On the Sources of Simple Modules in Nilpotent Blocks

Adam Salminen Department of Mathematics University of Evansville 1800 Lincoln Avenue Evansville, IN 47722

as341@evansville.edu

June 9, 2008

Abstract

Let G be a finite group and let k be an algebraically closed field of characteristic p. If b is a nilpotent block of kG with defect group P, then there is a unique isomorphism class of simple kGb-modules and Puig proved that the source of this module is an endo-permutation kP -module. It is conjectured that the image of this source is always torsion in the Dade group.

Let H be a finite group and let P be a p-subgroup of Aut(H). Also let c be a defect zero block of kH. If c is P -stable and Br_{P} (c) ≠ 0, then c is a nilpotent block of k(H ⋊P) and k(H ⋊P)c has P as a defect group. In this paper, we will investigate the sources of the simple k(H ⋊ P)c-modules when P ≅ C_{p} × C_{p}. Suppose that we can find an H and c as above such that a source of a simple k(H ⋊ P)c-module is not torsion in the Dade group. Then we can find H and c as above with H a central p′-extension of a simple group. When p ≥ 3 we show that H can be found in a quite restrictive subset of simple groups.

1 Introduction

Puig showed that the source algebra of a nilpotent block is isomorphic to End_{k}(L) ⊗ kP where P is a defect group of b and L is an indecomposable endo-permutation module with vertex P. We are interested in investigating which endo-permutation modules L can show up in this way.

Before we state the main result, we fix some notation. For this entire paper, p will be a fixed prime and k will be an algebraically closed field of characteristic p. Let G be a finite group. A block of kG is a primitive central idempotent of kG. If b is a block of kG, the algebra kGb is a block algebra of kG.

Let V be an indecomposable kG-module. A vertex of V is a minimal subgroup Q such that V is a direct summand of Ind_{Q}^{G}(W) for some kQ-module W. Since V is indecomposable, W can be taken to be indecomposable and is then called a source of V.

Associated with any block b of kG is a conjugacy class of p-groups called the defect groups of b. If 1 is a defect group of b we say that b is a defect zero block. Let b be a defect zero block of kG and let P be a p-subgroup of Aut(G). If b is stable under the action of P, then b remains a central idempotent of k(G ⋊ P). It is easy to check that b is a block of k(G ⋊ P). If we also assume that Br_{P} (b) ≠ 0, then P will be a defect group of b as a block of k(G ⋊ P). The main result of this paper is the following.

Theorem 1.1. Suppose that G is a finite group such that P ≅ C_{p} × C_{p} ≤ Aut(G), and that b is a P -stable defect 0 block of kG such that Br_{P} (b) ≠ 0. Suppose also that the source V of a simple k(G ⋊ P)b-module M is a finitely generated non-torsion endo-permutation kP -module. Then we can find G, b, V with the above properties where G is a central p′-extension of a simple group.

If we restrict to the case of odd p we have the following corollary.

Corollary 1.2. Assume the setup of Theorem 1.1. If p is odd, then we can find G, b, V where G is one of the following:

(i) (a) G is a central p′-extension of A_{n}(q) with p|(n + 1, q − 1, f) where q = r^{f}; or

(b) G is a central p′-extension of ^{2}A_{n}(q) with p|(n + 1, q + 1, f) where q = r^{f}; or

(ii) p = 3, q = r^{f} and G is a central extension of D_{4}(q) with 3|f.

This result is a reduction, for the above blocks, of the following conjectures of Puig and Feit to a central p′-extension of a simple group.

Conjecture 1.3 (Puig). Let P be a finite p-group. There are only finitely many isomorphism classes of interior P -algebras which are the source algebra of a block with P as a defect group.

Conjecture 1.4 (Feit). Let P be a finite p-group. There are only finitely many isomorphism classes of kP -modules which are sources of a simple module.

Puig has proved both of these conjectures for p-solvable groups [15]. The dimensions of sources of simple modules of p-solvable groups with C_{p} × C_{p} as a p-Sylow subgroup are calculated in Muncke [11].

This paper is organized as follows. Section 2 recalls some basic definitions and theorems. Sections 3 and 4 contain some general reduction theorems. Section 5 contains the main reduction to central p′-extensions of simple groups establishing Theorem 1.1. In section 6 we eliminate most of the simple groups. Finally, in section 7, we eliminate E_{6}(q) and ^{2}E_{6}(q) and obtain Corollary 1.2.

2 Preliminaries

Let G be a finite group. An interior G-algebra is a pair (A, φ) where A is a kalgebra and φ : G ⟶ A^{×} is a group homomorphism. φ is called the structural homomorphism of A and when φ is understood we will simply write A for the interior G-algebra (A, φ). We also use ^{g}a to denote φ(g)a(φ(g))^{−1}. If H is a subgroup of G and A is an interior G-algebra, the set of H fixed points of A is {a ∈ A |^{h} a = a for all h ∈ H} and is denoted A^{H}. The main examples of interior G-algebras are the group algebra kG and the block algebras kGb. These are interior G-algebras with structural homomorphisms sending g to its image, still denoted g, in kG and g to gb respectively.

Suppose that H is a subgroup of a finite group G and A is an interior Galgebra. The relative trace Tr_{H}^{G} is a map from A^{H} to A^{G} defined by a ↦ ∑_{g∈[G/H]}^{g}a. Also note that Im(Tr_{H}^{G}) is an ideal of A^{G} which we denote as A_{H}^{G}. If

P is a p-subgroup of G, the Brauer quotient A(P) is defined to be A^{P} / ∑_{Q⫋P} A_{Q}^{P}

and the Brauer homomorphism Br_{P}^{A} : A^{P} ⟶ A(P) is defined to be the obvious map. If A is understood we will simply write Br_{P} for the Brauer homomorphism.

Let G be a finite group and let b be a block of kG. A defect group of b is a minimal subgroup P such that b ∈ Im(Tr_{P}^{G}). This is equivalent to a maximal p-subgroup P such that Br_{P} (b) ≠ 0.

Now let b be a block of kG for a finite group G with defect group P. A primitive idempotent i of (kGb)^{P} such that Br_{P} (i) ≠ 0 is a source idempotent of b. The algebra ikGi is an interior P -algebra with structural homomorphism u ↦ iui and is called a source algebra of b. Blocks are Morita equivalent to their source algebras and most of the local structure of the block can be recovered from a source algebra.

We now quickly recall some results on endo-permutation modules. Let P be a finite p-group. A kP -module M is an endo-permutation kP -module if End_{k}(M) is a permutation kP -module. Dade showed that for a given endo-permutation module M there is at most one isomorphism class of summands with vertex P [5]. If such a summand exists we say M is capped and we call that summand a cap of M. Since the cap is unique up to isomorphism, we can define a relation on the set of capped endo-permutation kP -modules by M ∼ N if and only if M and N have isomorphic caps. It is easy to see that this is an equivalence relation and we denote the equivalence class of a capped endo-permutation kP -module M by [M].

Dade also showed that if M and N are capped endo-permutation kP -modules then so is M ⊗ N. So we can place an abelian group structure on the set of equivalence classes by defining [M]+[N] to be [M ⊗N]. This group is the Dade group of P and is denoted D(P). Puig showed that D(P) is finitely generated [13]. This implies that the torsion part of D(P) is finite.

Next we recall some definitions and results on Brauer pairs. Let G be a finite group. A Brauer pair is a pair (P, e) where P is a p-subgroup of G and e is a block of kC_{G}(P). If (P, e) and (Q, f) are Brauer pairs, then we say (Q, f) is contained in (P, e) and write (Q, f) ⊆ (P, e) if Q ⊆ P and for any primitive idempotent i ∈ (kG)^{P}, if Br_{P} (i)e ≠ 0, then Br_{Q}(i)f ≠ 0. This provides a partial ordering on the set of Brauer pairs. Let b be a block of kG. A Brauer pair (P, e) is called a b-Brauer pair if and only if Br_{P} (b)e ≠ 0.

For a Brauer pair (P, e), the normalizer of (P, e) in G is defined to be N_{G}(P, e) = {g ∈ N_{G}(P) | ^{g}e = e}. A block b is said to be nilpotent if N_{G}(Q, e)/C_{G}(Q) is a p-group for every b-Brauer pair (Q, e). The main theorem on nilpotent blocks that we make use of is the following.

Theorem 2.1 (Puig, [14]). Let G be finite group, let P be a finite p-group and let b be a nilpotent block of kG with P a defect group. If i ∈ (kGb)^{P} is a source idempotent, then the source algebra ikGi is isomorphic to S ⊗_{k} kP as interior P algebras where S is a k-simple algebra isomorphic to End_{k}(L) and L is an indecomposable endo-permutation kP -module with vertex P.

Note that this implies there is a unique isomorphism class of simple kGbmodules and the sources of these modules will be isomorphic to L.

3 Reduction Theorems

In this section, we will state some of the main reduction theorems we will need. The following is well known and will allow us to make a significant reduction. A more general theorem that contains the following is 16.6 of [16].

Proposition 3.1. Let G be a finite group, let N be a normal subgroup of G, let c be a block of kN, let b be a block of kG such that bc ≠ 0 and let H be the stabilizer of c in G. Then

(i) The map Tr_{H}^{G} : Z(kHc) ⟶ Z(kG Tr_{H}^{G}(c)) is an algebra isomorphism whose inverse is given by multiplication by c. In particular, we have a unique block d of kH such that Tr_{H}^{G}(d) = b, dc = d and cbc = d;

(ii) kHc is Morita equivalent to kG(Tr_{H}^{G}(c));

(iii) Every defect group of d is a defect group of b;

(iv) (Puig) Every source algebra of kHd is a source algebra of kGb;

(v) If b is nilpotent then d is nilpotent.

Assume the setup of the previous proposition with b, and therefore d, nilpotent. If V is the unique (up to isomorphism) simple kHd-module, then kGc⊗_{kHc} V ≅ kG ⊗_{kH} V = Ind_{H}^{G}(V) is the unique (up to isomorphism) simple kGbmodule. So if W is a source of V, then W will be a source of Ind_{H}^{G}(V).

Now let S be a finite dimensional matrix algebra over k on which a finite group G acts as algebra automorphisms. Details of the following can be found in section 3 of [8]. The Skolem-Noether theorem tells us that for each x ∈ G we can choose an s_{x} ∈ S^{×} such that ^{x}t = s_{x}ts_{x}^{−1} for all t ∈ S. Now we fix a choice of s_{x} for each x ∈ G. If x, y ∈ G, then s_{x}s_{y} and s_{xy} will have the same action on S. So there must be a λ(x, y) ∈ k^{×} such that s_{x}s_{y} = λ(x, y)s_{xy}. Defining λ(x, y) as above for each x, y ∈ G, the map λ : G × G ⟶ k^{×} is a 2-cocycle of G with values in k^{×}. If λ is any 2-cocycle of G with values in k^{×}, the twisted group algebra k_{λ}G is defined to be the free k-module with basis {x̂}_{x∈G} and multiplication being the bilinear extension of x̂ŷ = λ(x, y)x̂y for x, y ∈ G.

Let N be a normal subgroup of a finite group G and let c be a G-stable defect zero block of kN. Since kNc is simple we can define a 2-cocycle λ of G as above. We will choose the s_{x} so that λ induces a 2-cocycle of G/N. First we choose {1_{G} = x_{1}, ..., x_{t}} = [G/N] to be a set of coset representatives of G/N and for each of the x_{i} we choose an s_{x_{i}} as above. For i = 1, ..., t and n ∈ N, define s_{x_{i}n} to be s_{x_{i}}nc. From this one can easily check that λ(g, n) = 1 and λ(n, g) = 1 for all g ∈ G and n ∈ N. This implies that λ(gn, hm) = λ(g, h) for all g, h ∈ G and n, m ∈ N. Therefore we can define a 2-cocycle λ̃(gN, hN) = λ(g, h). We will use λ to denote λ̃. The next result is essentially Puig’s algebra theoretic version of Fong reduction of characters.

Theorem 3.2. Let G be a finite group, let N be a normal subgroup of G and let c be a defect zero block of kN which is G-stable. Let S = kNc and define the 2-cocycle λ of G and G/N as above. Then there is an algebra isomorphism

φ : kGc ⟶ S ⊗_{k} k_{λ^{−1}}(G/N)

mapping xc to s_{x} ⊗ x̂N with inverse map ψ mapping s ⊗ x̂N to ss_{x}^{−1}x. Where x ∈ G, s ∈ S and s_{x} are defined as above for each x ∈ G.

Proof. This is a routine verification that the above maps are homomorphisms and inverses of each other.

Since we are interested in source algebras, we want the isomorphisms in Theorem 3.2 to be an isomorphism of interior P -algebras. With the setup above, suppose that the p-group P is a subgroup of G such that P ∩ N = {1}. Since P is a p-group, we know its action on S = kNc is induced from a group homomorphism φ : P ⟶ S^{×} by 21.4 of [16]. We can then define ψ : P N ⟶ S^{×} by ψ(rn) = φ(r)nc for r ∈ P and n ∈ N. Since φ(r)n(φ(r))^{−1} = rnr^{−1} for all r ∈ P and n ∈ N, one can easily check that ψ is a group homomorphism.

Now pick {1 = x_{1}, ..., x_{t}} = [G/P N] to be a set of coset representatives. Choose an s_{x_{i}} for each x_{i} as we did above and define s_{x_{i}rn} to be s_{x_{i}}ψ(rn) = s_{x_{i}}φ(r)nc for all r ∈ P and n ∈ N. We will again get a 2-cocycle of G with values in k^{×} which induces a 2-cocycle of G/N with values in k^{×}. With this new choice of s_{g} the map r ↦ s_{r} is a homomorphism from P to S^{×}. Also the map r ↦ r̂N from P to (k_{λ^{−1}}(G/N))^{×} is a group homomorphism. By defining the structural homomorphism r ↦ s_{r} ⊗_{k} r̂N for r ∈ P, we obtain an interior P -algebra structure on S ⊗_{k} k_{λ^{−1}}(G/N). The following proposition is then immediate.

Proposition 3.3. Let G be a finite group, let N be a normal subgroup of G, let c be a defect zero block of N which is G-stable and let S = kNc. Suppose that the p-group P is a subgroup of G such that P ∩ N = {1} and λ is a 2-cocycle defined as above. Then the k-algebra isomorphism φ : kGc ⟶ S ⊗_{k} k_{λ^{−1}}(G/N) defined in Theorem 3.2 is an isomorphism of interior P -algebras.

To eliminate the twisted algebra from the above theorems we will use the following theorem which can be found in 10.5 of [16]. We give a slightly more explicit proof.

Theorem 3.4. If λ is a 2-cocycle of a finite group G with values in k^{×}, then there exists a central extension of finite groups 1 → Z → H → G → 1 where Z is cyclic of order prime to p such that k_{λ}G ≅ kHe for some idempotent e of kZ. If G is perfect, then H can be chosen to be perfect.

The proof uses the following well known lemma whose proof can be found in [9].

Lemma 3.5. With the setup of the theorem, there is a 2-cocycle α with values in the |G|_{p′}-th roots of unity of k^{×} and a map β : G ⟶ k^{×} such that λ(x, y) = α(x, y)β(x)^{−1}β(y)^{−1}β(xy).

Proof of Theorem 3.4. Let Z be the group of |G|_{p′}-th roots of unity of k^{×}. The lemma tells us there exists a 2-cocycle α and a map β : G → k^{×} such that λ(x, y) = α(x, y)β(x)β(y)β(xy)^{−1} with the values of α in Z. We define a central extension of G by Z by letting H = G × Z as a set and defining (x, η)(y, ζ) = (xy, ηζα(x, y)). This is a group since α is a 2-cocycle. Let e = \frac{1}{|Z|} ∑_{ζ∈Z} ζ^{−1}(1, ζ). Then

e^{2} = \frac{1}{|Z|^{2}} ∑_{ζ,ζ′∈Z}(ζζ′)^{−1}(1, ζζ′) = e.

So e is an idempotent and it is clearly central. We now show that k_{λ}G ≅ kHe. Define φ : kH → k_{λ}G to be the linear extension of φ((x, η)) = β(x)ηx̂. Then

φ((x, η))φ((y, ξ)) = β(x)β(y)ηξx̂ŷ
= β(x)β(y)ηξλ(x, y)x̂y

and φ((xy, ηξα(x, y)) = β(xy)ηξα(x, y)x̂y so φ is an algebra homomorphism and it is clearly onto. It is also easy to see φ(e) = 1 and so restricting φ to kHe is still onto. Now define ψ : k_{λ}G → kHe by ψ(ĝ) = β(g)^{−1}(g, 1)e. Note that if

ζ′ ∈ Z and g ∈ G, then ζ′(g, 1)e = \frac{1}{|Z|} ∑_{ζ∈Z} ζ′ζ^{−1}(g, ζ) = \frac{1}{|Z|} ∑_{ζ∈Z} ζ^{−1}(g, ζ′ζ) =

\frac{1}{|Z|} ∑_{ζ∈Z} ζ^{−1}(g, ζ′)(1, ζ) = (g, ζ′)e. From this we can verify that φ and ψ are

inverses.

Then ψ(x̂ŷ) = ψ(λ(x, y)x̂y) = λ(x, y)β(xy)^{−1}(xy, 1)e and

ψ(x̂)ψ(ŷ) = β(x)^{−1}(x, 1)β(y)^{−1}(y, 1)e
= β(x)^{−1}β(y)^{−1}(xy, α(x, y))e
= β(x)^{−1}β(y)^{−1}α(x, y)(xy, 1)e.

So ψ is a homomorphism. Also notice that

φ(ψ(x̂)) = φ(β(x)^{−1}(x, 1)e)
= β(x)^{−1}β(x)x̂
= x̂

and

ψ(φ((x, η)e)) = ψ(β(x)ηx̂)
= β(x)η(x, 1)β(x)^{−1}e
= (x, η)e

so φ and ψ are inverse homomorphisms. Therefore ψ and φ are isomorphisms.

Now suppose that G is perfect. We show that H can be chosen to be perfect. So we need H = [H, H] where [H, H] is the commutator subgroup of H. To do this, we let L = [H, H] and note that H = LZ(H), Z(L) ⊆ Z(H) = Z and [H, H] = [L, L]. This gives us L/Z(L) ≅ H/Z(H) so that L is still a central extension of G. Now for each g ∈ G we can choose a η_{g} ∈ Z such that (g, η_{g}) ∈ L. Let f = ∑_{ζ∈Z(L)} ζ^{−1}(1, ζ). Then f is a central idempotent of kL.

Using φ : kL → k_{λ}G defined by φ((g, η)) = β(g)^{−1}ηĝ as above, we have φ(f) = 1 and φ is onto so that φ restricted to kLf is onto. Define ψ : k_{λ}G → kLf by ψ(ĝ) = β(g)η_{g}^{−1}(g, η_{g})f. Then as above we check that φ and ψ are inverse maps. So we may assume that H = L is perfect.

Combining Theorem 3.2 and Theorem 3.4, kGc ≅ S ⊗_{k} kHe as k-algebras where H is a central extension of the group G/N. If we assume that N is a maximal normal subgroup, then we can assume that H is quasi-simple. If we further assume that P is a p-subgroup of G such that P ∩ N = {1}, then we will again have an interior P -algebra homomorphism as in Proposition 3.3. We denote the image of P in G/N by \overline{P} and note that P ≅ \overline{P}.

Assume the setup of the above theorems. Recall that H is defined to be a central p′-extension of G/N by a finite cyclic p′-group Z ≤ k^{×}, where H = ((G/N) × Z) as sets and multiplication is given by (xN, η)(yN, ζ) = (syN, ηζα(x, y)) where α is a 2-cocycle with values in k^{×}. Also recall that we

defined e to be \frac{1}{|Z|} ∑_{η∈Z} η^{−1}(1, η) and saw that e was a central idempotent of kH.

Let τ be the map from kHe to k_{λ^{−1}}(G/N) induced from (xN, η)e ↦ β(x)^{−1}ηx̂N. For every r ∈ P we denote by r̃ the unique element (rN, η) ∈ H whose order is a power of p. If r̃ = (rN, η), then τ(r̃) = β(r)^{−1}ηr̂N. Since r̃ has prime power order, β(r)^{−p^{m}}η^{p^{m}} = 1 for some positive integer m and since k has characteristic p, β(r)^{−1}η = 1. Thus τ(r̃) = r̂N.

Now let ψ be the map from S ⊗_{k} k_{λ^{−1}}(G/N) to kGc defined by s ⊗ ĝN ↦ ss_{g}^{−1}g. Then define ϕ : S ⊗_{k} kHe ⟶ kGc by s ⊗ he ↦ ψ(s ⊗ τ(he)). We give S ⊗_{k} kHe the structure of an interior P -algebra by defining the structural homomorphism to be the map r ↦ s_{r} ⊗r̃e. With this structural homomorphism it is easy to check that ϕ is an isomorphism of interior P -algebras.

4 More Reduction Results

Let G be a finite group and let b be a nilpotent block of kG with defect group P. Let N be a normal subgroup of G and suppose c is a defect zero block of kN such that c is G-stable and such that cb ≠ 0. This implies that cb = b. In particular, we can write c = b_{1} + ... + b_{t} with b_{1} = b and the b_{i} all blocks of kG. If P is a defect group of b, then P ∩ N = 1. Since c is defect zero, S = kNc is isomorphic to a matrix algebra. So we can write S = End_{k}(U) for some k-vector space U. Let R be a maximal p-subgroup of G containing P such that Br_{R}(c) ≠ 0. Note that Br_{R}(c) ≠ 0 if and only if Br_{R}(b_{i}) ≠ 0 for some i. Also since c is a defect zero block, we have R ∩ N = 1.

In section 3, we showed that there is an interior R-algebra isomorphism ψ : S ⊗_{k} kHe ⟶ kGc. Where S = kNc and H is a central p′-extension of G/N by a finite cyclic p′-group Z ≤ k^{×}. Recall that H = ((G/N) × Z) as sets and multiplication is given by extending (xN, η)(yN, ζ) = (xyN, ηζα(x, y)) where α

is a 2-cocycle of G with values in Z. We also defined e = \frac{1}{|Z|} ∑_{ζ∈Z} ζ^{−1}(1, ζ) and

saw that e is a central idempotent of kH.

By our choice of λ in section 3, the map r ↦ s_{r} from R to S^{×} is a group homomorphism. We also have a group homomorphism r ↦ r̃ from R to H. So for any subgroup Q of R we have isomorphic groups \overline{Q} and Q̃ = \{r̃ | r ∈ Q ⊂ R\} in G/N and H respectively. Identify \overline{Q} and Q̃ with Q through these isomorphisms.

Define ψ : S ⊗_{k} k_{λ^{−1}}(G/N) ⟶ kGc to be the homomorphism which sends s ⊗ ĝN to ss_{g}^{−1}g for s ∈ S and g ∈ G. Let τ be the homomorphism from kHe to k_{λ^{−1}}(G/N) which extends (sN, η) ↦ β^{−1}(xN)ηx̂N. Finally, define ϕ : S ⊗_{k} kHe ⟶ kGc by extending ϕ(s, he) = ψ(s⊗τ(h)). We showed that ϕ is a homomorphism of interior Q algebras for any Q ≤ R since R∩N = 1. For any Q ≤ R we have an isomorphism of Brauer quotients ϕ_{Q} : S(Q)⊗_{k} (kHe)(Q) ⟶ (kGc)(Q) such that ϕ_{Q}(Br_{Q}(x)⊗_{k} Br_{Q}(y)) = Br_{Q}(ϕ(x⊗y)) by Proposition 28.3 of [16]. Also since S is simple we have a bijection between the blocks of kH that appear in e and those of kG which appear in c, given by u ↦ ϕ(1_{S} ⊗ u). Let d be the block of kH which corresponds to b.

By the definition of R we have S(Q) ≠ 0 for all Q ≤ R. So Br_{Q}(b) ≠ 0 if and only if BrQ̃(d) ≠ 0. Theorem 5.16 in chapter 5 of [12] implies that \overline{R} is a Sylow p-subgroup of G/N. Therefore R̃ is a Sylow p-subgroup of H. So R̃ must contain a defect group of d, but the largest subgroup of R̃ with non zero image under the Brauer homomorphism is P̃. Thus P̃ ≅ P is a defect group of d. This yields the following proposition.

Proposition 4.1. With the above notation, the block d of kH that corresponds to b has a defect group isomorphic to P.

Since S is simple, kGb and kHd are Morita equivalent. Therefore there is a correspondence between the simple modules of kHd and those of kGb ≅ S ⊗_{k} kHd given by V ↦ U ⊗ V. Also since b is nilpotent, kHb has a unique isomorphism class of simple modules. So if M is the unique simple kGb module and V is the unique simple kHd-module, then M ≅ U ⊗ V. A vertex of V is contained in P̃ since P̃ is a defect group of d. Suppose V has vertex Q̃ ⊆ P̃ with source X. By our choice of R, S(Q) ≠ 0. So we know that U must contain a summand of vertex Q. Therefore U ⊗ X has a summand of vertex Q, which implies that U ⊗X is a direct summand of Res_{Q}(U ⊗V) ≅ Res_{Q}(M) and U ⊗X has a summand of vertex Q. So Q contains a vertex of M. Therefore Q̃ = P̃ and the source of M is isomorphic to a direct summand of the kP -module U ⊗ X.

Finally we show that d is nilpotent. This will allow us to reduce our problem from G to H. The following can be found in [10]. If Q is a subgroup of R, then S(Q) ≠ 0. So S(Q) is a matrix algebra by 28.6 of [16]. This means that ϕ_{Q} is an isomorphism from S(Q) ⊗_{k} (kHe)(Q) to (kGc)(Q). Also notice that (kHe)(Q) ≅ kC_{H}(Q) Br_{Q}(e) and (kGc)(Q) ≅ kC_{G}(Q) Br_{Q}(c). Therefore f ↦ ϕ_{Q}(1⊗f) gives a bijection between the set of blocks of kC_{H}(Q) Br_{Q}(e) and the set of blocks of kC_{G}(Q) Br_{Q}(c). If we consider this map over all Q ≤ P, then we get a bijection between the set of Brauer pairs of blocks of kH appearing in e and the set of blocks of kG appearing in c. In particular, we have a bijection between the d-Brauer pairs and the b-Brauer pairs. The following theorem shows that d is nilpotent and a proof can be found in 3.5 of [10].

Theorem 4.2. Let (Q, v) be a d-Brauer pair with Q ≤ P and let u = ϕ(1 ⊗ v) so that (Q, u) is a b-Brauer pair. Then N_{H}(Q, v) and N_{G}(Q, u) have the same image in Aut(Q). In particular d is nilpotent.

5 Main Reduction

We start by proving the following.

Theorem 5.1. Let P be a finite p-group. Suppose that there is a finite group H with P a subgroup of Aut(H). Also suppose that c is a defect zero block of kH such that c is P -stable and Br_{P} (c) ≠ 0. Finally, suppose that, when viewed as a block of k(H ⋊ P), c has a non-torsion kP -module as a source of its unique (up to isomorphism) simple module. Then H can be chosen to be a central p′-extension of a direct product of isomorphic simple groups.

The proof requires the following.

Lemma 5.2. Assume the notation of Theorem 5.1. Let N be a normal subgroup of H which is maximal with respect to being P -stable. Then we can find a P - stable block f of kN such that cf ≠ 0.

Proof. It is clear that there exists a block d of kN such that cd ≠ 0. Let G = Stab_{(H⋊P)}(d) = {x ∈ H ⋊ P | xdx^{−1} = d} and let e = Tr_{G}^{H⋊P} (d). Then e is the sum of distinct blocks of kN. So e is a central idempotent of k(H ⋊ P). Therefore e is a sum of blocks of k(H ⋊ P) and one of these blocks is c. So ce = c which implies that Br_{P} (e) ≠ 0, since Br_{P} (e) Br_{P} (c) = Br_{P} (c) ≠ 0. Mackey’s Theorem tells us that e = Tr_{G}^{H⋊P} (d) = ∑_{y∈[P \(H⋊P)/G]} Tr_{P ∩^{y}G}^{P}(^{y}d). If

P ∩^{y}G ⊊ P for all y ∈ [P \(H ⋊ P)/G], then Br_{P} (e) = 0. So we must have a y such that P ∩^{y}G = P. By letting f = ^{y}d we are done since Stab_{(H⋊P)}(f) = yStab_{H⋊P} (d).

Proof of Theorem 5.1. We assume the setup of Theorem 5.1 and let N be a normal subgroup of H which is maximal with respect to being P stable. Let d be a block of kN such that cd ≠ 0. The lemma allows us to assume that P ⊆ Stab_{(H⋊P)}(d). This implies that Stab_{(H⋊P)}(d) = Stab_{H}(d)P ≅ Stab_{H}(d) ⋊ P. Now we can apply Proposition 3.1 to c as a block of k(H ⋊ P). We get a block e of kStab_{(H⋊P)}(d) that satisfies all the conditions of the proposition and the source of the simple module of kStab_{(H⋊P)}(d)e is the same as that of k(H ⋊P)c. View e as a block of k(Stab_{H}(d) ⋊ P). Then it is easy to see that e is a block of k(Stab_{H}(d)). So we may assume that H = Stab_{H}(d). Thus we can assume that d is stable in H ⋊ P.

Having d be H ⋊ P stable allows us to apply our main reduction Theorem 3.2 and its generalizations at the end of section 3. So we can find a central extension G of (H ⋊ P)/N with a cyclic p′ center Z and a block e of kG with the following properties. The block e is nilpotent with defect P and the source of the simple kGe-module is a source of the simple k(H ⋊ P)c-module. Finally we need to show that G is isomorphic to a semi-direct product of H/N with P.

Let G be as above and let λ be the 2-cocycle described in section 3. We let G̃ be the group whose underlying set is (H/N) × Z and with multiplication defined by (hN, η)(gN, ζ) = (hgN, λ(h, g)ηζ). We also define an action of P on G̃ by ^{u}(hN, η) = (uhu^{−1}N, λ(u, h)η) for u ∈ P. Now let φ : G ⟶ G̃ ⋊ P be defined by extending (hpN, η) ↦ (hN, η)p. This is clearly a bijection. So we just need to show φ is a homomorphism. To do this observe that for h, g ∈ H, p, q ∈ P and η, ζ ∈ Z

φ((hpN, η)(gqN, ζ)) = φ((hpgp^{−1}pqN, λ(hp, gq)ηζ))
= (hpgp^{−1}N, λ(hp, gq)ηζ)pq

and that

φ((hpN, η))φ((gqN, ζ)) = (hN, η)p(gN, ζ)q
= (hN, ζ)(pgp^{−1}N, λ(p, g)ζ)pq
= (hpgp^{−1}N, λ(h, pgp^{−1})λ(p, gηζ)pq.

So to prove Theorem 5.1 we need λ(hp, gq) = λ(h, pgp^{−1})λ(p, g). Recall that by our choice of λ we have λ(g, p) = 1 for all g ∈ H and all p ∈ P. The rest follows from the structural equations for 2-cocycles. So we can replace G by G̃ ⋊ P.

Finally, since N was maximal with respect to being normal in H and P - stable, we have that H/N is a minimal normal subgroup of (H/N)⋊P. Therefore H/N is a direct product of isomorphic simple groups.

If H/N is a direct product of cyclic groups, then the central extension of H/N is p-solvable. A result of Puig shows that the source of a simple module of a block of a p-solvable group is torsion in D(P). So we may assume that H/N is a direct product of nonabelian simple groups. In this case the simple groups will be transitively permuted by P.

Now we eliminate the direct product from Theorem 5.1. Assume that H is a central p′-extension of S_{1} × ⋯ × S_{t} by Z with all of the S_{i} isomorphic to the nonabelian simple group S_{1}. We also suppose that c is a defect zero block of kH which is P -stable for a p-group P ⊆ Aut(H). Finally assume that a simple kHc-module has a non-torsion source when viewed as a k(H ⋊ P)c-module. Let φ be the canonical surjection of H onto S_{1} × ⋯ × S_{t} and let D_{i} = φ^{−1}(S_{i}). Then each D_{i} is isomorphic to a central p′-extension of S_{i} by Z. It is also easy to see that for x ∈ D_{i} and y ∈ D_{j} with i ≠ j we have [x, y] = xyx^{−1}y^{−1} ∈ Z. Fix an x ∈ D_{i} for some i. Then the map from D_{j} to Z defined by y ↦ [x, y] for j ≠ i is a group homomorphism. It is also easy to see that this map factors through S_{j}. Since S_{j} is simple, the map must be 1. Therefore [D_{i}, D_{j}] = 1 for all i ≠ j. So if x ∈ D_{i} and y ∈ D_{j} with i ≠ j, we have xy = yx.

Define a surjective map φ : D_{1} × ⋯ × D_{t} ⟶ H by (d_{1}, ..., d_{t}) ↦ d_{1}...d_{t} for d_{i} ∈ D_{i}. Since elements of D_{i} and D_{j} commute for i ≠ j, φ is a homomorphism. We now pull everything back to D_{1} × ⋯ × D_{t}. By a general lifting theorem on idempotents, Theorem 3.2 in [16], we can find a block f of k(D_{1} × ⋯ × D_{t}) whose image in kH is c. It is easy to see that ker(φ) is a p′-group. So we can apply Theorem 8.8 of Chapter 5 in [12] and see that k(D_{1} × ⋯ × D_{t})f ≅ kHc. Moreover, if M is a simple kHc-module, then M will be a simple k(D_{1} × ⋯ × D_{t})f-module with the action via the above homomorphism.

Note that k(D_{1}×⋯×D_{t}) ≅ (kD_{1})⊗_{k}⋯⊗_{k}(kD_{t}). So there are unique defect zero blocks f_{i} of kD_{i} such that (kD_{1}f_{1})⊗_{k} ⋯⊗_{k} (kD_{t}f_{t}) ≅ k(D_{1} ×⋯×D_{t})f ≅ kHc. We can define a P algebra structure on (kD_{1}f_{1}) ⊗_{k} ⋯ ⊗_{k} (kD_{t}f_{t}) which agrees with the action of P on kHc. Also notice that, since Z commutes with the action of P and the S_{i} are transitively permuted by P, the D_{i} are transitively permuted by P.

Set Q = Stab_{P} (D_{1}) = Stab_{P} (S_{1}). Then t = [P : Q]. We choose {u_{1}, ..., u_{t}} = [P/Q] with u_{1} = 1_{Q}. Also assume that the action of u_{i} maps D_{1} to D_{i}. From this we find a P -algebra structure on (kD_{1}f_{1}) ⊗_{k} ⋯ ⊗_{k} (kD_{t}f_{t}) such that the action of P permutes the 1⊗⋯⊗(kD_{i}f_{i})⊗⋯⊗1. We show that as P -algebras (kD_{1}f_{1}) ⊗_{k} ⋯ ⊗_{k} (kD_{t}f_{t}) is isomorphic to Ten_{Q}^{P}(kD_{1}f_{1}).

Let Q be a subgroup of a p-group P and let B be a Q-algebra. Recall Ten_{Q}^{P}(B) = B ⊗_{k} ⋯ ⊗_{k} B as k-algebras, where the number of B’s above is [P : Q]. The action of P on Ten_{Q}^{P}(B) is defined as follows. Let b_{1}, ..., b_{t} ∈ B and v ∈ P. Then ^{v}(b_{1}, ..., b_{t}) = (^{q_{τ^{−1}(1)}}b_{τ^{−1}(1)}, ...,^{q_{τ^{−1}(t)}}b_{τ^{−1}(t)}) where τ ∈ S_{t} and q_{i} ∈ Q are defined by vu_{i} = u_{τ(i)}q_{i}. Now let A = A_{1} ⊗_{k} ⋯ ⊗_{k} A_{t} be a P -algebra such that P transitively permutes the 1 ⊗ ⋯ ⊗ A_{i} ⊗ ⋯ 1 and identify 1 ⊗ ⋯ ⊗ A_{i} ⊗ ⋯ 1 with A_{i}. Let Q = Stab_{P} (A_{1}) and let {u_{1}, ..., u_{t}} = [P/Q] be a set of coset representatives such that the action of u_{i} maps A_{1} to A_{i}. Next identify A_{i} with A_{1} via the isomorphism induced by u_{i}. In this way we can assume that ^{u_{i}}(a ⊗ 1 ⊗ ⋯ ⊗ 1) = (1 ⊗ ⋯ ⊗ a ⊗ ⋯ ⊗ 1) where a is in the i’th position.

Let v ∈ P and let 1 ⊗ ⋯ ⊗ a ⊗ ⋯ ⊗ 1 be an arbitrary element of A_{j} for some j. We can write vu_{j} = u_{i}w for unique i and w ∈ Q. Then the action of v on 1 ⊗ ⋯ ⊗ a ⊗ ⋯ ⊗ 1 is the same as the action of vu_{j} on a ⊗ 1 ⊗ ⋯ ⊗ 1. This is the same as the action of u_{i}w which sends a ⊗ 1 ⊗ ⋯ ⊗ 1 to 1 ⊗ ⋯ ⊗^{w}a ⊗ ⋯ ⊗ 1 where ^{w}a is in the i’th position. This implies that A ≅ Ten_{Q}^{P}(A_{1}).

Letting B = kD_{1}f_{1} and A = kHc, the above yields kHc ≅ Ten_{Q}^{P}(kD_{1}f_{1}) as P -algebras. We know that kD_{1}f_{1} is a Q-algebra and it is isomorphic to End_{k}(M) where M is a simple kD_{1}f_{1}-module. Then by Lemma 2.1 of [2] we know that Ten_{Q}^{P}(End_{k}(M)) is isomorphic to End_{k}(Ten_{Q}^{P}(M)) as P -algebras. Thus we have that a simple kHc-module is isomorphic to Ten_{Q}^{P}(M). Note that if P = Q, then H is a central p′-extension of a simple group.

Now we assume that P ≅ C_{p} × C_{p}. Suppose that Q ⊊ P and let V be a kQ-module which is a source of M as a k(D_{1} ⋊ Q)f_{1}-module. Then V is torsion in D(Q) since Q is trivial or cyclic and D(Q) is finite in these cases. Since V is a source of M, M ≅ V ⊕ X as kQ-modules for some kQ-module X, and we know that V has vertex Q. Then by Proposition 3.15.2 of [1], Ten_{Q}^{P}(M) ≅ Ten_{Q}^{P}(V ⊕ X) ≅ Ten_{Q}^{P}(V) ⊕ Ten_{Q}^{P}(X) ⊕ U as kP -modules, where U is the direct sum of modules induced from proper subgroups of Q. We also know that Ten_{Q}^{P}(V) has a summand of vertex P. So if W is a source of a simple kHcmodule, then [W] = [Ten_{Q}^{P}(V)] in D(P). Also since Ten_{Q}^{P} : D(Q) ⟶ D(P) is a group homomorphism we conclude that W must also be torsion in D(P), since V is torsion in D(Q). Therefore we can assume that H is a central p′-extension of a finite simple group. Thus we have established the following reduction.

Theorem 5.3. Let P ≅ C_{p} × C_{p} where p is a prime. Suppose there is a finite group H with P a subgroup of Aut(H). Also suppose that c is a defect zero block of kH such that c is P -stable and Br_{P} (c) ≠ 0. Finally, suppose that, when viewed as a block of k(H ⋊ P), c has a non-torsion kP -module as a source of its unique (up to isomorphism) simple module. Then H can be chosen to be a central p′-extension of a simple group.

6 Simple Groups

Suppose that H is a finite group and P is a subgroup of Aut(H). Also assume that c is a P -stable defect zero block of kH such that Br_{P} (c) ≠ 0 and that u ∈ P ∩ Inn(H) is nontrivial. Then there exists an x ∈ H such that ^{u}h = ^{x}h for all h ∈ H. Let c = ∑_{h∈H} λ_{h}h. Since Br_{P} (c) ≠ 0, there exists an h_{0} ∈ H such

that ^{u}h_{0} = h_{0} and λ_{h_{0}} ≠ 0. Therefore ^{x}h_{0} is also equal to h_{0}. Since |u| = p^{e} for some positive integer e, we have x^{p^{e}} ∈ Z(H) and |x| = p^{e}m for some positive integer m. Then |x^{m}| = p^{e} and x^{m} stabilizes h_{0}. So < x^{m} > is a p-group and Br_{<x^{m}>}(c) ≠ 0. This implies that < x^{m} > is contained in a defect group of c, but c is a defect zero block. Therefore no such u exists and this yields following.

Proposition 6.1. Let H be a finite group, let P be a p-group in Aut(H) and let c be a defect zero block of kH which is P -stable and such that Br_{P} (c) ≠ 0. Then P ∩ Inn(H) = 1. In particular, P will be isomorphic to a subgroup of Out(H).

So we can detect P inside of Out(H). From Section 5 we may assume that H is a central p′-extension of a simple group. Also if H is a central p′-extension of the finite group G it is known that Out(H) is a subgroup of Out(G). So we consider Out(G) for the finite simple groups G. We are concerned with when C_{p} × C_{p} will be a subgroup of Out(G).

We now run through the classification of the finite simple groups and check to see when C_{p} × C_{p} is a subgroup of Out(G) for a simple group G. We will see that when p is odd, the set of groups where this can happen is very small. First consider the alternating groups. It is well known that for n ≥ 5 and n ≠ 6, Aut(A_{n}) ≅ S_{n}. So | Out(A_{n})| = 2. We also note that A_{6} ≅ P SL_{2}(9) and we will deal with this case later. Therefore C_{p} × C_{p} is not a subgroup of Out(A_{n}) for any n ≥ 5 and n ≠ 6. We can also eliminate the sporadic groups from the list of groups we need to check since | Out(G)| ≤ 2 for all 26 sporadic groups. So if C_{p} × C_{p} is a subgroup of Out(G) for a simple group G, then G must be a finite group of Lie type.

We now look at the finite simple groups of Lie type. A more complete treatment of the following can be found in [7]. The Chevalley groups, also known as the untwisted groups of Lie type, are denoted by A_{n}(q), B_{n}(q)(n ≥ 2), C_{n}(q)(n ≥ 3), D_{n}(q)(n ≥ 4), E_{6}(q), E_{7}(q), E_{8}(q), F_{4}(q) and G_{2}(q), where q = r^{f} for some prime r and positive integers f and n satisfying the given condition, if any. The twisted Chevalley groups, also called the Steinberg groups, are denoted by ^{2}A_{n}(q)(n ≥ 2), ^{2}D_{n}(q)(n ≥ 3), ^{3}D_{4}(q) and ^{2}E_{6}(q) with q and n as above. Finally the Suzuki-Ree groups are denoted by ^{2}B_{2}(2^{2f+1}), ^{2}F_{4}(2^{2f+1}) and ^{2}G_{2}(3^{2f+1}). With this notation we have the following.

Theorem 6.2 ([7]). Let G be the adjoint version of a finite group of Lie type. Then G is a nonabelian simple group except when

G = A_{1}(2), A_{1}(3),^{2}A_{2}(2),^{2}B_{2}(2), B_{2}(2), G_{2}(2),^{2}F_{4}(2), and ^{2}G_{2}(3)

and for G = B_{2}(2), G_{2}(2),^{2}F_{4}(2) and ^{2}G_{2}(3) the commutator [G, G] is the unique nontrivial proper normal subgroup. Moreover, in these cases [G, G] is a nonabelian simple group.

Three of the simple groups which show up as commutators in the above theorem are isomorphic to other simple groups in the list of groups of Lie type via the following isomorphisms A_{1}(8) ≅ (^{2}G_{2}(3))′, A_{1}(9) ≅ B_{2}(2)′, and ^{2}A_{2}(3) ≅ G_{2}(2)′. However, there is no such isomorphism for (^{2}F_{4}(2))′ and this is a simple group that doesn’t show up elsewhere in the classification. This leads to the following definition.

Definition 6.3 ([7]). The finite simple groups of Lie type are (^{2}F_{4}(2))′ and the finite groups of Lie type excluding

G = A_{1}(2), A_{1}(3),^{2}A_{2}(2),^{2}B_{2}(2), B_{2}(2),^{2}F_{4}(2) and ^{2}G_{2}(3).

We first note that Out(^{2}F_{4}(2)′) = C_{2}. Thus P is not a subgroup of Out(^{2}F_{4}(2)′). So we will only need to look at the finite simple groups of Lie type which are actually finite groups of Lie type.

If G is an adjoint version of a finite group of Lie type, then Out(G) will have order dfg where d, f and g are the orders of certain subgroups of Out(G). d is the order of the subgroup of diagonal automorphisms. This group is cyclic or elementary abelian of order 4. The field automorphisms will form a subgroup of order f and this subgroup will always be cyclic. Finally, the subgroup of order g is the group of graph automorphisms which is cyclic of order 2 or trivial (unless G = D_{4}(q) in which case it is isomorphic to S_{3}). In Table 1, we list d, f and g for the groups of Lie type. A similar table is given in [4].

Next we look at the question of when is a non-cyclic p-group P a subgroup of Out(G). Since P is not cyclic and p is odd we must have p dividing at least two of the d, f and g. Let q = r^{f} where r is the prime over which the group of Lie type is defined. We now restrict to the case of p being odd. When p ≥ 5 there are only two types of groups where this happens. First we could have A_{n}(q) with p | (n + 1, q − 1) (notice that this means p ≠ r) and p | f. The second case that we can’t eliminate is ^{2}A_{n}(q) with p | (n + 1, q + 1) (again this means p ≠ r) and p | f. When p = 3 we can again eliminate most cases. In this case there are only five types of groups that can show up. As above we can have A_{n}(q) with p | (n + 1, q − 1) and p | f or ^{2}A_{n}(q) with p | (n + 1, q + 1) and p | f. We also have the cases E_{6}(q) with p | (q − 1) and p | f and ^{2}E_{6}(q) with p | (q + 1) and p | f. In all four of these cases we must have p ≠ r. Our final possibility is D_{4}(q) with p | f and in this case we cannot assume p ≠ r. In the next section we eliminate the cases of G = E_{6}(q) and G = ^{2}E_{6}(q) by using Aut(G) instead of just Out(G).

7 E_{6}(q^{3}) and ^{2}E_{6}(q^{3})

Let G be the adjoint versions of E_{6}(q^{3}) or^{2}E_{6}(q^{3}) and let Ĝ denote the universal versions. In this section, we show that if 3|(q − 1) for E_{6}(q^{3}) or 3|(q + 1) for

Conditions  G  d  f  g
A_{1}(q)  (2, q − 1)  q = r^{f}  1
n ≥ 2  A_{n}(q) (n + 1, q − 1) q = r^{f}  2
n ≥ 2  ^{2}A_{n}(q) (n + 1, q + 1) q^{2} = r^{f}  1
B_{2}(q)  (2, q − 1)  q = r^{f} 2 if r = 2
f odd  ^{2}B_{2}(q)  1  q = 2^{f}  1
n ≥ 3  B_{n}(q)  (2, q − 1)  q = r^{f}  1
n ≥ 3  C_{n}(q)  (2, q − 1)  q = r^{f}  1
D_{4}(q)  (2, q − 1)^{2}  q = r^{f}  3!
3D_{4}(q)  1  q^{3} = r^{f}  1
n > 4, even D_{n}(q)  (2, q − 1)^{2}  q = r^{f}  2
n > 4, odd D_{n}(q)  (4, q^{n} − 1)  q = r^{f}  2
n ≥ 4  ^{2}D_{n}(q) (4, q^{n} + 1) q^{2} = r^{f}  1
G_{2}(q)  1  q = r^{f} 2 if r = 3
f odd  ^{2}G_{2}(q)  1  q = 3^{f}  1
F_{4}(q)  1  q = r^{f} 2 if r = 2
f odd  ^{2}F_{2}(q)  1  q = 2^{f}  1
E_{6}(q)  (3, q − 1)  q = r^{f}  2
2E_{6}(q)  (3, q + 1)  q^{2} = r^{f}  1
E_{7}(q)  (2, q − 1)  q = r^{f}  1
E_{8}(q)  1  q = r^{f}  1

Table 1: Orders of Out(G) for the adjoint versions for the groups of Lie type [4] 2E_{6}(q^{3}), then P ≅ C_{3}×C_{3} is never a subgroup of Aut(G) such that P ∩Inn(G) = 1. Our first step is to find an element h of Ĝ such that h is not fixed by the standard field automorphism σ of order 3 but the image of h in G is fixed by σ. In order to do this we need to introduce the Cartan subgroup Ĥ of Ĝ. This can be done for all finite groups of Lie type, but we will just introduce the facts that we need. First we will state some known results. It is known that Z(Ĝ) ⊆ Ĥ (see Proposition 2.5.9 of [7]). Let {α_{1}, ..., α_{s}} be a set of fundamental roots. So s = 6 if G = E_{6}(q^{3}) and s = 4 if G = ^{2}E_{6}(q^{3}). Then there are subgroups Ĥ_{i} =< h_{α_{i}}(t) > of Ĥ such that the following hold. When G = E_{6}(q^{3}), Ĥ_{i} ≅ (GF (q^{3}))^{×} ≅ C_{(q^{3}−1)} for i ∈ {1, ..., 6} and when G = ^{2}E_{6}(q^{3}), Ĥ_{1} ≅ Ĥ_{2} ≅ (GF (q^{3∗2}))^{×} ≅ C_{(q^{3∗2}−1)} and Ĥ_{3} ≅ Ĥ_{4} ≅ (GF (q^{3}))^{×} ≅ C_{(q^{3}−1)}. Also by Theorem 2.4.7 of [7], Ĥ ≅ {(h_{1}, ..., h_{s}) : h_{i} ∈ Ĥ_{i}}. One of the main results we need describes how the standard field automorphism of order 3 will act on Ĥ.

Theorem 7.1 ([7] 2.5.1). Let σ be the standard field automorphism of order 3. We identify each Ĥ_{i} with the appropriate Galois field. Then σ acts on Ĥ as σ(h_{1}, ..., h_{s}) = (h_{1}^{q}, ..., h_{s}^{q}).

We also need the following lemma.

Lemma 7.2. Let Z(Ĝ) =< z >. There exists w ∈ Ĥ such that w^{−1}σ(w) = z.

Proof. Proposition 2.5.9 of [7] tells us that z = (z_{1}, ..., z_{s}) ∈ Ĥ. We also know that z has order 3. If 3 | (q + 1), then 3 does not divide q^{3} − 1. So in this case z_{3} = z_{4} = 1. Note that 3 | (q − 1) implies 3(q − 1) | q^{3} − 1 and 3 | q + 1 implies 3(q − 1) | q^{6} − 1. This means that for each z_{i} ∈ H_{i} we can find a w_{i} ∈ H_{i} such that w_{i}^{(q−1)} = z_{i}. Letting w = (w_{1}, ..., w_{s}) completes the proof of the lemma since w^{−1}σ(w) = w^{−1}(w_{1}^{q}, ...w_{s}^{q}) = (w_{1}^{(q−1)}, ..., w_{s}^{(q−1)}) = z.

The important fact about w is that w will not be fixed by σ in Ĝ, but the image of w in G will be fixed by σ. Now that we have found w we will use some known facts on the centralizers of σ in Aut(G) and Aut(Ĝ) to obtain our result. In an earlier section we said that Out(G) will have a cyclic subgroup of order 3 which we called the diagonal subgroup. The pre-image of the diagonal subgroup is the group of inner diagonal automorphisms of G and is denoted by Inndiag(G). So Inndiag(G) ≅ G < γ > where γ ∈ Aut(G) \ Inn(G). We can now state the following.

Theorem 7.3 ([7] 4.9.1). Let K = E_{6}(q^{3}) or ^{2}E_{6}(q^{3}), let σ be a field automorphism of K of order 3 and let K_{σ} = E_{6}(q) or ^{2}E_{6}(q) respectively. Here we do not restrict the versions of K. Then the following hold

(i) If y ∈ Inndiag(K)σ has order 3, then σ and y are Inndiag(K)-conjugate,

(ii) If K is adjoint, then so is K_{σ} and C_{Inndiag(K)}(σ) ≅ Inndiag(K_{σ}),

(iii) If K is universal, then C_{K}(σ) ≅ K_{σ} and K_{σ} is universal.

Now we can state the main result of this section.

Theorem 7.4. Let G = E_{6}(q^{3}) with 3 | (q − 1) or ^{2}E_{6}(q^{3}) with 3 | (q + 1). We view G ≅ Inn(G) as a subgroup of A = Aut(G). Then there is no subgroup P of A with P ≅ C_{3} × C_{3} and P ∩ G = 1.

Proof. Suppose that P is a subgroup of A such that P ≅ C_{p} ×C_{p} and P ∩G = 1. Since Out(G) has 3-rank 2, P ∩ Gσ must be non-empty. Now by (i) of Theorem 7.3 there is a g ∈ Inndiag(G) such that σ ∈ ^{g}P and ^{g}P ∩ G = 1. So by replacing P with ^{g}P we may assume that σ ∈ P.

Now applying (iii) of Theorem 7.3 CĜ(σ) ≅ Ĝ_{σ} and Ĝ_{σ} is isomorphic to the universal version of E_{6}(q) or ^{2}E_{(}q). By our comments on Inndiag we have that Inndiag(G_{σ}) = G_{σ} < γ > for some γ ∈ Aut(G_{σ}) \ Inn(G_{σ}). Then by Lemma 7.2 we found an element w ∈ Ĝ which is not fixed by σ but the image of w, denoted w̄, in G is fixed by σ. So by passing from Ĝ to G we find a w̄ ∈ G such that w̄ ∈ C_{G}(σ) and w̄ ∉ G_{σ}. Now by (ii) of Theorem 7.3 we have C_{Inndiag(G)}(σ) ≅ Inndiag(G_{σ}) and since Inndiag(G_{σ}) = G_{σ}(γ) for some gamma and we know that w̄ ∈ G \ G_{σ}. So Inndiag(G_{σ}) ≅ G_{σ} < w̄ >. Therefore C_{Inndiag(G)}(σ) ⊆ G. Therefore P ∩ G is not empty. So no such P exists and the result is shown.

Combining this result with section 6 we have the following corollary to Theorem 5.3.

Corollary 7.5. Assume that p is odd. Suppose that P = C_{p} × C_{p} ≤ Aut(G) where G is a finite group. Suppose that b is a P -stable defect 0 block of kG such that Br_{P} (b) ≠ 0. Finally, suppose that the source V of a simple k(G ⋊ P)bmodule M is a finitely generated endo-permutation kP -module whose image is non-torsion in D(P). Then we can find G, b, V where G is one of the following.

(i) (a) G is a central p′-extension of A_{n}(q) with p|(n + 1, q − 1, f) where q = r^{f}; or

(b) G is a central p′-extension of ^{2}A_{n}(q) with p|(n + 1, q + 1, f) where q = r^{f}; or

(ii) p = 3, q = r^{f} and G is a central extension of D_{4}(q) with 3|f.

Acknowledgements

I would like to express my gratitude to my thesis advisor Dr. Markus Linckelmann for all of his guidance with this work. I would also like to thank Dr. Radha Kessar and Dr. Ronald Solomon for their assistance with various parts of this paper.

References

[1] D.J. Benson, “Representations and cohomology Vol. I,” Cambridge University Press, Cambridge, 1991.

[2] S. Bouc and J. Thévenaz, The group of endo-permutation modules, Invent. Math. 139 (2000), 275-349.

[3] M. Broué and L. Puig, A Frobenius theorem from blocks, Invent. Math. 56 (1980), 117-128.

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

[5] E.C. Dade, Endo-permutation modules over p-groups I, Ann. Math. 107 (1978), 459-494.

[6] W. Feit, Some consequences of the classification of finite simple groups, Proc. Symp. Pure Math. 37, 175-181.

[7] D. Gorenstein, R. Lyons and R.M. Solomon “The Classification of the Finite Simple Groups, Number 3,” Amer. Math. Soc. Surveys and Monographs 40, #3, 1998.

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

[9] I.M. Isaacs, “Character Theory of Finite Groups,” Academic Press, New York, 1976.

[10] R. Kessar, The Solomon system F_{SOL}(3) does not occur as fusion system of a 2-block, J. Algebra 296 (2006), 409-425.

[11] A. Muncke, The sources of simple modules of p-solvable groups with elementary abelian p-Sylow group of order p^{2}, Comm. Algebra 23(2) (1995), 569-600.

[12] H. Nagao and Y. Tsushima, “Representations of Finite Groups,” Academic Press, San Diego, 1989.

[13] L. Puig, Affirmative answer to a question of Feit, J. Algebra 131 (1990), 513-526.

[14] L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1988), 77-116.

[15] L. Puig, Notes sur les algébres de Dade, unpublished manuscript.

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