥ 2 + l. If l ≥ 3, then all unipotent charecters have 2-defect greater than or equal to 5. A_{2}(q) has one unipotent character, χ^{(1,2)} of 2-defect 2d, all other unipotent characters have 2-defect 2d + e ≥ 4. The 2-defect of any unipotent character of A_{1}(q) is d + e ≥ 3. (ii) Every unipotent character of ^{2}A_{l}(q) has 2-defect greater than or equal to el. If l + 1 is not a triangular number, then the 2-defect of any unipotent character of ^{2}A_{l}(q) is at least el + d ≥ 2 + l. If l ≥ 3, then all unipotent characters have 2-defect greater than or equal to 6. ^{2}A_{2}(q) has one unipotent character, χ^{(1,2)} of 2-defect 2e, all other uniptent characters have 2-defect 2e + d ≥ 4. The 2-defect of any unipotent character of ^{2}A_{1}(q) is d + e ≥ 3. (iii) Every unipotent character of B_{l}(q) or C_{l}(q), l ≥ 2 has 2-defect at least 2l ≥ 4. (iv) Every unipotent character of D_{l}(q) or ^{2}D_{l}(q), l ≥ 4 has 2-defect at least 2l − 1 ≥ 7.

Proof. We note that d + e ≥ 3.

(i) The unipotent characters of A_{l}(q) are parametrized by the partitions of l + 1. If α is a partition of l and χ^{α} is the corresponding unipotent character, then

|χ^{α}(1)|_{r′} = \frac{(q − 1)|A_{l}(q)|_{r′}}{∏_{h}(q^{h} − 1)}

where h runs over the set of hook lengths of α (see for instance [50, pp.152-153]). Thus the 2-defect of χ^{α} is f where 2^{f} = \frac{∏_{h}(q^{h}−1)_{+}}{(q−1)_{+}} . Since α has l + 1 hooks, the first assertion follows. If l + 1 is not a triangular number, then at least one hook of α is of even length. So, (q^{2} −1) divides q^{h} −1 for at least one hook length h of α, whence f ≥ (d + e) + d(l − 1). Since 5 is not a triangular number, it is immediate from the first two assertions that any unipotent character of A_{l}(q), l ≥ 4 has 2-defect at least 6. Any partition of 4 has two hooks of even length, hence f ≥ 2(d + e) ≥ 6. If α = (1, 1, 1) or α = (3), then the hook lengths of α are 1, 2 and 3 and it follows that f = 2d + e ≥ 4. If α = (2, 1), then the hook lengths of α are 1, 1 and 3 and the 2-defect is 2d. Finally, if α = (1, 1) or (2), then the hook lengths are 1 and 2 and the 2-defect is d + e.

(ii) The unipotent characters of ^{2}A_{l}(q) are also parametrised by partitions of l + 1. If χ^{α} is the character labelled by α, then

|χ^{α}(1)|_{r′} = \frac{(q + 1)|^{2}A_{l}(q)|_{r′}}{∏_{h}((−q)^{h} − 1)} ,

where h runs over the set of hook lengths of α (see [49]). The assertions of (ii) follow as in (i).

(iii) and (iv) Let G be one of the groups B_{l}(q), C_{l}(q), l ≥ 2, D_{l}(q), ^{2}D_{l}(q), l ≥ 4. In Lusztig’s description, the unipotent characters are labelled using symbols. A symbol is an equivalence class [X, Y] of unordered pairs [X, Y] with X, Y (possibly empty) subsets of the non-negative integers. The pair [X, Y] is equivalent to [X′, Y′] if and only if there exists an integer t such that either X = X^{′+t} and Y = Y^{′+t} or X = Y^{′+t} and Y = X^{′+t}. Let [X, Y] be a symbol, X = {a_{1}, a_{2}, ⋯, a_{k}}, Y = {b_{1}, b_{2}, ⋯, b_{r}}, with a_{1} > a_{2} ⋯ > a_{k} ≥ 0 and b_{1} > b_{2} > ⋯ b_{r} ≥ 0. The rank of [X, Y] is defined by

rk[X, Y] = ∑_{i} a_{i} + ∑_{j} b_{j} − ⌊(\frac{k + r − 1}{2})^{2}⌋

where ⌊x⌋ denotes the greatest integer less than or equal to x. Let c(X, Y) be defined by

c(X, Y) = \{ \begin{matrix}⌊\frac{k+r−1}{2}⌋ − |X ∩ Y | : & X ≠ Y \\ 0 : & X = Y. \end{matrix}

If χ_{[X,Y]} is the unipotent character of G indexed by the symbol [X, Y], then rk[X, Y] = l and

|χ_{[X,Y]}(1)|_{r′} = \frac{|G|_{r′}}{2^{c(X,Y)} ∏_{h}(q^{h} − 1) ∏_{h′}(q^{h′} + 1)}

where h runs over the set of hooks of [X, Y] and h′ runs over the set of cohooks of [X, Y] (see [59, Proposition 5]). Let h^{+}[X, Y] be the number of hooks of [X, Y] and let h^{−}[X, Y] denote the number of cohooks of [X, Y]. Then by the character formula above it suffices to prove that c(X, Y) + h^{+}[X, Y] + h^{−}[X, Y] = 2(rk[X, Y]) in case G = B_{l}(q) or G = C_{l}(q) and that c(X, Y) + h^{+}[X, Y] + h^{−}[X, Y] = 2(rk[X, Y]) − δ, where δ ∈ {0, 1} in case G = D_{l}(q) or G = ^{2}D_{l}(q). By [59, Equations 15, 16],

(6)  h^{+}[X, Y] = ∑_{i} a_{i} + ∑_{j} b_{j} − \binom{k}{2} − \binom{r}{2}

and

(7)  h^{−}[X, Y] = ∑_{i} a_{i} + ∑_{j} b_{j} − kr + |X ∩ Y |.

From these it is straighforward to check that

c(X, Y) + h^{+}[X, Y] + h^{−}[X, Y] − 2(rk[X, Y]) = 0

if either k − r is odd or X = Y and

c(X, Y) + h^{+}[X, Y] + h^{−}[X, Y] − 2(rk[X, Y]) = −1

if k − r is even and X ≠ Y. The result follows since if G = B_{l}(q) or G = C_{l}(q) then k − r is odd and if G = D_{l}(q) or G = ^{2}D_{l}(q) then k − r is even (see [17]). □

Proposition 15.2. Let q be an odd prime power.

(i) G_{2}(q) has two unipotent characters of 2-defect 0, (labelled G_{2}[θ], G_{2}[θ^{2}]); if q ≡ 1 (mod 4) then G_{2}(q) has two unipotent characters of 2-defect 3 (labelled G_{2}[1], G_{2}[−1]), and if q ≡ 3 (mod 4) then G_{2}(q) has one unipotent character of 2-defect 3 (labelled Φ_{2,2}). In both cases, all remaining unipotent characters of G_{2}(q) have 2-defect at least 4. (ii) ^{2}G_{2}(q) has two cuspidal unipotent characters of 2-defect 0, four cuspidal unipotent characters of 2-defect 2, and the remaining two unipotent characters have 2-defect at least 4. (iii) F_{4}(q) has two unipotent characters of 2-defect 0 (labelled F_{4}[θ], F_{4}[θ^{2}]), and two unipotent characters of 2-defect 5 (labelled F_{4}[i], F_{4}[−i]). All other unipotent characters of F_{4}(q) have 2-defect at least 7. (iv) E_{6}(q) and ^{2}E_{6}(q) each have two unipotent characters of 2-defect 0 (labelled E_{6}[θ], E_{6}[θ^{2}] in the case of E_{6}(q)), and all other unipotent characters have 2-defect at least 8. The unipotent characters of defect 0 of E_{6}(q) each have degree \frac{1}{3}q^{7}Φ_{1}^{6}(q)Φ_{2}^{4}(q)Φ_{4}^{2}(q)Φ_{5}(q)Φ_{8}(q) and the unipotent characters of defect 0 of ^{2}E_{6}(q) each have degree \frac{1}{3}q^{7}Φ_{1}^{4}(q)Φ_{2}^{6}(q)Φ_{4}^{2}(q)Φ_{8}(q)Φ_{10}(q).

(v) E_{7}(q) has four unipotent characters of 2-defect d ≥ 3, labelled (E_{6}[θ], 1), (E_{6}[θ^{2}], 1), (E_{6}[θ], ϵ), (E_{6}[θ^{2}], ϵ), where 2^{d} is the largest power of 2 dividing q^{2} − 1. These characters each have degree \frac{1}{3}q^{7}Φ_{1}^{6}(q)Φ_{2}^{6}(q)Φ_{4}^{2}(q)Φ_{5}(q)Φ_{7}(q)Φ_{8}(q)Φ_{10}(q)Φ_{14}(q). Furthermore, E_{7}(q) has exactly two unipotent characters of 2-defect 8 (labelled E_{7}[ξ], E_{7}[−ξ] if q ≡ 1 mod 4 and Φ_{512,11}, Φ_{512,12} if q ≡ 3 mod 4); all other unipotent characters have 2-defect at least 14. (vi) E_{8}(q) has four unipotent characters of 2-defect 0 (labelled E_{8}[ζ], E_{8}[ζ^{2}], E_{8}[ζ^{3}], E_{8}[ζ^{4}]), four unipotent characters of 2-defect 3 (labelled E_{8}[θ], E_{8}[θ^{2}], E_{8}[−θ], E_{8}[−θ^{2}] if q ≡ 1 mod 4, and (E_{8}[θ], Φ_{2,1}), (E_{8}[θ^{2}], Φ_{2,1}), (E_{8}[θ], Φ′_{2,2}), (E_{8}[θ^{2}], Φ′_{2,2}) if q ≡ 3 mod 4); all other unipotent characters have 2-defect at least 5. (vii) ^{3}D_{4}(q) has two unipotent characters with 2-defect 3, two unipotent characters with 2-defect at least 5, and four unipotent characters with 2-defect at least 6.

Proof. One proves this by first expressing the group orders given in [17, pp. 75-76] as products of cyclotomic polynomials evaluated at q. Then, for any of the above groups G and any unipotent character Φ of G one calculates the 2-part of \frac{|G|}{Φ(1)} by running through Lusztig’s lists of character degrees of unipotent characters in [17, pp. 477–488], observing that the only cyclotomic polynomials Φ_{d} which occur in these character degrees for which Φ_{d}(q) is even, are those for d ∈ {1, 2, 4, 8}. While the 2-part of Φ_{4}(q) and Φ_{8}(q) is exactly 2, the 2-part of Φ_{1}(q)Φ_{2}(q) = q^{2} − 1 is at least 8, possibly bigger, which accounts for the inequalities in the above statements. □

Some of the statements in the above Proposition are not needed later but included here for the sake of completeness.

16. Type F_{4}

We denote again by O a complete discrete valuation ring with an algebraically closed residue field k of characteristic 2 and a quotient field K of characteristic zero which is assumed to be large enough for all finite groups under consideration. Let q be an odd prime power.

Proposition 16.1. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is a simple group of type F_{4}(q). If kG has a block b with an elementary abelian defect group P of order 8, then either |Irr_{K}(G, b)| = 8 or OGb is Morita equivalent to a block of OL of a finite group L such that |L/Z(L)| < |G/Z(G)|.

Proof. Here Z(G) = 1 and G = G^{F}, where G is a simple algebraic group of type F_{4} over 𝔽̄_{q} and F : G → G is a Frobenius morphism with respect to an 𝔽_{q}-structure on G. Let [t] be the semi-simple label of b (see §9) and let C(t) := C_{G^{∗}}(t). Suppose first that C(t) is contained in a proper Levi subgroup of G^{∗}. Then by the Jordan decomposition of blocks (see §9), there exists a proper F -stable Levi subgroup, L such that the block OGb is Morita equivalent to some block of OL^{F} . Since L is a proper Levi subgroup of G, |L(t)^{F} /Z(L(t)^{F})| is strictly smaller than |G/Z(G)|. Hence, we may assume that t is a quasi-isolated element of G^{∗}, i.e. that C(t) is not contained in any proper Levi subgroup of G^{∗}. Since Z(G) is trivial, C(t) = C^{∘}(t), hence t is even isolated in G^{∗}, i.e C^{∘}(t) is not contained in any proper Levi subgroup of G^{∗}. From the description of centralisers of semi-simple elements in groups of type F_{4} given by by Shoji [68], (see also [24]) one sees that C̄^{∘}(t)^{F^{∗}} is isomorphic to one of F_{4}(q), B_{4}(q), C_{3}(q) × A_{1}(q), A_{3}(q) × A_{1}(q), ^{2}A_{3}(q) × A_{1}(q), A_{2}(q) × A_{2}(q) or ^{2}A_{2}(q) × ^{2}A_{2}(q) (and Z^{∘}(C^{∘}(t)) = 1). By (1),

(4) and Propositions 15.1 and 15.2, the 2-defect of any element of E(G^{F} , [t]) = E(G^{F} , (t)) is at least 4, whence E(G^{F} , (t)) ∩ Irr_{K}(G, b) = ∅, a contradiction. □

17. On characters of small defect in type E groups

We use the notation of §9.

Proposition 17.1. Suppose that G is simple, simply connected of type E_{6}, and G^{F} is a group of type E_{6}(q). Let s be a semi-simple element of G^{∗F^{∗}} and let χ ∈ Irr_{K}(G) ∩ E(G^{F} , (s)). Let λ, λ′ and λ̄ correspond to χ as in (3). Suppose that the 2-defect d(χ) of χ is at most 3. Then a_{χ} is 1 or 3, α_{χ} = 0 (see notation after (4) in §9) and s, χ and λ̄ satisfy one of the rows in the following table.

Δ_{s} C^{∘}(s)^{′F^{∗}} z(s)  λ̄  a_{χ}χ(1)_{r′}  d(χ) cond. on q
(i)  -  -  Φ_{1}^{2}Φ_{3}^{2}  1  Φ_{1}^{4}Φ_{2}^{4}Φ_{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12} 2  q ≡ 3 mod 4
(ii)  -  -  Φ_{1}^{2}Φ_{5}  1  Φ_{1}^{4}Φ_{2}^{4}Φ_{3}^{3}Φ_{4}^{2}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12}  2  q ≡ 3 mod 4
(iii)  -  -  Φ_{1}Φ_{2}Φ_{3}^{2}  1  Φ_{1}^{5}Φ_{2}^{3}Φ_{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12} 3 (q^{2} − 1)_{+} = 8
(iv)  -  -  Φ_{1}Φ_{2}Φ_{5}  1  Φ_{1}^{5}Φ_{2}^{3}Φ_{3}^{3}Φ_{4}^{2}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12}  3 (q^{2} − 1)_{+} = 8
(v)  -  -  Φ_{1}Φ_{2}Φ_{3}Φ_{6}  1  Φ_{1}^{5}Φ_{2}^{3}Φ_{3}^{2}Φ_{4}^{2}Φ_{5}Φ_{6}Φ_{8}Φ_{9}Φ_{12} 3 (q^{2} − 1)_{+} = 8
(vi)  -  -  Φ_{3}^{3}  1  Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12}  0
(vii)  -  -  Φ_{2}^{2}Φ_{3}Φ_{6}  1  Φ_{1}^{6}Φ_{2}^{2}Φ_{3}^{2}Φ_{4}^{2}Φ_{5}Φ_{6}Φ_{8}Φ_{9}Φ_{12} 2  q ≡ 1 mod 4
(viii)  -  -  Φ_{3}Φ_{12}  1  Φ_{1}^{6}Φ_{2}^{4}Φ_{3}^{2}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}  0
(ix)  -  -  Φ_{9}  1  Φ_{1}^{6}Φ_{2}^{4}Φ_{3}^{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{12}  0
(x)  -  -  Φ_{3}Φ_{6}^{2}  1  Φ_{1}^{6}Φ_{2}^{4}Φ_{3}^{2}Φ_{4}^{2}Φ_{5}Φ_{8}Φ_{9}Φ_{12}  0
(xi)  A_{2}  A_{2}(q) Φ_{3}^{2}  χ^{(2,1)}  Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12}  2  q ≡ 3 mod 4
(xii)  A_{2}  ^{2}A_{2}(q) Φ_{3}Φ_{6}  χ^{(2,1)}  Φ_{1}^{6}Φ_{2}^{2}Φ_{3}Φ_{4}^{2}Φ_{5}Φ_{8}Φ_{9}Φ_{12}  2  q ≡ 1 mod 4
(xiii) D_{4}  ^{3}D_{4}(q) Φ_{3}  ϕ_{2,2}  \frac{1}{2} Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}Φ_{9}Φ_{12}  3  q ≡ 3 mod 4
(xiv) D_{4}  ^{3}D_{4}(q) Φ_{3}  ϕ_{2,1}  \frac{1}{2} Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}  3  q ≡ 3 mod 4
(xv)  D_{4}  ^{3}D_{4}(q) Φ_{3}  ^{3}D_{4}[−1]  \frac{1}{2} Φ_{1}^{6}Φ_{2}^{2}Φ_{3}^{2}Φ_{4}^{2}Φ_{5}Φ_{8}Φ_{9}  3  q ≡ 1 mod 4
(xvi) D_{4}  ^{3}D_{4}(q) Φ_{3}  ^{3}D_{4}[1]  \frac{1}{2} Φ_{1}^{6}Φ_{2}^{2}Φ_{4}^{2}Φ_{5}Φ_{8}Φ_{9}Φ_{12}  3  q ≡ 1 mod 4
(xvii) 3A_{2} A_{2}(q^{3}) 1  χ^{(2,1)}  Φ_{1}^{4}Φ_{2}^{4}Φ_{3}Φ_{4}^{2}Φ_{5}Φ_{8}Φ_{12}  2  q ≡ 3 mod 4
(xviii) E_{6}  E_{6}(q) 1  E_{6}[θ], E_{6}[θ^{2}] \frac{1}{3} Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}  0

26

Here Φ_{d} = Φ_{d}(q) denotes the d-th cyclotomic polynomial over ℚ evaluated at q. The non-blank entries of the third column are to be interpreted as C^{∘}(s)^{′F^{∗}} being a group of the given type, and do not specify the isomorphism type of C^{∘}(s)^{′F^{∗}}.

Proof. Let s, χ, λ be as in the statement. The group C(s)/C^{∘}(s) is isomorphic to a subgroup of Irrℚ̄_{2′} (Z(G)/Z^{∘}(G)) (see for instance [22, Lemma 13.14(iii), Remark 2.4]). The first two assertions follow since Z(G)/Z^{∘}(G) is a cyclic group of order 3.

We will use (4), and Propositions 15.1 and 15.2 in conjunction with the tables giving connected centralisers of semi-simple elements in groups of type E_{6}(q) in [24] in order to identify the Δ_{s}, C^{∘}(s)^{F^{∗}}, z(s) and λ̄ entries of a row in the table. Once these have been identified, the \frac{χ(1)_{r′}}{α_{χ}} entry of the row is calculated using (3), the degree formulae for unipotent characters as given in [17, §13.8] and the order formulae for the finite groups of Lie type (see [17, pp.75-76]). The last two entries are obtained by comparing \frac{χ(1)_{r′}}{α_{χ}} with |G^{F} |. We note here that

|G^{F} | = q^{36}Φ_{1}^{6}Φ_{2}^{4}Φ_{3}^{3}Φ_{4}^{2}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{9}Φ_{12}.

The connected components of Δ_{s} are of type A_{l}, l ≥ 1, D_{4}, D_{5}, or E_{6}. If Δ_{s} has a component of type E_{6} (and in particular is a single component), then s is central in G^{∗}, z(s) = 1 and λ̄ is one of the unipotent characters E_{6}[θ] or E_{6}[θ^{2}] of E_{6}(q) as in Proposition 15.2. Both E_{6}[θ] and E_{6}[θ^{2}] have degree \frac{1}{3}q^{7}Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}, hence χ, s, and λ̄ are as in the last row of the table.

Assume from now on that the components of Δ_{s} are of classical type. So C̄^{∘}(s)^{F^{∗}} is a direct product of groups A_{l}(q^{j}), l ≥ 1, ^{2}A_{l}(q^{j}), l ≥ 2, D_{l}(q^{j}), ^{2}D_{l}(q^{j}), l ≥ 4 or ^{3}D_{4}(q). By Propositions 15.1 and 15.2, the 2-defect of any unipotent character of a group A_{l}(q^{j}), l ≥ 1, ^{2}A_{l}(q^{j}), l ≥ 2, D_{l}(q^{j}), ^{2}D_{l}(q^{j}), l ≥ 4 or ^{3}D_{4}(q) is at least 2. On the other hand, by (4), λ̄ has 2-defect at most 3 (note that α_{χ} = 0). Thus, the connected components of Δ(s) form one orbit under the action of F^{∗}, and in particular are all pairwise isomorphic. It also follows from Proposition 15.1 that the connected components are not of type D_{5} or of type A_{l}, l ≥ 3.

Let j be the number of connected components of Δ_{s}. We claim that the connected components of Δ_{s} are not of type A_{1}. Indeed, assume otherwise, so C^{∘}(s)^{′F^{∗}} = A_{1}(q^{j}). By Proposition 15.1(i)-(ii), every unipotent character of A_{1}(q^{j}) or ^{2}A_{1}(q^{j}) has 2-defect at least 3. Thus, by (4), ζ_{s} = 0. But by [24], whenever Δ_{s} is of type jA_{1}, ζ_{s} ≥ 1, proving the claim.

Suppose that the connected components of Δ_{s} are of type A_{2}, 1 ≤ j ≤ 3. If j = 3, then by [24], C^{∘}(s)^{′F^{∗}} = A_{2}(q^{3}), z(s) = 1, and by Proposition 15.1(i), λ̄ is the character χ^{(2,1)}. By the hook-length degree formula for unipotent characters of groups of type A (see the proof of Proposition 15.1(i)),

λ̄(1)_{r′} = \frac{(q^{3} − 1)|A_{2}(q^{3})|}{(q^{9} − 1)(q^{3} − 1)^{2}} = \frac{|A_{2}(q^{3})|}{Φ_{1}^{2}Φ_{3}^{2}Φ_{9}} ,

we see that s and χ are as in row (xvii) of the table. If j = 2, then by [24], C^{∘}(s)^{′F^{∗}}(s) = A_{2}(q^{2}), whereas by Proposition 15.1 any unipotent character of A_{2}(q^{2}) has 2-defect at least 6 (note that (q^{2} − 1)_{+} ≥ 8). Thus this case does not occur.

If j = 1, and C^{∘}(s)^{′F^{∗}} = A_{2}(q), then ζ_{s} ≤ 1, hence by [24], z(s) is (q^{2} + q + 1) and s and χ are as in row (xi) of the table. If j = 1, and C^{∘}(s)^{′F^{∗}} = ^{2}A_{2}(q), then ζ_{s} ≤ 1, hence by [24], z(s) is (q^{4} + q^{2} + 1) and s and χ are as in row (xii) of the table.

Suppose that the connected components of Δ_{s} are of type D_{4}, so j = 1. By Proposition 15.1(iv), and by [24], C^{∘}(s)^{′F^{∗}} = ^{3}D_{4}(q) and z(s) = q^{2} + q + 1. By Proposition 15.2(vii) λ̄ is one of ϕ_{2,2} or ϕ_{2,1} if q ≡ 3 (mod 4) and λ̄ is one of ^{3}D_{4}[−1] or ^{3}D_{4}[1] if q ≡ 1 (mod 4). Hence, χ, s and λ are as in lines (xiii)-(xvii) of the table.

Finally assume that C^{∘}(s) is a torus, so C^{∘}(s)′ and λ̄ are trivial. By (4)

z(s) = d(χ) ≤ 3.

An inspection of [24] yields that z(s) is as in the first ten rows of the table. □

Proposition 17.2. Suppose that G is simple, simply connected of type E_{6}, and G^{F} is a group of type ^{2}E_{6}(q). Let s be a semi-simple element of G^{∗F^{∗}} and let χ ∈ Irr_{K}(G) ∩ E(G^{F} , (s)). Let λ̄ correspond to χ as in (3). Suppose that the 2-defect d(χ) of χ is at most 3. Then a_{χ} is 1 or 3, α_{χ} = 0 and s, χ and λ̄ satisfy one of the rows in the following table.

Δ_{s} C^{∘}(s)^{′F^{∗}} z(s)  λ̄  a_{χ}χ(1)_{r′}  d(χ) cond. on q
(i)  -  -  Φ_{2}^{2}Φ_{6}^{2}  1  Φ_{2}^{4}Φ_{1}^{4}Φ_{6}Φ_{4}^{2}Φ_{10}Φ_{3}^{2}Φ_{8}Φ_{18}Φ_{12}  2  q ≡ 1 mod 4
(ii)  -  -  Φ_{2}^{2}Φ_{10}  1  Φ_{2}^{4}Φ_{1}^{4}Φ_{6}^{3}Φ_{4}^{2}Φ_{3}^{2}Φ_{8}Φ_{18}Φ_{12}  2  q ≡ 1 mod 4
(iii)  -  -  Φ_{2}Φ_{1}Φ_{6}^{2}  1  Φ_{2}^{5}Φ_{1}^{3}Φ_{6}Φ_{4}^{2}Φ_{10}Φ_{3}^{2}Φ_{8}Φ_{9}Φ_{12}  3  (q^{2} − 1)_{+} = 8
(iv)  -  -  Φ_{2}Φ_{1}Φ_{10}  1  Φ_{2}^{5}Φ_{1}^{3}Φ_{6}^{3}Φ_{4}^{2}Φ_{3}^{2}Φ_{8}Φ_{18}Φ_{12}  3  (q^{2} − 1)_{+} = 8
(v)  -  -  Φ_{2}Φ_{1}Φ_{6}Φ_{3}  1  Φ_{2}^{5}Φ_{1}^{3}Φ_{6}^{2}Φ_{4}^{2}Φ_{10}Φ_{3}Φ_{8}Φ_{18}Φ_{12}  3  (q^{2} − 1)_{+} = 8
(vi)  -  -  Φ_{6}^{3}  1  Φ_{2}^{6}Φ_{1}^{4}Φ_{4}^{2}Φ_{10}Φ_{9}^{2}Φ_{8}Φ_{18}Φ_{12}  0
(vii)  -  -  Φ_{1}^{2}Φ_{6}Φ_{3}  1  Φ_{2}^{6}Φ_{1}^{2}Φ_{6}^{2}Φ_{4}^{2}Φ_{10}Φ_{3}Φ_{8}Φ_{18}Φ_{12}  2  q ≡ 3 mod 4
(viii)  -  -  Φ_{6}Φ_{12}  1  Φ_{2}^{6}Φ_{1}^{4}Φ_{6}^{2}Φ_{4}^{2}Φ_{10}Φ_{3}^{2}Φ_{8}Φ_{18}  0
(ix)  -  -  Φ_{18}  1  Φ_{2}^{6}Φ_{1}^{4}Φ_{6}^{3}Φ_{4}^{2}Φ_{10}Φ_{3}^{2}Φ_{8}Φ_{12}  0
(x)  -  -  Φ_{6}Φ_{3}^{2}  1  Φ_{2}^{6}Φ_{1}^{4}Φ_{6}^{2}Φ_{4}^{2}Φ_{10}Φ_{8}Φ_{18}Φ_{12}  0
(xi)  A_{2}  ^{2}A_{2}(q) Φ_{6}^{2}  χ^{(2,1)}  Φ_{2}^{4}Φ_{1}^{4}Φ_{4}^{2}Φ_{10}Φ_{3}^{2}Φ_{8}Φ_{18}Φ_{12}  2  q ≡ 1 mod 4
(xii)  A_{2}  A_{2}(q)  Φ_{6}Φ_{3}  χ^{(2,1)}  Φ_{2}^{6}Φ_{1}^{2}Φ_{6}Φ_{4}^{2}Φ_{10}Φ_{8}Φ_{18}Φ_{12}  2  q ≡ 3 mod 4
(xiii)  D_{4}  ^{3}D_{4}(q) Φ_{6}  ^{3}D_{4}[1]  \frac{1}{2} Φ_{2}^{4}Φ_{1}^{4}Φ_{4}^{2}Φ_{10}Φ_{8}Φ_{18}Φ_{12}  3  q ≡ 1 mod 4
(xiv)  D_{4}  ^{3}D_{4}(q) Φ_{6}  ^{3}D_{4}[−1]  \frac{1}{2} Φ_{2}^{4}Φ_{1}^{4}Φ_{4}^{2}Φ_{10}Φ_{3}^{2}Φ_{8}Φ_{18}  3  q ≡ 1 mod 4
(xv)  D_{4}  ^{3}D_{4}(q) Φ_{6}  ϕ_{2,1}  \frac{1}{2} Φ_{2}^{6}Φ_{1}^{2}Φ_{6}^{2}Φ_{4}^{2}Φ_{10}Φ_{8}Φ_{18}  3  q ≡ 3 mod 4
(xvi)  D_{4}  ^{3}D_{4}(q) Φ_{6}  ϕ_{2,2}  \frac{1}{2} Φ_{2}^{6}Φ_{1}^{2}Φ_{4}^{2}Φ_{10}Φ_{8}Φ_{18}Φ_{12}  3  q ≡ 3 mod 4
(xvii) 3A_{2}  ^{2}A_{2}(q^{3}) 1  χ^{(2,1)}  Φ_{2}^{4}Φ_{1}^{4}Φ_{6}Φ_{4}^{2}Φ_{10}Φ_{8}Φ_{12}  2  q ≡ 1 mod 4
(xviii) E_{6}  ^{2}E_{6}(q) 1  ^{2}E_{6}[θ], ^{2}E_{6}[θ^{2}] \frac{1}{3} Φ_{2}^{6}Φ_{1}^{4}Φ_{4}^{2}Φ_{10}Φ_{8}  0

Proof. This is exactly as in the proof of Proposition 17.1, replacing q by −q and noting that this switches the roles of the pairs Φ_{1} and Φ_{2}, Φ_{3} and Φ_{6}, Φ_{5} and Φ_{10}, and Φ_{9} and Φ_{18}. □

18. Type E in odd characteristic

We denote again by O a complete discrete valuation ring with an algebraically closed residue field k of characteristic 2 and a quotient field K of characteristic zero which is assumed to be large enough for all finite groups under consideration. Let q be an odd prime power. We do not know whether there are any blocks with elementary abelian defect group of order 8 in type E over a field of odd characteristic - the next result says that if there is such a block, it is either nilpotent or its inertial quotient has order 7.

Proposition 18.1. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is a simple group of type E_{6}(q), ^{2}E_{6}(q), E_{7}(q) or E_{8}(q). If kG has a block b with an elementary abelian defect group P of order 8, then the inertial quotient of b is either trivial (in which case b is nilpotent) or has order 7.

Proof. It suffices to show that if u ∈ P is an involution and if e is a block of kC_{G}(u) with defect group P then e is nilpotent. Indeed, there are non-nilpotent (G, b)-Brauer elements if and only if the inertial quotient of b has order 3 or 21. The pattern of the proof is this: using the lists of centralisers in [36, 4.5.1, 4.5.2], we show that in ‘most’ cases, C_{G}(u) has a direct factor of the form 2.H.2 for some finite group H. As in [20], this means that 2.H is a central extension of H by an involution such that 2.H is contained as a subgroup of index 2 in 2.H.2, and an automorphism of 2.H induced by an element in 2.H.2 − 2.H has an image in the outer automorphism group of 2.H of order 2. Using [20, 6.4] and again [36, 4.5.1, 4.5.2], we will then show that the acting outer 2-automorphism stabilises all blocks of H, hence is contained in a defect group of any block, and so any block of 2.H.2 with an elementary abelian defect group P is necessarily nilpotent, because its image in H.2 is a block with a Klein four defect group which is also a 2-extension of a block of H with a defect group of order 2. Here are the details for the different groups; this follows arguments in [20, §12]. All group theoretic facts about centralisers of involutions are from [36, §4].

Suppose that G/Z(G) is of type E_{6}(q). There are two classes of involutions in G, with representatives t_{1}, t_{2}. If u = t_{1} then C_{G}(u) has a central cyclic subgroup of order gcd(4, q − 1), hence q ≡ 3 (mod 4) since P has exponent 2. Then C_{G}(u) ≅ 2. PΩ_{10}(q).2 × C_{(q−1)/2}. The acting 2-automorphism of PΩ_{10} is inner diagonal, hence [20, 6.4] implies that it is contained in a defect group of any block of C_{G}(u). Thus any block of C_{G}(u) with P as defect group is nilpotent by the argument outlined at the beginning of the proof. If u = t_{2} then C_{G}(u) ≅ 2.(PSL_{2}(q) × Z(N). PSL_{6}(q)).2. Again, the acting 2-automorphism is inner diagonal on both factors, and so the same argument using [20, 6.4] shows that any block of C_{G}(u) with defect group P is nilpotent.

Suppose that G/Z(G) is of type ^{2}E_{6}(q). There are two classes of involutions in G, with representatives t_{1}, t_{2}. If u = t_{1} then C_{G}(u) has a central cyclic subgroup of order gcd(4, q + 1), hence q ≡ 1 (mod 4) since P has exponent 2. Then C_{G}(u) ≅ 2. PΩ_{10}^{−}(q).2 × C_{(q+1)/2}. If u = t_{2} then C_{G}(u) ≅ 2.(PSL_{2}(q) × (3, q + 1). PSU_{6}(q)).2, and the same argument as for E_{6}(q) shows that in both cases, any block of kC_{G}(u) with defect group P is nilpotent.

Suppose that G/Z(G) is of type E_{7}(q). In that case Z(G) = {1}, so G is simple, but the finite group of Lie type E_{7}(q) is a central extension of G by an involution. Let E = Inndiag(G); then |E : G| = 2. Table 4.5.1 in [36] describes C_{E}(u), for any involution u in G, and involutions in G which are E-conjugate will have isomorphic centralisers in G. There are five Econjugacy classes of involutions in G, represented by t_{1}, t_{4}, t′_{4}, t_{7} and t′_{7}. If u = t_{1} then C_{E}(u) ≅ 2.(PSL_{2}(q) × PΩ_{12}(q)).(C_{2} × C_{2}), so our standard argument implies that any block of C_{G}(u) with P as defect group is nilpotent. If u = t_{4} then q ≡ 5 (mod 8) (else t_{4} is non-inner), and C_{E}(u) ≅ 2. PSL_{8}(q).C_{gcd(8,q−1)}.γ, where γ is a graph automorphism. It follows from considering the table 4.5.2 in [36] applied to the inverse image of u in E_{7}(q) that γ is not in G, hence C_{G}(u) ≅ 2. PSL_{8}(q).C_{gcd(8,q−1)}. Since gcd(8, q − 1) is at least 2, we get again that any block of C_{G}(u) with P as defect group is nilpotent. If u = t′_{4} then q ≡ 3 (mod 4) (or else t′_{4} is non-inner) and C_{E}(u) ≅ 2. PSU_{8}(q).C_{gcd(8,q−1)}.γ. As in the previous case we get C_{G}(u) ≅ 2. PSU_{8}(q).C_{gcd(8,q−1)}, and thus that any block of C_{G}(u) with P as defect group is nilpotent. If u = t_{7} then q ≡ 1 (mod 4) and C_{E}(q) ≅ (gcd(3, q − 1).E_{6}(q).3 ∗ C_{q−1}).2, and hence, by an argument as before (using [36, 4.5.2]), we get that C_{G}(u) ≅ (gcd(3, q − 1).E_{6}(q).3 ∗ C_{q−1}). This shows that C_{G}(u) has a normal cyclic subgroup of order 4, hence has no block with P as defect group. Similarly, if u = t′_{7} then q ≡ 3 (mod 4) and C_{E}(q) ≅ (gcd(3, q + 1).^{2}E_{6}(q).3 ∗ C_{q+1}).2; as above we get C_{G}(u) ≅ (gcd(3, q + 1).^{2}E_{6}(q).3 ∗ C_{q+1}). This shows that C_{G}(u) has again a normal cyclic subgroup of order 4, hence has no block with P as defect group.

Suppose finally that G/Z(G) is of type E_{8}(q). Then Z(G) = {1} and G has two conjugacy classes of involutions, with representatives t_{1} and t_{8}. Their centralisers are isomorphic to 2. PΩ_{16}(q).2 and 2.(PSL_{2}(q)×Ē_{7}(q)).2, where Ē_{7}(q) is the simple quotient of E_{7}(q). In both cases our standard argument shows that every block with defect group P is nilpotent. □

Notation. For each n > 2, let p_{n} be a Zsigmondy prime for the pair (q, n), i.e., p_{n} is a prime number such that p_{n} divides q^{n} −1 but p_{n} does not divide q^{m} −1 for any m < n. Such p_{n} exist by Zsigmondy’s theorem (see [65, Theorem 3]), and note that q is odd and that n is assumed greater than 2. By definition, p_{n} is a divisor of Φ_{n}(q). We note that if p_{n} divides Φ_{d}(q), d a natural number, then p_{n} divides gcd(q^{d} − 1, q^{n} − 1) = q^{gcd(d,n)} − 1, hence n|d. Further, we note that since n ≥ 3, p_{n} ≠ 2, 3.

Proposition 18.2. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is a simple group of type E_{6}(q). If kG has a block b with an elementary abelian defect group P of order 8, then either |Irr_{K}(G, b)| = 8 or OGb is Morita equivalent to a block of OL of a finite group L such that |L/Z(L)| < |G/Z(G)|.

Proof. The centre Z(G) has order 1 or 3 and we may assume without loss of generality that Z(G) has order 3. Hence G = G^{F}, where G is a simply connected simple algebraic group of type E_{6} over 𝔽̄_{q} and F : G → G is a Frobenius morphism with respect to an 𝔽_{q}-structure on G. Keeping the notation of the previous section let [t] be the semi-simple label of b. Arguing as for Proposition 16.1, we may and we will assume that t is quasi-isolated. Suppose, if possible that |Irr_{K}(G, b)| ≠ 8. By Propositions 2.6 and 18.1,

|Irr_{K}(G, b)| = 5 and Irr_{K}(G, b) = {χ_{j}, 1 ≤ j ≤ 5}

where χ_{1} has height one and χ_{j} has height zero for 2 ≤ j ≤ 5. For each j, 1 ≤ j ≤ 5, let s_{j} be a semi-simple element in G^{∗F^{∗}} such that χ_{j} ∈ E(G, [s_{j}]). By [14], for any 1 ≤ j ≤ 5, s_{j} can be chosen to have the form s_{j} = tu_{j}, where u_{j} is a 2-element of C(t)^{F^{∗}}. Since C(t)/C^{∘}(t) has exponent dividing the order of t, (see for example [22, Remark 13.15 (i)]), u_{j} ∈ C^{∘}(t)^{F^{∗}}. In particular, the connected components of Δ_{s_{j}} are subdiagrams of the extended connected components of Δ_{t}.

Since t is quasi-isolated and has odd order, by [7, Table III], Δ_{t} is one of E_{6}, A_{2} × A_{2} × A_{2}, or D_{4}. On the other hand, since E(G, [t]) ∩ Irr_{K}(G, b) ≠ ∅, E(G, [t]) and hence E(G, (t)) contains an element of 2-defect 2 or 3. By the discussion Proposition 17.1, it follows that C_{G}(t)^{F^{∗}} contains a unipotent character of 2-defect at most 3 and at least 2. Now if Δ_{t} is of type E_{6}, then t is central in G^{∗}, i.e. C_{G}(t)^{F^{∗}} = E_{6}(q). But by Proposition 15.2 the 2-defect of a unipotent character of a group of type E_{6}(q) is either 0 or at least 8, a contradiction.

Suppose that Δ_{t} = 3A_{2}. Then, since C_{G}(t)^{∘F^{∗}} contains a unipotent character of 2-defect 2 or 3, we see that t is as in row (xvii) of Table 17.1. But since the characters corresponding to (xvii) have 2-defect 2, it follows by Proposition 2.6 that u_{1} = 1 and that u_{j} is non-trivial for all j, 2 ≤ j ≤ 5.

Let 2 ≤ j ≤ 5. Since the 2-defect of s_{j} is 3 and since Δ_{s_{j}} is not of type D_{4} or E_{6}, C_{G^{∗}}(s_{j})^{∘} is a torus corresponding to rows (iii), (iv) or (v) of Table 17.1. In particular, Φ_{9}(q) and hence p_{9} is a divisor of χ_{j}(1) for all j, 2 ≤ j ≤ 5. On the other hand, χ_{1} corresponds to row (xvii) of Table 17.1 and we see that p_{9} does not divide χ_{1}(1). Hence, p_{9} does not divide 2χ_{1}(1) − ∑_{2≤j≤5} δ_{j}χ_{j}(1). In particular, 2χ_{1}(1) − ∑_{2≤j≤5} δ_{j}χ_{j}(1) ≠ 0, contradicting Proposition 2.6(ii).

So Δ_{t} = D_{4}. Let j_{0} be such that s_{j_{0}} = t, 1 ≤ j_{0} ≤ 2. Then t corresponds to one of rows (xiii)- (xvi) of Table 17.1. Since the characters corresponding to these rows have 2-defect 2, j_{0} ≥ 2, say j_{0} = 2. Let us first consider the case that χ_{2} corresponds to row (xiii) of the table, so q ≡ 3 (mod 4) and let l, 3 ≤ l ≤ 5 be such that (s_{l}, χ_{l}) also corresponds to row (xiii) of the table. Then semisimple part of C_{G^{∗}}(s_{l})^{∘}, and hence of C_{C_{G^{∗}} (t)^{∘}}(u_{l}) is of type D_{4}. Thus u_{l} is central in C_{G^{∗}}(t)^{∘}. Since u_{l} is a 2-element, and ζ_{t} is odd, we get that u_{l} is a central element of [C_{G^{∗}}(t)^{∘}, C_{G^{∗}}(t)^{∘}]. But the group ^{3}D_{4}(q) has a trivial centre, hence u_{1} = 1. Thus, if (s_{l}, χ_{l}) corresponds to row (xiii) of Table 17.1, then [s_{l}] = [t]. By [7], t is quasi-isolated but not isolated and C_{G^{∗}}(t)^{∘} is of index 3 in C_{G^{∗}}(t). Since ϕ_{2,1} is the unique unipotent character of its degree in ^{3}D_{4}(q), ϕ_{2,1} is stable under C_{G^{F^{∗}}} (t) and hence there are three possibilities for χ_{l}, each of degree

δ = \frac{1}{6}q^{33}Φ_{1}^{4}(q)Φ_{2}^{4}(q)Φ_{4}^{2}(q)Φ_{5}(q)Φ_{8}(q)Φ_{9}(q)Φ_{12}(q).

Let χ_{1} − ∑_{2≤i leq5} δ_{i}χ_{i} be the element of L^{0}(G, b) as in Proposition 2.6. Then, δ_{l} = δ_{2}. Thus, from the degree formula above, it follows that ∑_{i} δ_{i}χ_{i}(1), where i ranges over the indices for which s_{i} corresponds to row (xiii) of Table 17.1 is not divisible by the Zsigmondy prime p_{6} (note that the number of such indices is at most 3).

Now let j, 1 ≤ j ≤ 5 be such that s_{j} does not correspond to row (xiii) of Table 17.1. Since q ≡ 3 (mod 4), and Δ_{s_{j}} is a subdiagram of the extended Dynkin diagram of Δ_{t}, (s_{j}, χ_{j}) corresponds to one of rows (i), (ii), (iii), (iv), (v), (xi), or (xiv) of Table 17.1. In particular, from the character degree column of Table 17.1, we see that χ_{j}(1) is divisible by p_{6} for any such j. Hence, 2χ_{1}(1) − ∑_{2≤i≤5} δ_{i}χ_{i}(1) is not divisible by p_{6}, a contradiction.

We get a similar contradiction if (t, χ_{2}) corresponds to row (xvi) of Table 17.1 and with p_{6} replaced by p_{12} if (t, χ_{2}) corresponds to row (xiv) or row (xv) of Table 17.1. □

Proposition 18.3. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is a simple group of type ^{2}E_{6}(q). If kG has a block b with an elementary abelian defect group P of order 8, then either |Irr_{K}(G, b)| = 8 or OGb is Morita equivalent to a block of OL of a finite group L such that |L/Z(L)| < |G/Z(G)|.

Proof. This is as the proof of Proposition 18.2, with the roles of q and −q suitably reversed. □

Proposition 18.4. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is a simple group of type E_{7}(q). Suppose that kG has a block b with an elementary abelian defect group P of order 8 such that |Irr_{K}(G, b)| ≠ 8. Then, there exists a finite group L with |L/Z(L)| < |G/Z(G)| and a block c of OL with elementary abelian defect groups of order 8 and such that |Irr_{K}(L, c)| ≠ 8.

Proof. First note that Z(G) = 1 and that G = G̃/⟨z⟩, where G̃ = G̃^{F} for a simply-connected simple algebraic group G̃ of type E_{7}, F : G̃ → G̃ is a Frobenius morphism with respect to an 𝔽_{q}-structure on G̃, and ⟨z⟩ is the central subgroup of order 2 of G̃. Let b̃ be the block of OG̃ lifting b and let [t] be the semi-simple label of b̃.

Suppose first that t is not a quasi-isolated element of G̃^{∗}. Then, arguing as for Proposition 16.1, there exists a proper F -stable Levi subgroup L̃ of G̃, and a block c̃ of OL̃^{F} such that OG̃^{F} b̃ and OL̃^{F} c̃ are Morita equivalent. Noting that z ∈ L̃^{F}, set L = L̃^{F} /⟨z⟩ and let c be the image in OL of c̃. By Proposition 2.8, the defect groups of c are elementary abelian and c does not satisfy Alperin’s weight conjecture. Since

|L/Z(L)| ≤ L = \frac{|L̃^{F} |}{2} < \frac{|G̃^{F} |}{2} = |G| = |G/Z(G)|,

the result follows.

We assume from now on that t is quasi-isolated in G̃^{∗} . Before proceeding we note that since t has odd order, CG̃^{∗}(t) is connected and hence t is in fact isolated. By table III of [7], and using that t is of odd order, we get that either t = 1 and the Dynkin diagram Δ_{t} corresponding to CG̃(t) is of type E_{7} or t has order 3 and Δ_{t} is of type A_{2} × A_{5}. Since there is an ordinary irreducible character of b̃ in the E(G̃^{F} , [t]), it follows from Equation (3) in §9 that (CG̃^{∗}(t)/Z(CG̃^{∗}(t)))^{F^{∗}} has a unipotent character of defect at most 4 (note that b̃ has defect groups of order 16). By Proposition 15.1 it follows that Δ_{t} is not of type A_{2} × A_{5}. So, Δ_{t} is of type E_{7}, i.e., b̃ is a unipotent block. But all non-principal unipotent blocks of G̃b have dihedral defect groups ([26, p.357]). On the other hand, a defect group of b̃ is a central extension of an elementary abelian group of order 8 by a group of order 2, a contradiction. □

Proposition 18.5. Let G be a quasi-simple finite group such that Z(G) has odd order and such that G/Z(G) is a simple group of type E_{8}(q). Suppose that kG has a block b with an elementary abelian defect group P of order 8 such that |Irr_{K}(G, b)| ≠ 8. Then, there exists a finite group L with |L/Z(L)| < |G/Z(G)| and a block c of OL with elementary abelian defect groups of order 8 and such that |Irr_{K}(L, c)| ≠ 8.

Proof. Note that G = G^{F} for a simply-connected simple algebraic group G of type E_{8}, and F : G → G a Frobenius morphism with respect to an 𝔽_{q}-structure on G. Let [t] be the semisimple label of b. Again, the Bonnafé-Rouquier result allows us to reduce to the case that t is a quasi-isolated (hence isolated) element of G^{∗}. By the tables in [23], one sees that Δ_{t} is one of A_{4} × A_{4}, A_{5} × A_{2} × A_{1}, A_{7} × A_{1}, A_{8}, D_{5} × A_{3} , D_{8}, E_{6} × A_{2}, E_{7} × A_{1} or E_{8}. Using as before that (C_{G^{∗}}(t)/Z(C_{G^{∗}}(t)))^{F^{∗}} has a unipotent character of defect at most 3 (see Equation (3) in §9) and Propositions 15.1 and 15.2 we have that Δ_{t} is one of E_{6} × A_{2}, or E_{8}. By [26, p.364], the defect groups of any unipotent block of positive defect of G have order at least 16 hence Δ_{t} is not of type E_{8}.

Suppose that Δ_{t} is of type E_{6} × A_{2} and set C = C_{G^{∗}}(t). Since C is connected and has centre of odd order, by Propositions 15.1 and 15.2, and the discussion preceding Proposition 17.1, C^{F^{∗}} has no unipotent character of defect 3, and at most one unipotent character of 2-defect 2 (corresponding to the product of the unipotent character of 2-defect 2 of A_{2}(q) or ^{2}A_{2}(q) and a unipotent character of 2-defect zero of E_{6}(q) or ^{2}E_{6}(q)). We have |Irr_{K}(G, b)| = 5 and can write Irr_{K}(G, b) = {χ_{j}, 1 ≤ j ≤ 5} as in Proposition 2.6. For each j, 1 ≤ j ≤ 5, let s_{j} and u_{j} be as in the proof of Proposition 18.2. Let j_{0}, 1 ≤ j_{0} ≤ 5 be such that χ_{j_{0}} is in the rational series indexed by [t]. By the above discussion j_{0} = 1 and χ_{1} is the unique character of b in the rational Lusztig series corresponding to [t]. In other words, u_{j} is a non-trivial 2-element for 2 ≤ j ≤ 5.

Let 2 ≤ j ≤ 5. Set C_{j} = C_{G^{∗}}(s_{j}) = C_{G^{∗}}^{∘}(s_{j}), Z_{j} = Z(C_{G^{∗}}(s_{j})) and C̄_{j} = C_{j}/Z_{j}. Set z_{j} = |Z_{j}^{F^{∗}}| and let ζ_{j} be defined by 2^{ζ_{j}} = z_{j}_{+}. Further, let λ̄_{j} be a unipotent character of C̄_{j}^{F^{∗}} corresponding to χ_{j} as in the discussion preceding Proposition 17.1, i.e., such that

(8)  χ_{j_{0}}(1) = \frac{|G|_{r′}}{z_{j}|C̄_{j}^{F^{∗}}|}λ̄_{j}(1)

and

(9)  ζ_{j} + d(τ_{j}) = 3

where d(τ_{j}) is the 2-defect of τ_{j}. We have that C := C_{G^{∗}}(t) = X.Y where X is a simply connected group of type A_{2} and Y is a simple group of type E_{6}, X and Y commute and intersect in a central subgroup of order 3. As u_{j} is a non-trivial F^{∗}-stable 2-element, there are unique F^{∗}-stable 2-elements v_{1} ∈ X, v_{2} ∈ Y, at least one of which is non-trivial and such that u_{j} = v_{1}v_{2},

C_{G^{∗}}(s_{j}) = C_{X}(v_{1})C_{Y}(v_{2})

and C_{G}(s_{j})^{F^{∗}} contains C_{X}(v_{1})^{F^{∗}}C_{Y}(v_{2})^{F^{∗}} as a normal subgroup of index 3.

In particular, ⟨v_{1}, v_{2}⟩ is a central subgroup of C_{G^{∗}}(s_{j})^{F^{∗}}. Since the centre of C_{G^{∗}}(s_{j})^{F^{∗}} is contained in the kernel of any unipotent character of C_{G^{∗}}(s_{j})^{F^{∗}} it follows that the product of the orders of v_{1} and v_{2} is at most 8.

First suppose that both C_{X}(v_{1}) and C_{Y}(v_{2}) are tori. Then, both v_{1} and v_{2} are non-trivial. On the other hand, the product of the orders of v_{1} and v_{2} is at most 8, hence at least one of v_{1} and v_{2} has order 2. But the centraliser of an element of order 2 in a simple algebraic group of type A_{2} or E_{6} is not a torus (see [36, Table 4.3.1]), a contradiction.

Suppose that neither C_{X}(v_{1}) nor C_{Y}(v_{2}) is a torus. We have

C̄_{j} = C_{X}(v_{1})/Z(C_{X}(v_{1})) × C_{Y}(v_{2})/Z(C_{Y}(v_{2})).

Hence, λ̄_{j} = ϕ_{1} × ϕ_{2}, where ϕ_{1} is a unipotent character of C_{X}(v_{1})/Z(C_{X}(v_{1}))^{F^{∗}} and ϕ_{2} is a unipotent character of C_{Y}(v_{2})/Z(C_{Y}(v_{2}))^{F^{∗}}, such that the sum of ζ_{j} and the 2-defects of ϕ_{1} and ϕ_{2} equals 3. Since the only subdiagrams of the extended Dynkin diagram of type A are of type A, it follows from [23] and Propositions 15.1 and 15.2 that either Δ_{j} is A_{1} × E_{6}, and ζ_{j} has odd order or Δ_{j} is A_{2} × E_{6} and ζ_{j} has even order. But by [23], there are no such s_{j}.

Thus, exactly one of C_{X}(v_{1}) or C_{Y}(v_{2}) is a torus. Then ζ_{j} > 0 (v_{i} is a 2-element) so by Equation (9), the 2-defect of λ̄_{j} is at most 2. From Propositions 15.1 and 15.2, it follows that either Δ_{j} is a product of copies of A_{2} transitively permuted by F^{∗} or Δ_{j} is of type E_{6}.

If Δ_{j} is a product of copies of A_{2} transitively permuted by F^{∗}, then by the tables in [23], one sees that either ζ_{j} = 0 or ζ_{j} ≥ 2, whence by Equation (9), λ̄_{j} has 2-defect 3 or at most 1. This is a contradiction as any unipotent character of A_{2}(q^{m}) or ^{2}A_{2}(q^{m}) has 2-defect 2 or greater than 3.

If Δ_{j} is of type E_{6}, then λ̄_{j} has 2-defect 0, whence the Sylow 2-subgroups of Z_{j}^{F^{∗}} have order 8. Since Δ_{j} is a subdiagram of the extended diagram associated to A_{2} × E_{6}, it follows that C_{X}(v_{1}) is a torus and that the Sylow 2-subgroups of C_{X}(v_{1})^{F^{∗}} have order 8. From [23] (or from the description of F^{∗}-stable maximal tori in type A_{2}), it follows that |C_{X}(v_{1})^{F^{∗}}| is one of (q^{2} − 1), (q ± 1)^{2}, or (q^{2} ± q + 1). Thus the only possibility is that |C_{X}(v_{1})^{F^{∗}}| = (q^{2} − 1) and 8 is the highest power of 2 dividing q^{2} − 1.

First suppose that C^{F^{∗}} is of type E_{6}(q) or A_{2}(q). Then C̄_{j}^{F^{∗}} is of type E_{6}(q) for all j, 2 ≤ j ≤ 5. By the formula for the character degrees of unipotent characters of 2-defect 0 of E_{6}(q), we get that for all j, 2 ≤ j ≤ 5,

(10)  χ_{j}(1) = \frac{|G|_{r′}}{3|E_{6}(q)|(q^{2} − 1)}q^{7}Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8} = \frac{|G|_{r′}}{3|E_{6}(q)|}q^{7}Φ_{1}^{5}Φ_{2}^{3}Φ_{4}^{2}Φ_{5}Φ_{8}.

Here, as before, we use Φ_{d} to denote Φ_{d}(q).

The unipotent character of defect 2 of A_{2}(q) is of degree q(q + 1) whence

(11)  χ_{1}(1) = \frac{|G|_{r′}}{3|E_{6}(q)||A_{2}(q)|}(q + 1)q^{8}Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8} = \frac{|G|_{r′}}{3|E_{6}(q)|} \frac{q^{8}Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}}{Φ_{3}^{2}} .

Consider the element 2χ_{1} − ∑_{2≤j≤5} δ_{j}χ_{j} of L^{0}(G, b) as in Proposition 2.6(ii). By the equation above, all χ_{j}, 2 ≤ j ≤ 5 have the same degree. Hence, by Proposition 2.6(iii), δ_{j} = δ_{i} for all i, j such that 2 ≤ i, j ≤ 5. Thus,

2χ_{1}(1) − ∑_{2≤j≤5} δ_{j}χ_{j}(1) = 0

implies that

qΦ_{2} − 2δ_{2}Φ_{1}Φ_{3}^{2} = 0,

but this is impossible since p_{3} does not divide Φ_{2}(q).

The case that C^{F^{∗}} is of type E_{6}(q)A_{2}(q) is similar with p_{3} replaced by p_{6}. □

19. Proof of Theorem 1.1

Proof of Theorem 1.1. By Theorem 4.1, and using its notation, it suffices to show that |Irr_{K}(G, b)| = 8. Arguing inductively, in order to prove Theorem 1.1 we may assume, by Theorem 3.1, that G is a quasi-simple finite group with a centre of odd order. We also may assume, by [44, Theorem 3.7], that b is a non-principal block. If G/Z(G) is a sporadic simple group then by Proposition 5.2 we have G ≅ Co_{3} and by Proposition 5.3, we have |Irr_{K}(G, b)| = 8. By the results in §§6 and 7, G/Z(G) is neither a finite simple group with an exceptional Schur multiplier nor an alternating group. If G/Z(G) is a finite group of Lie type in characteristic 2 then, by Proposition 8.1, we have G ≅ PSL_{2}(8) and b is the principal block; in particular, we have again |Irr_{K}(G, b)| = 8. Let q be an odd prime power and n a positive integer. By Theorems 11.1 and 12.1, the group G/Z(G) cannot be isomorphic to PSL_{n}(q) or PSU_{n}(q); alternatively, |Irr_{K}(G, b)| = 8 holds in these cases as a consequence of [6]. If G/Z(G) is isomorphic to one of PSp_{2n}(q), (n ≥ 2), PΩ_{2n+1}(q), (n ≥ 3), or PΩ_{2n}^{±}(q), (n ≥ 4) then, by Theorem 13.1, the block b is nilpotent; in particular, |Irr_{K}(G, b)| = 8. By Proposition 14.1, G/Z(G) cannot be isomorphic to a simple group of type G_{2}(q). Since b is assumed to be non-principal, G/Z(G) cannot be isomorphic to a simple group of type ^{2}G_{2}(q), and by Proposition 14.3, G/Z(G) cannot be isomorphic to a simple group of type ^{3}D_{4}(q). If G/Z(G) is isomorphic to one of the remaining exceptional simple groups F_{4}(q), E_{6}(q), ^{2}E_{6}(q), E_{7}(q) or E_{8}(q) then b is Morita equivalent to a block of a finite group L such that |L/Z(L)| is smaller than |G/Z(G)|, by Propositions 16.1, 18.2, 18.3, 18.4 and 18.5, respectively. Theorem 1.1 follows inductively. □

Remark 19.1. We do not know whether there are actually any blocks with an elementary abelian defect group of order 8 if G/Z(G) is of one of the exceptional types F or E. If not, one could avoid the rather tedious calculations from §15 onwards.

20. Appendix

We provide a proof for a result announced by Rouquier in [67]. The notation is as in [46, Appendix]. Let O be a complete discrete valuation ring having an algebraically closed residue field k of characteristic 2. For A, B two symmetric O-algebras, a bounded complex X of A-B-bimodules which are projective as left A-modules and as right B-modules is said to induce a stable equivalence if there are isomorphisms of complexes of bimodules X ⊗_{B} X^{∗} ≅ A ⊕ Y and X^{∗} ⊗_{A} X ≅ B ⊕ Z with Y and Z homotopy equivalent to bounded complexes of projective A-A-bimodules and B-B-bimodules, respectively. If Y and Z are homotopic to zero then X is called a Rickard complex.

Theorem 20.1 (cf. [67, Theorem 6.10]). Let G be a finite group, let b be a block of OG with an elementary abelian defect group of order 8, set H = N_{G}(P) and denote by c the block of OH satisfying Br_{ΔP} (b) = Br_{ΔP} (c). Let i ∈ (OGb)^{ΔP} and j ∈ (OHc)^{ΔP} be source idempotents such that Br_{ΔP} (i) = Br_{ΔP} (j). There is a bounded complex of OGb-OHc-bimodules whose components are finite direct sums of summands of the bimodules OGi ⊗_{OQ} jOH, with Q running over the subgroups of P, such that X induces a stable equivalence.

Proof. The proof follows the lines of [67, 6.3]. For any subgroup Q of P denote by e_{Q} and f_{Q} the unique blocks of kC_{G}(Q) and kC_{H}(Q), respectively, satisfying Br_{ΔQ}(i)e_{Q} ≠ 0 and Br_{ΔQ}(j)f_{Q} ≠ 0. Since P is abelian, the fusion systems on P determined by i and by j are equal to that of kN(P, e_{P})e_{P}. Hence, for any subgroup Q of P, we have

N_{G}(Q, e_{Q})/C_{G}(Q) ≅ N_{H}(Q, f_{Q})/C_{H}(Q)

and both sides have odd order (either 1 or 3 in case Q is a proper subgroup of P). The blocks e_{Q}, f_{Q} lift to unique blocks ê_{Q}, f̂_{Q} of OC_{G}(Q), OC_{H}(Q), respectively. The images of ê_{Q}, f̂_{Q} in OC_{G}(Q)/Q, OC_{H}(Q)/Q are blocks, denoted by ē_{Q}, f̄_{Q}, respectively. Suppose now that Q has order 2. Then ē_{Q}, f̄_{Q} have the Klein four group P/Q as defect group, and C_{H}(Q)/Q is the normaliser in C_{G}(Q)/Q of P/Q. Thus f̄_{Q} is in fact the Brauer correspondent of ē_{Q}. By [20, Theorem 1.1], the source algebras of blocks with a Klein four defect group V_{4} are either OV_{4}, or OA_{4}, or OA_{5}b_{0}, where b_{0} is the principal block of OA_{5}. By [64, §3], there is an explicitly described two-term splendid Rickard complex between OA_{4} and OA_{5}b_{0}. Thus there is a Rickard complex of OC_{G}(Q)/Qē_{Q}-OC_{H}(Q)/Qf̄_{Q}-bimodules C̄_{Q} of the form

C̄_{Q} = \xymatrix{⋯ \ar[r] & 0 \ar[r] & N̄_{Q} \ar[r]^{Φ̄_{Q}} & ē_{Q}C_{G}(Q)/Qf̄_{Q} \ar[r] & 0 \ar[r] & ⋯}

for some (possibly zero) projective bimodule N̄_{Q}. By [66, 10.2.11], this complex lifts to a Rickard complex of OC_{G}(Q)ê_{Q}-OC_{H}(Q)f̂_{Q}-bimodules of the form

C_{Q} = \xymatrix{⋯ \ar[r] & 0 \ar[r] & N_{Q} \ar[r]^{Φ_{Q}} & ê_{Q}C_{G}(Q)f̂_{Q} \ar[r] & 0 \ar[r] & ⋯}

where N_{Q} is a projective O(C_{G}(Q) × OC_{H}(Q))/ΔQ-module lifting N̄_{Q}, inflated to O(C_{G}(Q) × C_{H}(Q)), and where Φ_{Q} lifts the map Φ̄_{Q}. By adapting arguments of Marcuş [51, 5.5] this complex extends to the group

T = N_{G×H}(ΔQ) ∩ (N_{G}(Q, e_{Q}) × N_{H}(Q, f_{Q}))

and T contains C_{G}(Q)×C_{H}(Q) as normal subgroup of odd index at most 3 by the above remarks. One can see this also directly: the modules of the complex C_{Q} are clearly T -stable, hence extend to T (cf. [66, 10.2.13]), and the map Φ̄ lifts to a T -stable map Ψ because N_{Q} remains projective when considered as OT/ΔQ-module. We set

V_{Q} = Ind_{T}^{G×H}(N_{Q}) .

The inclusion C_{G}(Q) ⊆ G induces an OT -homomorphism e_{Q}OC_{G}(Q)f_{Q} ⟶ bOGc, which, by adjunction, yields a homomorphism of O(G × H)-modules

α_{Q} : Ind_{T}^{G×H}(e_{Q}OC_{G}(Q)f_{Q}) ⟶ bOGc .

Set ψ_{Q} = α_{Q} ∘ Ind_{T}^{G×H}(Φ_{Q}) : V_{Q} → bOGc and define the complex X by

X = \xymatrix{⋯ \ar[r] & 0 \ar[r] & ⊕_{Q}V_{Q} \ar[r]^{⊕_{Q}Ψ_{Q}} & bOGc \ar[r] & 0 \ar[r] & ⋯}

with bOGc in degree zero, where Q runs over a set of representatives of the N_{G}(P, e_{P})-conjugacy classes of subgroups of order 2 of P. One checks that if Q, R are two subgroups of order 2 which are not N_{G}(P, e_{P})-conjugate then V_{R}(ΔQ) = {0}. This implies that e_{Q}X(ΔQ)f_{Q} ≃ C_{Q} ⊗_{O} k, and this is a Rickard complex of kC_{G}(Q)e_{Q}-C_{H}(Q)f_{Q}-bimodules. Moreover, bOGc is a direct summand of OGi ⊗_{OP} jOH, and V_{Q} is a direct sum of summands of OGi ⊗_{OQ} jOH because it is obtained from lifting, inflating and inducing a projective bimodule. Thus another result of Rouquier (with a proof given in [46, Theorem A.1]) applies, showing that X induces a stable equivalence. □

Remark 20.2. We have used [20, Theorem 1.1] for the description of Rickard complexes for blocks with a Klein four group, which requires the classification of finite simple groups. One can avoid this by making use of another technique of Rickard, replacing one of the endo-permutation sources of a simple module by a p-permutation resolution (see e.g. [45, Theorem 1.3]). The only difference this makes is that the resulting complex may have more than two non-zero components. One can be more precise in Theorem 20.1 if E is either trivial or has order 7. If E is trivial, b is nilpotent, and its source algebra is of the form End_{O}(V) ⊗_{O} OP for some indecomposable endo-permutation OP -module V with P as vertex. Using again [20, Theorem 1.1] we get that V ≅ Ω_{P}^{n} (O) for some integer n. Equivalently, if E is trivial then OGb is Morita equivalent to OP via an OGb-OP -bimodule which, when viewed as O(G × P)-module, has vertex ΔP = {(u, u) | u ∈ P } and source Ω_{ΔP}^{n} (O) for some integer n. By applying Ω_{O(G×P)}^{−n} to M one gets a stable equivalence of Morita type given by a bimodule with vertex ΔP and trivial source. Similarly, if E is cyclic of order 7, then by a result of Puig [61], there is a stable equivalence of Morita type between OGb and OP given by a bimodule with vertex ΔP and trivial source.

Acknowledgements. The authors would like to thank Felix Noeske for bringing to their attention the work in [56] and Gunter Malle for pointing the authors to the references [5], [6]. The second author made several visits to the University of Aberdeen in 2008 and 2009 during which parts of this work were carried out. He is grateful to the Institute of Mathematics, University of Aberdeen for its kind hospitality. For this research the second author was partially supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)20540008, 2008– 2010.

References

[1] J. L. Alperin, Weights for finite groups, Proc. Symp. Pure Math. 47 (1987), 369–379.

[2] J. An, 2-weights for classical groups, J. Reine Angew. Math. 439 (1993), 159–204.

[3] H. Bender, On groups with abelian Sylow 2-subgroups, Math. Z. 117 (1970), 164–176.

[4] D. J. Benson, Representations and Cohomology II, Camridge studies in advanced mathematics 31, Camridge University Press, Cambridge (1991).

[5] T. R. Berger, R. Knörr, On Brauer’s height 0 conjecture, Nagoya Math. J. 109 (1988), 109–116.

[6] H. I. Blau, H. Ellers, Brauer’s height zero conjecture for central quotients of special linear and special unitary groups, J. Algebra 212 (1999), 591–612.

[7] C. Bonnafé, Quasi-isolated elements in reductive groups, Comm. Algebra 33 (2005), 2315–2337.

[8] C. Bonnafé, R. Rouquier, Catégories dérivées et variétés de Deligne-Lusztig , Publ. Math. Inst. Hautes Études Sci. 97 (2003), 1–59.

[9] R. Brauer, Some applications of the theory of blocks of characters of finite groups IV, J. Algebra 17 (1971), 489–521.

[10] R. Brauer, On 2-blocks with dihedral defect groups, Symp. Math. XIII, Academic Press (1974), 367–393.

[11] M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 180–181 (1990), 61–92.

[12] M. Broué, Isométries de caractères et equivalences de Morita ou dérivées, Publ. Math. Inst. Hautes Études Sci. 71 (1990), 45–63.

[13] M. Broué, Equivalences of blocks of group algebras, in: Finite Dimensional Algebras and Related Topics (Ottawa 1992), NATO Adv. Sci. Inst., Ser C, Math. Phys. Sci. 424, Kluwer Acad. Publ., Dordrecht, 1994, 1–26.

[14] M. Broué, J. Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395 (1989), 56-67.

[15] M. Broué, L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117–128.

[16] M. Cabanes, M. Enguehard, Unipotent blocks of finite reductive groups of a given type, Math. Z. 213 (1993), 479–490.

[17] R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley Classics Library, John Wiley and Sons, Ltd., Chichester, 1993.

[18] R. W. Carter, P. Fong, The Sylow 2-subgroups of the finite classical groups, J. Algebra 1 (1964), 139–151.

[19] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

[20] D. A. Craven, C. W. Eaton, R. Kessar, M. Linckelmann, The structure of blocks with a Klein four defect group, preprint (2008).

[21] E. C. Dade, Block extensions, Illinois. J. Math. 17 (1973), 198–272.

[22] F. Digne and J. Michel, Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts 21, 1991.

[23] D. I. Deriziotis, The centralisers of semisimple elements of the Chevalley groups E_{7} and E_{8}, Tokyo J. Math. 6 (1983), 191–216.

[24] D. I. Deriziotis, Conjugacy Classes and Centralizers of Semisimple Elements in Finite Groups of Lie Type, Lecture Notes in Mathematics at the University of Essen 11, Universität Essen, Fachbereich Mathematik, Essen, 1984.

[25] D. I. Deriziotis, G. O. Michler, Character table and blocks of finite simple triality groups ^{3}D_{4}(q), Trans. Amer. Math. Soc. 303 (1987), 39–70.

[26] M. Enguehard, Sur les l-blocs unipotents des groupes réductifs finis quand l est mauvais, J. Algebra 230 (2000), 334–377.

[27] M. Enguehard, Vers une décomposition de Jordan des blocs groupes réductifs finis, J. Algebra 319 (2008), 1035–1115.

[28] D. Fendel, A characterization of Conway’s .3, J. Algebra 24 (1973), 159–196.

[29] P. Fong, On the characters of p-solvable groups, Trans. Amer. Math. Soc. 98 (1961), 263–284.

[30] P. Fong, M. E. Harris, On perfect isometries and isotypies in finite groups, Invent. Math. 114 (1993), 139–191.

[31] P. Fong, B. Srinivasan, The blocks of finite general and unitary groups, Invent. Math. 69 (1982), 109–153.

[32] P. Fong, B. Srinivasan, The blocks of finite classical groups, J. Reine Angew. Math. 396 (1989), 122–191.

[33] M. Geck, G. Hiss, Basic sets of Brauer characters of finite groups of Lie type, J. Reine Angew. Math. 418 (1991), 173–188.

[34] D. Gorenstein, K. Harada, Finite simple groups of low 2-rank and the families G_{2}(q), D_{4}^{2}(q), q odd, Bull. Amer. Math. Soc. 77 (1971), 829–862.

[35] D. Gorenstein, R. Lyons, R. Solomon, The Classification of Finite Simple Groups, Number 1, Mathematical Surveys and Monographs 40, Amer. Math. Soc., Providence, 1998.

[36] D. Gorenstein, R. Lyons, R. Solomon, The Classification of Finite Simple Groups, Number 3, Mathematical Surveys and Monographs 40, Amer. Math. Soc., Providence, 1998.

[37] M. E. Harris, M. Linckelmann, On the Glauberman and Watanabe correspondences for blocks of finite p-solvable groups, Trans. Amer. Math. Soc. 354 (2002), 3435–3453.

[38] G. Hiss, J. Shamash, 2-blocks and 2-modular characters of the Chevalley groups G_{2}(q), Mathematics of Computation 59, No. 200 (1992), 645–672.

[39] Z. Janko, A new finite simple group with abelian Sylow 2-subgroups and its characterization, J. Algebra 3 (1966), 147–186.

[40] R. Kessar, Scopes reduction for blocks of finite alternating groups, Quart. J. Math. 53 (2002), 443–454.

[41] R. Kessar, M. Linckelmann, On stable equivalences and blocks with one simple module, J. Algebra 323 (2010), 1607–1621.

[42] B. Külshammer, Morita equivalent blocks in Clifford theory of finite groups, Astérisque 181–182 (1990), 209–215.

[43] P. Landrock, Non-principal 2-blocks of sporadic simple groups, Communications in Algebra 6 (1978), 1865–1891.

[44] P. Landrock, On the number of irreducible characters in a 2-block, J. Algebra 68 (1981), 426–442.

[45] M. Linckelmann, On derived equivalences and local structure of blocks of finite groups, Turkish J. Math. 22 (1998), 93–107.

[46] M. Linckelmann, Trivial source bimodule rings for blocks and p-permutation equivalences, Trans. Amer. Math. Soc. 361 (2009), 1279–1316.

[47] G. Lusztig, Characters of reductive groups over finite fields Annals of math. studies 107 Princeton University Press (1984).

[48] G. Lusztig, On the representations of groups with disconnected center Astérisque 168 (1988), 157–166.

[49] G. Lusztig, B. Srinivasan, The characters of the finite unitary groups J. Algebra 49 (1977), 167-171.

[50] I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Univ. Press, (Clarendon), London/New York, 1979.

[51] A. Marcuş, On equivalences between blocks of group algebras: reduction to simple components, J. Algebra 184 (1996), 372–396.

[52] The Modular Atlas, http://www.math.rwth-aachen.de/ MOC/

[53] W. Müller, Symmetrische Algebren mit injektivem Zentrum, Manuscripta Math. 11 (1974), 283–289.

[54] M. Murai, Block induction, normal subgroups and characters of height zero, Osaka J. Math. 31 (1994), 9–25.

[55] H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic Press, New York, 1988.

[56] F. Noeske, ADGC for Sporadic Groups, http://www.math.rwth-aachen.de/∼Felix.Noeske/tabular.pdf [57] T. Okuyama, Y. Tsushima, Local properties of p-block algebras of finite groups, Osaka J. Math. 20 (1983),

33–41.

[58] J. Olsson, On 2-blocks with quaternion and quasidihedral defect groups, J. Algebra 36 (1975), 212–241.

[59] J. Olsson, Remarks on symbols, hooks and degrees of unipotent characters, J. Combinatorial Theory, Series A 42 (1986), 223-238.

[60] L. Puig, Local block theory in p-solvable groups, Proc. Symp. Pure Math. 37 (1980), 385–388.

[61] L. Puig, Une correspondance de modules pour les blocs à groupes de défaut abéliens, Geom. Dedicata 37 (1991), 9–43.

[62] L. Puig, On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks, Progress in Math. 178, Birkhäuser Verlag, Basel-Boston-Berlin (1999).

[63] W. F. Reynolds, Blocks and normal subgroups of finite groups, Nagoya Math. J. 22 (1963), 15–32.

[64] J. Rickard, Splendid equivalence: derived categories and permutation modules, Proc. London Math. Soc. 72 (1996), 331–358.

[65] M. Roitman, On Zsigmondy primes, Proc. Amer. Math. Soc. 125 (1997), 1913–1919.

[66] R. Rouquier, The derived category of blocks with cyclic defect groups, in: Derived Equivalences for Group Rings (S. König, A. Zimmermann), Lecture Notes in Math. 1685, Springer Verlag, Berlin-Heidelberg, 1998, 199–220.

[67] R. Rouquier, Block theory via stable and Rickard equivalences, in: Modular representation theory of finite groups (Charlottesville, VA 1998), (M. J. Collins, B. J. Parshall, L. L. Scott), DeGruyter, Berlin, 2001, 101-146.

[68] T. Shoji, The conjugacy classes of Chevalley groups of type (F_{4}) over finite fields of characteristic p ≠ 2, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 1–17.

[69] J. Thévenaz, G-Algebras and Modular Representation Theory, Oxford Science Publications, Clarendon, Oxford, 1995.

[70] J. H. Walter, The chracterization of finite groups with abelian Sylow 2-subgroups, Ann. of Math. 89 (1969), 405–514.

[71] H. N. Ward, On Ree’s series of simple groups, Trans. Amer. Math. Soc. 121 (1966), 62–89.

[72] A. Watanabe, Notes on p-blocks of characters of finite groups, J. Algebra 136 (1991), 109–116.

University of Aberdeen, Institute of Mathematics, Fraser Noble Building, Aberdeen, AB24 3UE, United Kingdom

Department of Mathematics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inageku, Chiba, 263-8522, Japan

University of Aberdeen, Institute of Mathematics, Fraser Noble Building, Aberdeen, AB24 3UE, United Kingdom