THE SOLOMON SYSTEM F_{SOL}(3) DOES NOT

OCCUR AS FUSION SYSTEM OF A 2-BLOCK

Radha Kessar

January 31, 2004

1. Introduction

Let p be a prime number. Fusion systems (referred to as full frobenius systems in [26], and as saturated fusion systems in [5]) on finite p-groups, were introduced by Puig in the context of local block theory. To each pair (G, P) where G is a finite group and P is a Sylow p-subgroup of G, is associated a fusion system F_{G}(P) on P called a p-fusion system of G, which reflects the p-fusion pattern of G. Similarly, to each quadruple (H, b, Q, e) where H is a finite group, b is a p-block of H, and (Q, e) is a maximal b-Brauer pair (that is Q is a defect group of G and e is a p-block of C_{G}(Q) in correspondence with b) is associated a fusion system F_{(H,b)}(Q, e) on P, called a fusion system of b.

An interesting fusion system is the Solomon system F_{Sol}(3) which is a system on a Sylow 2-subgroup of the classical group Spin_{7}(3). F_{Sol}(3) is defined and studied in [19], but was implicitly considered by Solomon in [28]. Solomon’s results imply that F_{Sol}(3) is “exotic”, that is there is no finite group having F_{Sol}(3) as a 2-fusion system. As a consequence, Solomon proved that that the only finite simple group having a Sylow 2-subgroup isomorphic to that of the Conway group .3 is .3, an important step in the classification of finite simple groups. In [3], Benson constructed a topological space which would have been the 2-completed classifying space of a finite group having F_{Sol}(3) as 2-fusion system (had such a group existed), and he predicted that this construction was an instance of a general theory, which insight was realized in the Broto-Levi-Oliver theory of p-local groups in [5].

In this paper we prove the following.

Theorem 1.1. The Solomon fusion system F_{Sol}(3) is not a fusion system of a 2-block of a finite group.

This theorem provides the first example of a fusion system which does not occur in a p-block. The proof relies on the classification of finite simple groups. Since by Brauer’s third main theorem a p-fusion system of a finite group G is a fusion system

Typeset by AMS-TEX

of the principal p-block of G, in the case of principal blocks Theorem 1.1. follows from [28]. Thus, we use Solomon’s result both directly and indirectly (via the classification).

If b is a non-principal block of G, then the defect groups of b are in general not Sylow p-subgroups of G and a fusion system of b is very far away from being a fusion system of G. On the other hand, if G is a finite group of Lie type or a finite Weyl group, then in several cases, there is a subgroup H of G (depending on b) such that a fusion system of b is a p-fusion system of H. It is tempting to speculate whether such a result holds always, that is if it is the case that fusion systems of blocks are all non-exotic. Part of the motivation for Theorem 1.1. came from this direction.

The proof of Theorem 1.1. proceeds in two steps. First, we prove a general theorem which for certain fusion systems F reduces the question of whether F occurs as a fusion system of a block of a finite group to the corresponding question for blocks of quasisimple groups. The theorem is trivial for the case of principal blocks -the general case requires the Dade Clifford theory of blocks. The second step of the proof consists of checking that F_{sol}(3) does not occur in blocks of quasi-simple groups. Here the complicated case is (surprisingly) that of groups of Lie type in odd characteristic. Given the phenomenon of Jordan decomposition of blocks ([13], [14] [7], [4]) and the many instances of control of fusion in unipotent blocks ([8]), philosophically, this case is easy. However, in order to cover every possible case, we needed to use ad hoc methods.

It would be interesting to prove Theorem 1.1. without invoking the classification in full but this appears to be very hard. Solomon’s proof of the exoticism of F_{Sol}(3) analyzed the mod 3-representation theory of a putative finite group with F_{Sol}(3) as 2-fusion system; there are no analogous methods in block theory.

The paper is divided into 5 sections. In section 2, we briefly recall relevant definitions. In section 3, we prove the reduction theorem. Section 4 contains a list of properties of F_{Sol}(3) that we will need for proving Theorem 1.1. In section 5, we run through the classification of finite simple groups to finish the proof.

2. Fusion systems and blocks

Let p be a prime number. If P, Q, R are subgroups of a finite group G, we denote by Hom_{P} (Q, R) the set of group homomorphisms φ : Q → R for which there is y ∈ P satisfying φ(u) = yuy^{−1} for all u ∈ Q; we write Aut_{P} (Q) = Hom_{P} (Q, Q). Thus Aut_{P} (Q) is canonically isomorphic to N_{P} (Q)/C_{P} (Q); in particular Aut_{Q}(Q) ≅ Q/Z(Q) is the group of inner automorphisms of Q.

Definition 2.1. A category on a finite p-group P is a category F whose objects are the subgroups of P and whose morphism sets Hom_{F}(Q, R) consist, for any two subgroups Q, R of P, of injective group homomorphisms with the following properties:

(i) if Q is contained in R then the inclusion Q ⊆ R is a morphism in F; (ii) for any φ ∈ Hom_{F}(Q, R), the induced isomorphism Q ≅ φ(Q) and its inverse are

morphisms in F;

(iii) composition of morphisms in F is the usual composition of group homomorphisms.

Definition 2.2. Let F be a category on a finite p-group P. A subgroup Q of P is called fully F-centralized if |C_{P} (R)| ≤ |C_{P} (Q)| for any subgroup R of P such that R ≅ Q in F, and Q is called fully F-normalized if |N_{P} (R)| ≤ |N_{P} (Q)| for any subgroup R of P such that R ≅ Q in F.

The following definition is due to Broto, Levi and Oliver [5].

Definition 2.3. Let F be a category on a finite p-group P, and let Q be a subgroup of P. For any morphism φ : Q → P in F, we set N_{φ} = {y ∈ N_{P} (Q)| there is z ∈ N_{P} (φ(Q)) such that φ(^{y}u) = ^{z}φ(u)for all u ∈ Q}.

In other words, N_{φ} is the inverse image in N_{P} (Q) of the group Aut_{P} (Q) ∩ (φ^{−1} ∘ Aut_{P} (φ(Q)) ∘ φ). Note that in particular QC_{P} (Q) ⊆ N_{φ} ⊆ N_{P} (Q). Broto, Levi and Oliver use the groups N_{φ} in [5] to give a definition of fusion systems (called saturated fusion systems in [5]) which is equivalent to Puig’s original definition (called full Frobenius systems there), which in turn has been simplified by Stancu [29]; we present here Stancu’s version:

Definition 2.4. A fusion system on a finite p-group P is a category F on P satisfying in addition the following properties.

(i) For any two subgroups Q, R of P we have Hom_{P} (Q, R) ⊆ Hom_{F}(Q, R).

(ii) Aut_{P} (P) is a Sylow-p-subgroup of Aut_{F}(P).

(iii) every morphism φ : Q → P in F such that φ(Q) is fully F-normalized extends to a morphism ψ : N_{φ} → P (that is, ψ|_{Q} = φ).

We will also need the following notion.

Definition 2.5. Let F be a fusion system on the p-group P and let Q be a subgroup of P.

(i) Q is F-centric if C_{P} (R) = Z(R) for any subgroup R of P such that R ≅ Q in F. (ii) Q is strongly F-closed, if for any subgroup R of P and any morphism φ : R → P

in F we have φ(R ∩ Q) ⊆ Q.

Definition 2.6. Let F and F′ be fusion systems on p-groups P and P′ respectively. F and F′ are equivalent if there is a group isomorphism f : P → P′ such that the induced functor F : F → F′ given by F (Q) = f(Q) and F (φ)(x) := f ∘ φ ∘ f^{−1}(x) for a subgroup Q of P and a morphism φ in F is an equivalence of categories.

We describe briefly fusion systems in the context of finite groups and blocks of finite groups.

Definition 2.7. Let G be a finite group, and P a Sylow p-subgroup of G. We denote by F_{P} (G) the category on P with morphisms Hom_{F_{P} (G)}(Q, R) := Hom_{G}(Q, R). It is easy to check that F_{P} (G) is a fusion system. Also, F_{P} (G) and F_{P′}(G) are equivalent for different Sylow p-subgroups P and P′ of G.

Let k be an algebraically closed field of characteristic p , let G be a finite group, let b be a p-block of G; that is, b is a primitive idempotent in Z(kG). A b-Brauer pair is a pair (Q, f) consisting of a p-subgroup Q of G and a block f of kC_{G}(Q) such that Br_{Q}(b)f = f. Here Br_{Q} : (kG)^{Q} → kC_{G}(Q) is the Brauer homomorphism mapping any element of C_{G}(Q) to itself and any non trivial Q-conjugacy class sum of elements in G to zero. By [1], the set of b-Brauer pairs admits a partial order “ ⊆ ” which is compatible with the action of G by conjugation on this set, such that the maximal b-Brauer pairs form a single G-conjugacy class. Given a maximal b-Brauer pair (P, e), for every subgroup Q of P there is a unique block e_{Q} of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e), and the group P is called a defect group of the block b. The choice of a maximal b-Brauer pair gives rise to a category on P.

Definition 2.8. Let G be a finite group, let b be a block of kG, and let (P, e) be a maximal b-Brauer pair. For any subgroup Q of P, denote by e_{Q} the unique block of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e). We denote by F_{(P,e)}(G, b) the category on P whose morphisms are the group homomorphisms φ : Q → R for which there is an element x ∈ G such that φ(u) = xux^{−1} for all u ∈ Q such that xe_{Q}x^{−1} = e_{xQx^{−1}}, or equivalently, such that ^{x}(Q, e_{Q}) ⊆ (R, e_{R}), where Q, R are subgroups of P.

If S is a Sylow-p-subgroup of G containing the defect group P of b, then clearly F_{(P,e)}(G, b) is a subcategory of F_{S}(G), but it is not in general a full subcategory, because the elements x in G used to define the morphisms in F_{(P,e)}(G, b) have to fulfill the additional compatibility property ^{x}(Q, e_{Q}) ⊆ (R, e_{R}). If b is the principal block of kG (that is, b is the unique block of kG not contained in the augmentation ideal of kG), then P is a Sylow-p-subgroup of G and e_{Q} is the principal block of kC_{G}(Q) for any subgroup Q of P, and hence F_{(P,e)}(G, b) = F_{P} (G) in this case. The following statement is essentially a reformulation of results in [1]; details of the proof may be found in [21].

Theorem 2.9. Let G be a finite group, let b be a block of kG, and let (P, e) be a maximal b-Brauer pair. For every subgroup Q of P, denote by e_{Q} the unique block of kC_{G}(Q) such that (Q, e_{Q}) ⊆ (P, e).

(i) The category F_{(P,e)}(G, b) is a fusion system on P.

(ii) If (P′, e′) is another a maximal b-Brauer pair, then F_{(P,e)}(G, b) is equivalent to F_{(P,e)}(G, b).

(iii) A subgroup Q of P is F-centric iff Z(Q) is a defect group of kC_{G}(Q)e_{Q}.

Definition 2.10. Let F be a fusion system over the p-group P, let G be a finite group and let b be a p-block of G. F is a p-fusion system of G if F is equivalent to the system F_{S}(G) for some (and hence any) Sylow p-subgroup ofG. Similarly, F is a fusion system of b if F is equivalent to the system F_{(D,e)}(G, b) for some maximal b-Brauer pair (D, e). In this situation, we will say that b is an F-block.

Let X be a finite group, and let D be a p-subgroup of X. We say D is of defect type in X if for any Sylow p-subgroup S of X containing D, there is x ∈ C_{X}(D) such that D = S ∩ ^{x}S. If D is a defect group of a p-block of G , then D is of defect type in X. We will need the following easy group theoretic fact.

Lemma 2.11. Let X be a finite group and Z a central subgroup of X. Set X̄ = X/Z. Denote by Ā the image of any subset A of X under the canonical mapping X → X̄. Let D be a p-subgroup of X containing the Sylow p-subgroup of Z.

If D is of defect type in X, then D̄ is of defect type in X̄. Conversely, if Z is a p′-group and if D̄ is of defect type in X̄, then D is of defect type in X.

Proof. Suppose that D is of defect type in X. Let T be a Sylow p-subgroup of X̄ containing D̄. There is a unique Sylow p-subgroup S of X such that S̄ = T. Also, S contains D since D contains the Sylow p-subgroup of Z. Thus there is x in C_{X}(D) such that D = S ∩ ^{x}S. Then x̄ ∈ CX̄(S̄) and D̄ = \overline{S ∩ ^{x}S} = S̄ ∩ x̄S̄. Hence D̄ is of defect type in X̄.

Now suppose that Z is a p′-group and that D̄ is of defect type in X̄. Let S be a Sylow p-subgroup of X containing D. There is an x ∈ X such that x̄ ∈ CX̄(D̄) and D̄ = S̄ ∩ x̄S̄ = \overline{S ∩ ^{x}S}. Since Z is a p′-group, a p-element of X̄ lifts uniquely to a p-element of X, hence D = S ∩ ^{x}S and x ∈ C_{X}(D).

3. A reduction Theorem

Theorem 3.1. Let F be a fusion system on a finite p-group P, G a finite group of having an F-block. If F has no strongly closed subgroups and if Aut(P) is a p-group, then there is a quasisimple group L such that Z(L) is a p′-group and such that L has an F-block.

Proof. We may assume that G is of minimal order with respect to having an F-block. Let b be an F-block of G. The minimality of |G| implies in particular that if H is a normal subgroup of G and c a block of H covered by b (that is with cb ≠ 0 ), then c is G-stable for otherwise we could replace G by the inertial group I := Stab_{G}(c) and b by the unique block d of I such that bd ≠ 0 and dc ≠ 0.

3.2. G is generated by the conjugates of P.

Proof. Let H be the subgroup of G generated by the conjugates of P, and let c be the unique block of H such that bc ≠ 0.

Conjugation by elements of G on kHc induces a group homomorphism from G to Aut(kHc) and hence from G to Out(kHc). Let K be the kernel of the latter homomorphism, that is K is the subgroup of G consisting of elements which act as inner automorphisms on kHc. By part (iii) of the Theorem in Section 5 of [17], G/K is a p′-group whose order divides |Out(P)|^{2}. Thus, G = K. Since P is a defect group of both b and c, it follows by Theorem 7 in [16], that OGb and OHc have isomorphic source algebras, and hence that c is a F block. By minimality of |G|, we have G = H.

3.3. Let N be a proper normal subgroup of G and d the block of N covered by b. Then d is of zero defect.

Proof. P ∩ N is a defect group of d. On the other hand, P ∩ N is strongly closed in F. If P ∩ N ≠ 1, then by assumption, P ∩ N = P. Hence P and all G-conjugates of P lie in N. By 3.2, N = G, a contradiction.

Let N and d be as in 3.3. A variation of Fong Reduction allows us to deduce that there is a p′-central extension G̃ of G/N and a block c of G̃ such that OGb is Morita equivalent to OG̃c. It is also well known that the block c is an F-block. We give the details for the convenience of the reader.

Denote the image of a subset A of G under the canonical map G → Ḡ by Ā. Let S be a p-subgroup of G containing P, maximal subject to Br_{S}(d) ≠ 0. Then S ∩ N = 1. G acts by conjugation on the algebra ONd. Since d is of defect 0, for each g ∈ G there is an element, i_{g} of (ONd)^{∗} such that ^{i_{g}}x = ^{g}x for all x in ONd. Further, the elements i_{g} may be chosen such that for g ∈ G, and n ∈ N, i_{gn} = i_{g}nd and such that the map s → i_{s} is a homomorphism of S into (ONd)^{∗}. Let α be the corresponding 2-cocycle on G, that is α(g, h) is defined via i_{g}i_{h} = α(g, h)i_{gh}. For any g, h ∈ G , and m, n ∈ N, we have α(gm, hn) = α(g, h), thus α factors to a 2-cocycle, ᾱ on Ḡ := G/N. We will denote also by α and ᾱ respectively the images of α and ᾱ in the cohomology groups H^{2}(G, O^{∗}) and H^{2}(Ḡ, O^{∗}). Let O_{}_{ᾱ^{−1}} ˆḠ be the twisted group algebra, corresponding to ᾱ^{−1}, that is O_{}_{ᾱ^{−1}} ˆḠ is the free O-module on {ˆḡ : ḡ ∈ G/N} with multiplication given by ˆḡˆh̄ = α^{−1}(g, h) ˆḡh.

Define a function

ϕ : ONd ⊗_{O} O_{}_{ᾱ^{−1}} ˆḠ → OGd

via the O-linear extension of the map x ⊗ ˆ ḡ → xi_{g}^{−1}g for x ∈ ONd and g ∈ G. This ϕ is an algebra isomorphism, with ϕ^{−1}(gd) = i_{g} ⊗ ˆ ḡ for g ∈ G.

There is a central extension 1 → Z → G̃ → Ḡ → 1, with Z a cyclic p′-group, an idempotent e of OZ and an algebra isomorphism τ : OG̃e → O_{}_{ᾱ^{−1}} ˆḠ with the following property: For g ∈ G, and g̃ any lift of ḡ in G̃, τ(g̃e) = αg̃ˆḡ for some element αg̃ of O^{∗} (see [30, Proposition 10.8 and its proof]).

For each s ∈ S, let s̃ denote the unique lift of s in G̃ which is also a p-element and for a subgroup Q of S, define Q̃ = {s̃: s ∈ Q}. Then s → s̃ is an isomorphism from S to S̃. We record the following fact.

3.4. Let g ∈ G be such that some (hence any) lift g̃ of ḡ to G̃ is in NG̃(Q̃). There exists n ∈ N such that h := ng ∈ N_{G}(Q). Further, ^{ h̃}x = g̃x̃ for all x ∈ Q.

Proof. Clearly, ḡ ∈ NḠ(Q̄), hence ^{g}Q ⊂ NQ. Since d is a G-stable block of N of defect 0, d is a block of NQ with defect group Q. Again, since d is G-stable, and since gQ ⊂ NQ, ^{g}Q is also a defect group of d as block of NQ. Hence there exists n ∈ N, q ∈ Q such that ^{qng}Q = Q, so qng and therefore ng is in N_{G}(Q). The remaining statement is immediate.

Henceforth, we identify the groups S, S̄ and S̃ via the maps s → s̄ → s̃.

Consider ONd ⊗_{O} OG̃e as interior S-algebra via the map s → i_{s} ⊗ se. Since for all s ∈ S, ϕ(i_{s} ⊗ ŝ) = sd and τ(ŝ) = se, the map

ψ : ONd ⊗_{O} OG̃e → OGd; (x ⊗ y → ϕ(x ⊗ τ(y)),

is an isomorphism of interior S-algebras.

Further, since ONd and OG̃e are permutation S-algebras, for each subgroup Q of S, ψ induces an algebra isomorphism

ψ_{Q} : ONd(Q) ⊗_{k} OG̃e(Q) ≅ OGd(Q)

satisfying

ψ_{Q}(Br_{Q}(x) ⊗ Br_{Q}(y)) = Br_{Q}(ψ(x ⊗ y))

for all x ∈ (ONd)^{Q} and all y ∈ (OG̃e)^{Q} (see [30, Proposition 28.3]).

Since ONd is simple, ψ defines a bijection between the blocks of OG̃e and the blocks of OGd given by u → ψ(1 ⊗ u). In other words, u → ψ(1 ⊗ u) is a bijection between the blocks of G̃ covering e and the blocks of G covering d. Further, for any subgroup Q of S, OGd(Q) is non-zero. Since ψ_{Q} is an isomorphism this means that ONd(Q) is non-zero, hence ONd(Q) is a matrix algebra over k (see [30, Proposition 28.2]). Thus ψ_{Q} induces a bijection between the blocks of kCG̃(Q)Br_{Q}(e) ≅ OGe(Q) and the blocks of kC_{G}(Q)Br_{Q}(d) ≅ OGd(Q) via f → ψ_{Q}(1 ⊗_{k} f). That is, the maps (Q, f) → (Q, ψ_{Q}(1 ⊗ f)) provide a bijection between the set of Brauer pairs associated to blocks of G̃ covering e and with first component contained in S and the set of Brauer pairs associated to blocks of G covering d and with first component contained in S and this bijection is inclusion preserving.

3.5. Let Q ≤ S and let (Q, v) be a Brauer pair for a block of G̃ covered by e. Set u := ψ_{Q}(1 ⊗ v). The groups NG̃(Q, v) and N_{G}(Q, u) have the same image in Aut(Q).

Proof. Let v_{0} ∈ (OG̃e)^{Q} with Br_{Q}(v_{0}) = v. Set u_{0} = ψ(1 ⊗ v_{0}). Then from the description of ψ_{Q}, we have Br_{Q}(u_{0}) = ψ_{Q}(Br_{Q}(1) ⊗ Br_{Q}(v_{0})) = ψ_{Q}(1 ⊗ v) = u. Suppose h ∈ N_{G}(Q), and let h̃ be a lift of h̄ in G̃. Clearly, h̃ ∈ NG̃(Q). Set t = ψ^{−1}(hd). Then t = i_{h} ⊗ αh̃e for some α ∈ O^{∗} and we have

hu = ^{h}Br_{Q}(u_{0})
= Br_{Q}( ^{h}u_{0})
= Br_{Q}(ψ( ^{t}(1 ⊗ v_{0}))
= Br_{Q}(ψ(1 ⊗ h̃v_{0})
= ψ_{Q}(1 ⊗ Br_{Q}( h̃v_{0})
= ψ_{Q}(1 ⊗ h̃Br_{Q}(v_{0}))
= ψ_{Q}(1 ⊗ h̃v).

Thus h stabilises u if and only if h̃ stabilizes u. The statement now follows from 3.4.

Let c be the block of G̃ corresponding to b under the algebra isomorphism ψ, that is such that b = ψ(1 ⊗ c).

3.6. The group P is a defect group of the block c of G̃.

Proof. Let Q be a subgroup of S. Then ONd(Q) is a non-zero matrix algebra over k. In particular, Br_{Q}(b) ≠ 0 if and only if Br_{Q}(c) ≠ 0. So, P is maximal among subgroups of S with Br_{P} (c) ≠ 0. On the other hand, since S is a Sylow p-subgroup of Ḡ [25, Chapter 6, Theorem 5.16] and therefore of G̃, S contains a defect group of c. Thus, P is a defect group of c.

3.7. The block c is an F-block.

Proof. This is immediate from 3.5, 3.6 and Alperin’s fusion theorem (see for instance [5, Theorem A.10]).

Now assume that N is a maximal proper normal subgroup of G, and let L = [G̃, G̃]. Then G̃ = LZ(G̃), L is quasisimple and Z(L) ⊂ Z(G̃) is a cyclic p′ group. Let c′ be the unique block of L covered by c. Then c′ is an F block. This completes the proof of Theorem 3.1.

4. The system F_{Sol}(3)

Proposition 4.1. Let P be a Sylow 2-subgroup of Spin_{7}(3).

(i) |P | = 2^{10}, |Z(P)| = 2, Aut(P) is a 2-group, P has rank 4 and P is generated by its involutions.

(ii) The dimension of a faithful representation of P over a field of characteristic different from 2 is at least 8.

Proof. (i) These can all be deduced easily from Lemma 2.1 of [27] and Lemma 2.1.of [28].

(ii) Let 𝔽 be a field of characteristic different from 2 and let n be the least positive integer such that GL_{n}(𝔽) contains a subgroup isomorphic to P. It suffices to show that n ≥ 8. We may assume that 𝔽 is algebraically closed. Let V be the natural module for GL_{n}(𝔽). Since the characteristic of 𝔽 is different from 2, P acts semi simply on V. In fact, since |Z(P)| = 2, the minimality of n allows us to assume that P acts irreducibly on V, and hence that n is a power of 2. Let Q be an elementary abelian subgroup of P of order 16. We may assume (after replacing P by a conjugate in GL_{n}(𝔽)) that Q is a subgroup of the diagonal matrices of GL_{n}(𝔽). The fact that Q is elementary abelian of order 16 now gives immediately that n ≥ 4. Hence, it suffices to show that n > 4. Suppose, if possible that n = 4. Then Q is the subgroup of GL_{n}(𝔽) consisting of all diagonal matrices with entries ±1. In particular, Q has an element x whose −1 eigen space (on V) has dimension 3 and whose 1-eigen space has dimension 1. Let Z(P) =< z >. Then since z ∈ Q, z is the element −I. Thus, the dimension of the −1-eigen space of zx is 1 and the dimension of the 1-eigen space of zx is 3. So, x and zx are not conjugate in GL_{n}(𝔽), and in particular x and zx are not conjugate in P. But every non-central involution σ of P is P -conjugate to σz (see [28], Lemma 2.1 (a)). This proves the result.

From now on till the end of the paper, P will be a Sylow 2-subgroup of Spin_{7}(q). We now state Solomon’s theorem ([28, Theorem 3.2]).

Theorem 4.2. There does not exist a finite group G satisfying (i) P is a Sylow 2-subgroup of G (ii) All involutions of G are G-conjugate (iii) For an involution z of G, a 2-fusion system of C_{G}(z) is a 2-fusion system of

Spin_{7}(3).

If G is a fusion system on a p-group D and if E is a subgroup of D, then C_{G}(E) denotes the category on C_{D}(E) whose morphism sets are defined by Hom_{C_{G}(E)}(R, R′) = {ϕ : R → R′|ϕ extends to a morphism ψ : ER → ER′ in G such that ψ_{E} = Id_{E}}. If E is a central subgroup of D, then C_{G}(E) is a fusion system.

The existence of a fusion system on P reflecting (ii) and (iii) is proved in [19]:

Proposition 4.3. There is a fusion system F_{Sol}(3) on P satisfying the following. (i) All involutions of P are F_{Sol}(3) conjugate. (ii) For the involution z of Z(P), C_{F_{Sol}(3)}(< z >) is a fusion system of Spin_{7}(3). (iii) There is an elementary abelian subgroup Q of P of order 16 which is F_{Sol}(3)-

centric and such that Aut_{F}(Q) is the full automorphism group of Q.

(iv) P has no proper non-trivial strongly closed F_{Sol}(3)- subgroups.

Proof. Statements (i) and (ii) are in Theorem 2.1 of [19]; Part (iii) follows from Proposition A.8. and Lemma 3.1 of [19]. Part (iv) is immediate from (i) and the fact that P is generated by its involutions.

If G is a fusion system on a p-group D, and E is a central subgroup of D then C_{G}(E) induces a unique fusion system, C_{G}(E)/E on D/E. Further, if G_{1} and G_{2} are fusion systems on p-groups D_{1} and D_{2} respectively, then the natural category G_{1} × G_{2} on D_{1} × D_{2} is a fusion system on D_{1} × D_{2} (see [5, Lemma 1.5]).

The proof of the next Proposition relies on a result of Linckelmann [22].

Proposition 4.4. Let G be a finite group, b an F_{Sol}(3)-block of G. Let

1 → Z → G̃ → G → 1

be a central extension of G and let S be the Sylow 2-subgroup of Z. Let b̃ be the block of G̃ whose image under the canonical surjection kG̃ → kG induced by the above central extension is b. Let P̃ be the Sylow p-subgroup of the inverse image of P in G̃, a defect group of the block b̃. Then b̃ is an F_{Sol}(3)×F_{S}(S)-block. In particular, P ×S is a defect group of b̃.

Proof. If Z is a p′- group, there is nothing to prove. Thus it suffices to prove the result when S = Z. Let P̃ be the inverse image of P in G̃. Then P̃ is a defect group of P. If b̃ is an F̃-block where F̃ is a fusion system on P̃, then clearly F̃ = CF̃(Z). Furthermore, F is equivalent to F̃/Z (see [15, Lemma 4.1]). Now F contains a 2-fusion system of Spin_{7}(3), Spin_{7}(3) is a perfect group and H^{2}(Spin_{7}(3), ℂ^{∗}) is a group of odd order. The proposition thus follows from Corollary 4.4 of [22].

5. Proof of Theorem 1.1.

Proof. Let G be a quasisimple finite group such that Z(G) is a group of odd order. Denote by Ḡ the simple group G̃/Z(G). As in the previous section P will be a Sylow 2 subgroup of Spin_{7}(3). Let F denote the fusion system F_{Sol}(3). By Proposition 4.1 (i) and Proposition 4.3 (iv), F satisfies the hypotheses of Theorem 3.1, hence in order to prove Theorem 1.1 it suffices to show that G has no F-block. So suppose that b is an F-block of G. By Theorem 4.2, we may assume that P is not a Sylow p-subgroup of G.

We consider various possibilities for Ḡ separately.

Alternating Groups. Suppose that Ḡ = A_{n} for some n ≥ 5. Then G = Ḡ = A_{n}. and it follows from the block theory of the symmetric and alternating groups, that P is isomorphic to a Sylow 2-subgroup of some alternating group A_{m} and this is impossible by Theorem 1.1 of [28].

Groups of Lie type in characteristic 2. Here we use the following fact, a proof of which may be found in [9]. Recall the definition of groups of defect type given at the end of section 2.

Lemma 5.1. Let p be a prime and let X be a finite group having a Tits system (B, N, S). Set T = B ∩ N and W = N/T. Suppose that the following conditions hold:

(i) The BN pair for X is split of characteristic p, that is B = U ⋊ T where U is a normal p-subgroup of B and T is a p′-group.

(ii) The BN pair for X is strongly split, that is for any J ⊂ S, and any n ∈ N such that the image of n in W is the longest element w_{J} of < J >, U ∩ U^{n} is a normal subgroup of U.

(iii) If Q is any p subgroup of X satisfying O_{p}(N_{X}(Q)) = Q, then there is an x ∈ X and an I ⊂ S such that ^{x}Q ≤ P_{I} := ∪_{w∈<I>}BwB,

(iv) W is of irreducible type, that is, if we denote by Δ = {α_{s} : s ∈ S} a basis of ℝ^{|S|} such that the element s of S acts as the reflection in the hyperplane perpendicular to α_{s} in the natural representation of W as a reflection group in ℝ^{|S|}, (see [11], section 2.2), then there is no partition of Δ into non-empty orthogonal subsets.

Then every p-subgroup of X which is of defect type in X is either the identity group or a Sylow p-subgroup of X.

Proof. This is contained in the proof of Theorem 3.38 of [9].

Suppose that Ḡ is a finite simple group of Lie type in characteristic 2. The simple group B_{2}(2)′ has Sylow 2-subgroups of order 8, hence Ḡ ≠ B_{2}(2)′. The 2-rank of ^{2}F_{4}(2) is 10 (see [24], Table 3.3.1), and the 2-part of the order of ^{2}F_{4}(2) is 2^{12} ([23], Page 8 ). Since the order of P is 2^{10} and the rank of P is 4, P is not a subgroup of ^{2}F_{4}(2) and Ḡ ≠ ^{2}F_{4}(2)′. The 2 part of the order of G_{2}(2) is 2^{6}, hence Ḡ ≠ G_{2}(2)′.

Since B_{2}(2)′ is the only group in characteristic 2 having a non-trivial 2′ part in the exceptional part of its Schur multiplier (see [24], Table 6.1.3), there is a central extension G̃ of G such that G̃ = G^{F} where G is a simply connected simple algebraic group and F is a Frobenius endomorphism of G (note that since Ḡ is not one of ^{2}F_{4}(2)′, B_{2}(2)′ or G_{2}(2)′, Ḡ is in Lie(2) in the sense of [24], definition 2.2.2). The group G̃ has a strongly split BN pair of characteristic 2 (see [11], Section 1.18), satisfying condition (ii) of Lemma 5.1 (see Proposition 2.6.4 of [11]) and conditions (iii) and (iv) of Lemma 5.1 (see Proposition 2.3.2 and Corollary 3.1.6 of [24]). Thus every 2-block of G̃ is either of full defect or of zero defect, but then the same is true for G. So, P is a Sylow 2-subgroup of G, which is impossible by Theorem 4.2.

Classical groups in odd characteristic. Let q be a power of an odd prime p and let Ḡ be one of the simple groups A_{n}(q), ^{2}A_{n}(q), B_{n}(q), (n ≥ 2), C_{n}(q), (n ≥ 2), D_{n}(q), (n ≥ 3), ^{2}D_{n}(q), (n ≥ 3). Note that if there is no simply connected algebraic group G with a Frobenius F such that G^{F} is a central extension of G, then Ḡ is one of A_{1}(9), 2A_{3}(3) or B_{3}(3). But none of these groups contains an elementary abelian subgroup of order 16. So, we may assume from now on that there is a simply connected algebraic group G with a Frobenius F such that G^{F} is a central extension of G. In what follows, we indicate a suitable choice of G and G^{F}, given Ḡ. We will use the notation G̃ for G^{F}. We will also define a classical group L containing G̃ as a normal subgroup.

• If Ḡ = A_{n}(q), let G̃ = SL(n, q) and L = GL_{n}(q).

• If Ḡ = ^{2}A_{n}(q), let G̃ = SU(n, q) and L = GU_{n}(q).

• If Ḡ = B_{n}(q), n ≥ 2, let G̃ = Ḡ and L = O_{2n+1}(q), the full orthogonal group.

• If Ḡ = C_{n}(q), n ≥ 2, let G̃ = L = Sp_{2n}(q).

• If Ḡ = D_{n}(q), n ≥ 3, let G̃ = Ω_{2n}^{+}(q) and L = O^{+}(2n, q).

• If Ḡ = ^{2}D_{n}(q), n ≥ 3, let G̃ = Ω_{2n}^{−}(q) and L = O^{−}(2n, q).

In all cases, G̃ is a central extension of G and G̃ is a normal subgroup of L and the kernel of the extension G̃ → G is cyclic. Thus Proposition 4.4 applies. Let S and b̃ be as in Proposition 4.4 and let P̃ = S × P. Let f be a block of L covering b̃. Then there is a defect group R of f such that P̃ = R ∩ G̃. On the other hand, it is well known that there is a semisimple element s of odd order in L such that R is a Sylow 2-subgroup of C_{L}(s) (see [6, Proposition 4.18] and [2, Section 5A]).

Note that if Ḡ is A_{n}(q) or ^{2}A_{n}(q), then L/G̃ is cyclic, and since S is cyclic, this means that the rank of P is at most 2 less than the rank of R. In the other four cases |R|/|P | ≤ 8, hence the rank of P differs from the rank of R by at most 3. Since the rank of P is 4, the rank of R is at most 7.

The structures of centralizers of semi-simple elements of classical groups are well known. In particular,

C_{L}(s) ≃ ∏_{i} L_{i}

such that each L_{i} is either a cyclic group or is isomorphic to one of the groups GL_{m}(q′), GU_{m}(q′), O_{m}(q′) ( m odd ), Sp_{m}(q′), (m even), or O_{m}^{±}(q′) (m even) for some m and some power q′ of q (see for instance [13], [14]). Now since P is a subgroup of C_{L}(s) and since Z(P) is cyclic, it follows that for some i, L_{i} contains a subgroup isomorphic to P. Let R_{i} be a Sylow 2-subgroup of L_{i}. By lemma 1.3, m ≥ 8. Thus, if L_{i} is one of the groups GL_{m}(q′), GU_{m}(q′), O_{m}(q′) ( m odd ), or O_{m}^{±}(q′) (m even), by [12], the rank of R_{i} is at least 8 which is impossible since the rank of R is at most 7. Thus L_{i} is a symplectic group Sp_{m}(q′), (m even, m ≥ 8). In particular, Ḡ is neither A_{n}(q) nor ^{2}A_{n}(q), so |R|/|P | ≤ 8. On the other hand, since m ≥ 8, we see easily that |R| ≥ |R_{i}| ≥ 2^{15}, a contradiction since |P | = 2^{10}.

Exceptional groups in odd characteristic. We will need the following lemma, an application of the results of [20].

Lemma 5.2. Let p be an odd prime, 𝔽̄ the algebraic closure of the field of p elements and q a power of p. Let X be a simply connected exceptional algebraic group over 𝔽̄ having an 𝔽_{q}-rational structure and let F : X → X be the corresponding Frobenius endomorphism. Suppose that Q is an elementary abelian subgroup of order 16 of X^{F} such that N_{X^{F}} (Q)/C_{X^{F}} (Q) ≃ Aut(Q). Then

(i) X is of type E_{7} or E_{8}.

(ii) There is an F -stable maximal torus T of X with Q ≤ T and such that T is normal in N_{X}(Q). In particular, N_{X^{F}} (Q)/C_{X^{F}} (Q) is isomorphic to a section of N_{X^{F}} (T)/T^{F}.

Proof. If X is of type G_{2}, then X^{F} has 2-rank 3 (see [24], Theorem 4.10.5 (e)), hence X^{F} does not have an elementary abelian subgroup of order 16. If X is of type F_{4}, then by Proposition 1.4 of [20], X has no elementary abelian elementary abelian subgroup of order 16 all of whose non-identity elements are X conjugate. Hence, X is of type E_{6}, E_{7} or E_{8}. Consequently, there is a simply connected group Y of type E_{8} over 𝔽̄ such that X may be identified with a closed subgroup of Y. All non-identity elements of Q are conjugate in X, hence in Y. By Proposition 1.3 (iv) and (vi) of [20], Q lies in a maximal torus U of Y. Furthermore, U is the connected centralizer C_{Y}(Q)^{∘} of Q in Y. In particular, U is a normal subgroup of N_{Y}(Q). Then T := (U ∩ X)^{∘} is a torus of X and is normal in N_{X}(Q). Also, T is a maximal torus of X, since if T′ were a torus of X containing T, then T′ ≤ C_{Y}(Q)^{∘} = U, hence T′ ≤ (U ∩ X)^{∘} = T. Since T is normal in N_{X}(Q) and T ≤ C_{X}(Q), N_{X}(Q)/C_{X}(Q) and hence N_{X^{F}} (Q)/C_{X^{F}} (Q) is isomorphic to a section of N_{X}(T)/T , the Weyl group of X. Since Aut(Q) is isomorphic to the group GL(4, 2), it follows that |N_{X}(T)/T| is divisible by 7, which is not the case if X is of type E_{6}. This proves part (i) of the lemma. We assume from now on that X is of type E_{7} or E_{8}. In particular, we may assume that Y has a 𝔽_{q}-rational structure such that the corresponding Frobenius endomorphism is an extension of F; we will denote this extension to Y also by F. Since Q lies in X^{F} ⊂ Y^{F}, C_{Y}(Q) is F -stable. Hence U and T are also F -stable. Since T^{F} ≤ C_{X^{F}} (Q), it follows that N_{X^{F}} (Q)/C_{X^{F}} (Q) is isomorphic to a section of N_{X^{F}} (T)/T^{F}. This completes the proof of the lemma.

As before, q will denote a power of an odd prime p.

• Suppose that Ḡ = E_{8}(q). Then G = Ḡ. Let G̃ = G = Ḡ = E_{8}(q) and let G be a simple algebraic group of type E_{8} with a Frobenius endomorphism F such that G^{F} = G̃. Let Q be as in Proposition 4.3 (iii) and let T be as in Lemma 5.2. (ii).

Since T^{F} is normal in C_{G}(Q), the Sylow 2 subgroup of T^{F} is contained in all psubgroups of C_{G}(Q) which are defect groups of blocks of C_{G}(Q). Since Q is F-centric, C_{G}(Q) has blocks with Q as defect group. Hence, the 2-part of the order of T^{F} is 2^{4}. The G-stable conjugacy classes of F -stable maximal tori of G are in 1−1 correspondence with the conjugacy classes of the Weyl group W of type E_{8} (see [11], Proposition 3.3.3 noting that F acts trivially on W); let T correspond to the class of the element w. We have N_{G}(T)/T^{F} ≃ C_{W} (w) ([11], Proposition 3.3.6). Thus it follows by Lemma 5.2 (ii) that Aut_{F}(Q) ≃ GL(4, 2) is involved in C_{W} (w). In particular, since |W | = 2^{14}.3^{5}.5^{2}.7 and |GL(4, 2)| = 2^{6}.3^{2}.5.7, the number of elements in the conjugacy class of w in W is at most 2^{8}.3^{3}.5. The conjugacy classes of W have been described in [10] in terms of admissible diagrams. Inspection of the last column on pages 54-58 of this article yields only four possibilities for w, namely w must correspond to one of the admissible diagrams ϕ, A_{1}, A_{1}^{7} or A_{1}^{8}. On the other hand, we have |T^{F} | = χ(q) where χ(x) is the characteristic polynomial of the action of w on Y_{0} ⊗ ℝ and Y_{0} is the cocharacter group corresponding to a maximally split torus T_{0} of G ([11], Proposition 3.3.3). Furthermore, the characteristic polynomials of elements of W on Y_{0} ⊗ ℝ are described in terms of the corresponding admissible diagrams in [10, Theorem B, Proposition 21 and Table 3] ; it is easy to see that if W corresponds to one of the diagrams ϕ, A_{1}, A_{1}^{7} or A_{1}^{8}, then T^{F} is divisible by 2^{7}, a contradiction.

• Suppose that Ḡ = E_{7}(q). Then Ḡ = G. Let G be the simply connected group of type E_{7} and F a Frobenius endomorphism of G such that G̃ := G^{F} is a central extension of G by a group of order 2. We apply Proposition 4.4 (i) and Lemma 5.2 with X = G, and let T be as in the conclusion of (ii) of Lemma 2.1. Arguing as in the previous case, we conclude that |T^{F} | is 2^{5}. Again, let W be the Weyl group of type E_{7} and let T correspond to the element w of W. Since the Weyl group of type E_{7} has order 2^{10}.3^{4}.5.7, and since by Lemma 2.1 (ii), NG̃((T ∩ G))/T^{F} ∩ G̃ involves GL(4, 2), by inspection of the tables on pages 51-53 of [10], we have that w = 1. In other words, T is a maximally split F -stable torus of G. Thus, |T^{F} | is divisible by 2^{7} a contradiction.

• Suppose Ḡ is one of ^{±}E_{6}(q), F_{4}(q), G_{2}(q), ^{2}G_{2}(q) (with q > 3 ) or ^{2}G_{2}(3)′. It follows by Lemma 5.2 that Ḡ, has no elementary abelian subgroups Q of order 16 such that NḠ(Q) ≅ Aut(Q), hence neither does G.

Sporadic groups. The non-principal 2-blocks of sporadic simple groups have been described in [18]; in particular, the order of the defect groups of a non-principal 2-block of a sporadic simple group is at most 2^{7}. Thus, we may assume that G ≠ Ḡ. Hence Ḡ is one of the groups M_{22}, J_{3}, Suz, Mc, O′Nan, F i_{22}, F i′_{24} (see [24], Page 313). But the 2 parts of the orders of M_{22}, J_{3}, Mc and O′Nan being respectively 2^{7}, 2^{7}, 2^{7} and 2^{9} (see [23], Page 8), Ḡ is one of Suz, F i_{22} and F i′_{24}, and Z(G) is cyclic of order 3. Let z be the unique involution in Z(P) and let e be the Brauer correspondent in C_{G}(z) of the block b of G. Then P is a defect group of e and e is an C := C_{F}(z) block. Let Q be as in Proposition 4.3 (iv). Clearly, Aut_{C}(Q) ≅ GL(3, 2). Thus, (N_{G}(Q) ∩ C_{G}(z))/C_{G}(Q) and hence (NḠ(Q̄) ∩ CḠ(z̄))/CḠ(Q̄) has a subgroup isomorphic to GL(3, 2). In particular, 7 divides the order of CḠ(z̄).

Suppose first that Ḡ = Suz. By Table 5.3o of [24], CḠ(z̄) is either the group Q_{8}^{3}#Ω_{6}^{−}(2) or (E_{2^{2}} × P SL_{3}(4)).2. Since |Q_{8}^{3}#Ω_{6}^{−}(2)| is not divisible by 7, CḠ(z̄) ≠ Q_{8}^{3}#Ω_{6}^{−}(2). On the other hand, the 2-part of the order of (E_{2^{2}} × P SL_{3}(4)).2 is 2^{9}, hence CḠ(z̄) ≠ (E_{2^{2}} × P SL_{3}(4)).2.

Now suppose that Ḡ = F i_{22}. Then CḠ(z̄) is either 2U_{6}(2) , (2 × D_{8}^{∗4})O_{6}^{−}(2) or a group of order 2^{13}.3^{3}.8 (see [24], Table 5.3t). In the latter two cases 7 does not divide the order of CḠ(z̄), hence CḠ(z̄) is 2U_{6}(2). By Lemma 2.11, P̄ ≃ P is a group of defect type in 2U_{6}(2) and P/ < z > is of defect type in U_{6}(2). Since SU_{6}(2) is an extension of U_{6}(2) by a group of order 3, by Lemma 2.11 again, there is a defect type subgroup of SU_{6}(2) isomorphic to P/ < z >. Thus, by Lemma 5.1 P/ < z > is isomorphic to the Sylow 2-subgroups of SU_{6}(2). This is a contradiction, as the 2-part of the order of SU_{6}(2) is 2^{15}.

Finally, consider the case Ḡ = F i′_{24}. Then CḠ(z̄) is either 2F i_{22}2 or D_{8}^{∗6}(3U_{4}(3).2). (see [24], Table 5.3v). Any 2-subgroup of defect type in D_{8}^{∗6}(3U_{4}(3).2) must contain the group D_{8}^{∗6}, hence is of order at least 2^{13}. In particular, P is not a subgroup of defect type in D_{8}^{∗6}(3U_{4}(3).2). Thus CḠ(z̄) = 2F i_{22}2. In this case C_{G}(z) is isomorphic to a direct product of a group of order 3 with 2F i_{22}2 (see [24], Table 5.3v). This structure of C_{G}(z) implies that 2F i_{22}2 has a block with defect group isomorphic to P, and hence that F i_{22}2 has a block with defect group isomorphic to P/ < z >. From the theory of covering blocks, it follows that F i_{22} has a block with defect groups isomorphic to P or to a subgroup of P of index 2. But by [18] this is impossible.

This concludes the proof of Theorem 1.1.

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Radha Kessar Department of Mathematics Ohio State University 231, W. 18th Avenue Columbus, OH 43210 USA