A NOTE ON THE SCHUR MULTIPLIER OF A FUSION SYSTEM

Markus Linckelmann

Abstract. The Schur multiplier of a fusion system F on a finite p-group P, being defined as the inverse limit of the Schur multipliers of the subgroups of P taken over the category F, has a series of properties analogous to some properties of Schur multipliers of finite groups. This is one of the tools used in Kessar [3] to show that the Solomon fusion system cannot arise as the Brauer category of a p-block of a finite group.

1 Introduction

The Schur multiplier of a finite group G is defined in the literature either as H_{2}(G; ℤ) or its Pontryagin dual H^{2}(G; ℂ^{×}); both are noncanonically isomorphic. Given the application to fusion systems which has motivated this note, we work here with the second definition.

Throughout this paper, p is a prime. A fusion system on a finite p-group P is a category F whose objects are the subgroups of P and whose sets of morphisms Hom_{F}(Q, R) consist of injective group homomorphisms fulfilling a list of axioms which we recall in the appendix; this concept is due to Puig. If F is a fusion system on a finite p-group P, a subgroup Q of P is called F-centric if C_{P} (Q′) = Z(Q′) for any subgroup Q′ of P which is isomorphic to Q in the category F, and Q is called F-radical if the largest normal p-subgroup of Aut_{F}(Q) is equal to the group Aut_{Q}(Q) of inner automorphisms of Q.

Definition 1.1. Let F be a fusion system on a finite p-group P. The Schur multiplier of F is defined to be the group

m(F) = lim_{⟵}_{Q∈F} H^{2}(Q, ℂ^{×}) ;

that is, m(F) is the inverse limit of the contravariant functor on F mapping each subgroup Q of P to its Schur multiplier. If F is the underlying fusion system of a p-local group L we set m(L) = m(F) and call m(L) the Schur multiplier of the p-local group L.

Typeset by AMS-TEX

Given an abelian group Z on which P acts trivially and a non negative integer n, an element ζ ∈ H^{n}(P, Z) is called F-stable if Res_{Q}^{P}(ζ) = Res_{φ}(ζ) for any subgroup Q of P and any group homomorphism φ : Q → P belonging to the category F. By the appropriate version of Alperin’s fusion theorem, an element ζ ∈ H^{n}(P, Z) is F-stable if and only if Res_{Q}^{P}(ζ) is Aut_{F}(Q)-stable in H^{n}(Q, Z) for any F-radical centric subgroup Q of P.

We frequently identify m(F) to the subgroup of H^{2}(P, ℂ^{×}) consisting of all F-stable elements in H^{2}(P, ℂ^{×}). Since H^{2}(P, ℂ^{×}) is a finite abelian p-group, so is m(F). By using a biset as in Broto, Levi, Oliver [1] we will show that m(F) is a direct factor of H^{2}(P, ℂ^{×}); if F = F_{P} (G) is the fusion category of a finite group G on a Sylowp-subgroup P of G then m(F) is well-known to coincide with the p-part of the Schur multiplier of G.

2 Background material on central extensions

The content of this section is well-known and included for the convenience of the reader. Given a fusion system F on a finite p-group P and a central subgroup Z of P there is a fusion system C_{F}(Z) on P which consists of all morphisms φ : Q → R in F which can be extended to a morphism ψ : QZ → RZ in F in such a way that ψ|_{Z} = Id_{Z}. Moreover, C_{F}(Z) induces a unique fusion system C_{F}(Z)/Z on P/Z. If F = C_{F}(Z), the connections between F and G = F/Z are well-understood; see the appendix below for more details.

Proposition 2.1. Let 1 → Z → P → Q → 1 be a central extension of finite p-groups and let ζ ∈ H^{2}(P̄, Z) be the cohomology class represented by this extension. Let G be a fusion system on P̄. If there is a fusion system F on P such that F = C_{F}(Z) and such that F/Z = G then the cohomology class ζ is G-stable.

The proof is an easy consequence of the following Lemma:

Lemma 2.2. Let 1 → Z → P →^{π} P̄ → 1 be a central extension of finite p-groups and let ζ ∈ H^{2}(P̄, Z) be the cohomology class represented by this extension. Let Q be a subgroup of P containing Z and let Q̄ be the image of Q in P̄. Let φ be an automorphism of Q. Then ResP̄ ; Q̄(ζ) = Res_{φ}(ζ) if and only if there is an automorphism ψ of Q which makes the following diagram commutative:

1 ⟶ Z ⟶ Q ⟶^{π|_{Q}} Q̄ ⟶ 1
∥  ^{ψ}↓  ↓^{φ}
1 ⟶ Z ⟶ Q ⟶_{π|_{Q}} Q̄ ⟶ 1

Proof. The cohomology classes ResP̄ ; Q̄ and Res_{φ}(ζ) are represented by the central extensions

1 ⟶ Z ⟶ Q ⟶^{π|_{Q}} Q̄ ⟶ 1 ,
1 ⟶ Z ⟶ Q ⟶^{φ∘π|_{Q}} Q̄ ⟶ 1 ,

respectively, and thus ResP̄ ; Q̄(ζ) = Res_{φ}(ζ) if and only if there is an automorphism ψ of Q making the diagram

1 ⟶ Z ⟶ Q ⟶_{φ∘π|_{Q}} Q̄ ⟶ 1
∥  ^{ψ}↓  ∥
1 ⟶ Z ⟶ Q ⟶^{π|_{Q}} Q̄ ⟶ 1

is commutative. But that is equivalent to the commutativity of the diagram in the statement. □

Proof of 2.1. If there is a fusion system F on P such that F = C_{F}(Z) and such that F/Z = G then in particular for any subgroup Q of P containing Z the induced map Aut_{F}(Q) → Aut_{G}(Q̄) is surjective. It follows from Lemma 2.2 and Alperin’s fusion theorem that ζ is G-stable. □

3 Some cohomological properties of m(F)

Given a finite p-group P and an abelian group A endowed with the trivial action of P, for any integer n, Tate cohomology Ĥ^{n}(P; A) is annihilated by |P |, hence uniquely p′-divisible. Equivalently, Ĥ^{n}(P; A) is a ℤ_{(p)}-module.

Let P, P′ be finite groups, and let X be a finite P′-P -biset on which P′, P act regularly on the left and on the right, respectively. Then every transitive subset of X is of the form P′ ×_{Q} _{φ}P for some subgroup Q of P′ and some injective group homomorphism φ : Q → P.

We extend the usual definitions of restriction and transfer maps on cohomology to define a transfer map

t_{X} : Ĥ^{∗}(P; A) ⟶ Ĥ^{∗}(P′; A)

in the obvious way as follows. If X is transitive, say X = P′ ×_{Q} _{φ}P with Q, φ as above,

we set t_{X} = t_{Q}^{P′} ∘ res_{φ}, where t_{Q}^{P′} is the usual transfer map. If X is the disjoint union of two non-empty P′-P -bisets Y, Y′, we set t_{X} = t_{Y} + t_{Y′}. The map t_{X} is clearly graded, and if no confusion arises, we use the same notation t_{X} for the induced map Ĥ^{n}(P; A) → Ĥ^{n}(P′; A), where n is any integer. We want to use this in conjunction with the following result, due to Broto, Levi, Oliver:

Proposition 3.1. ([1, 5.5]) Let F be a fusion system on a finite p-group P. There is a finite P -P -biset X with the following properties:

(i) Every transitive subbiset of X is isomorphic to P ×_{Q} _{φ}P for some subgroup Q of

P and some group homomorphism φ : Q → P belonging to the category F. (ii) |X|/|P | is prime to p. (iii) For any subgroup Q of P and any φ ∈ Hom_{F}(Q, P) the Q-P -bisets _{Q}X and _{φ}X

are isomorphic.

Note that (i) implies automatically that P acts regularly on the left and on the right of X. The set _{Q}X is the Q-P biset obtained from restricting the action on X on the left to Q via the inclusion Q ⊆ P, and the set _{φ}X is the Q-P -biset obtained from restricting the action on X on the left to Q via the group homomorphism φ : Q → P. The next Proposition is a trivial consequence of the properties of X.

Proposition 3.2. Let F be a fusion system on a finite p-group P and let A be an abelian group. Let X be a finite P -P -biset as in 3.1. Then the transfer map t_{X} : Ĥ^{n}(P, A) → Ĥ^{n}(P, A) has the following properties:

(i) Ĥ^{n}(P, A) = Im(t_{X}) ⊕ Ker(t_{X}) .

(ii) Im(t_{X}) = lim_{⟵}_{Q∈F} Ĥ^{n}(Q, A) .

(iii) t_{X} acts as multiplication by |X|/|P | on Im(t_{X}) .

Proof. Let ζ ∈ Ĥ^{n}(P, A) for some integer n. Let Q be a subgroup of P and let φ : Q → P be a morphism in F. Since the Q-P -bisets _{Q}X and _{φ}X are isomorphic, we have res_{φ}(t_{X}(ζ)) = t_{_{φ}X}(ζ) = t_{_{Q}X}(ζ) = res_{Q}^{P}(ζ). Thus t_{X}(ζ) is F-stable. If ζ is F-stable then t_{P ×_{Q} _{φ}P} (ζ) = [P : Q]ζ, hence t_{X}(ζ) = \frac{|X|}{|P |}ζ. Since Ĥ^{n}(P, A) is uniquely p′-divisible,

multiplication by |X|/|P | induces an automorphism on Ĥ^{n}(P, A) by property (iii) in 3.1. The result follows. □

Corollary 3.3. The functor sending an abelian group A to the ℤ_{(p)}-module lim_{⟵}_{Q∈F} Ĥ^{∗}(Q, A) is cohomological.

Proof. Let 0 → A′ → A → A′′ → 0 be a short exact sequence of abelian groups. This gives rise to a long exact sequence of functors from F to ℤ_{(p)}-modules

⟶ Ĥ^{1}(−, A) ⟶ Ĥ^{1}(−, A′′) ⟶ Ĥ^{2}(−, A′) ⟶ Ĥ^{2}(−, A) ⟶

Evaluating this at P yields a long exact sequence. The transfer map t_{X} commutes with the morphisms in this sequence. It follows from 3.2 that applying t_{X} to all components in this sequence preserves exactness, thus yielding a long exact sequence of the required inverse limits by 3.2.(ii). □

Corollary 3.4. We have m(F) = t_{X}(H^{2}(P, ℂ^{×})); in particular, m(F) is a direct factor of the Schur multiplier H^{2}(P, ℂ^{×}) of the p-group P.

4 Perfect fusion systems

Let F be a fusion system on a finite p-group P. For any subgroup Q of P we denote by [Q, F] the subgroup of P generated by the set of elements of the form uφ(u^{−1}), where u runs over the set of elements of Q and φ ∈ Hom_{F}(< u >, P). Note that [Q, F] ⊆ Q if and only if Q is strongly F-closed, and that in this case [Q, F] is again a strongly F-closed subgroup of P. We say that the fusion system F is perfect if [P, F] = P. If F = F_{P} (G) is the fusion system of a finite group G on one of its Sylow-p-subgroups P then [P, F] is the focal subgroup of P. In particular, if G is perfect (that is, G = [G, G]) then F_{P} (G) is perfect (cf. [2]).

Proposition 4.1. Let F be a perfect fusion system on a finite p-group P and let Z be a cyclic p-subgroup of ℂ^{×}. Then the map lim_{⟵}_{Q∈F} H^{2}(Q, Z) ⟶ m(F) induced by the

inclusion Z ⊆ ℂ^{×} is injective.

Proof. Raising every element of ℂ^{×} to the |Z|-th power gives rise to a short exact sequence of abelian groups

1 ⟶ Z ⟶ ℂ^{×} ⟶ ℂ^{×} ⟶ 1 .

Applying H^{∗}(P, −) yields a long exact sequence of ℤ_{(p)}-modules

⋯ ⟶ H^{1}(P, ℂ^{×}) ⟶ H^{2}(P, Z) ⟶ H^{2}(P, ℂ^{×}) ⟶ ⋯

By 3.3 and 3.2, applying t_{X} to the terms in this sequence we get a long exact sequence

⋯ ⟶ lim_{⟵}_{Q∈F} H^{1}(Q, ℂ^{×}) ⟶ lim_{⟵}_{Q∈F} H^{2}(Q, Z) ⟶ m(F) ⟶ ⋯

Now lim_{⟵}_{Q∈F} H^{1}(Q, ℂ^{×}) can be identified with the subset of Hom(P, ℂ^{×}) consisting of all

group homomorphisms α : P → ℂ^{×} such that [P, F] ⊆ Ker(α). Since F is perfect it follows that lim_{⟵}_{Q∈F} H^{1}(Q, ℂ^{×}) = {0} The long exact sequence above inplies that the map

in 4.1 is injective. □

Corollary 4.2. If F is a perfect fusion system on a finite p-group P such that m(F) = {0} then for any finite abelian p-group Z we have lim_{⟵}_{Q∈F} H^{2}(Q, Z) = {0}, or equivalently,

H^{2}(P, Z) contains no non zero F-stable element.

Proof. If Z is cyclic this is clear by 4.1, and in general this follows from exact sequences and 3.3. □

Corollary 4.3. Let G be a perfect finite group such that H^{2}(G, ℂ^{×}) is a p′-group, let P be a Sylow-p-subgroup of G and let F be a fusion system on P such that F_{P} (G) ⊆ F. Then F is perfect, we have m(F) = {0} and hence lim_{⟵}_{Q∈F} H^{2}(Q, Z) = {0} for any finite

abelian p-group Z.

Proof. Since G is perfect the set of commutators [u, x] with u ∈ P and x ∈ G such that [u, x] ∈ P generates P, and thus since F_{P} (G) ⊆ F the fusion system F is perfect, too. Again since F_{P} (G) ⊆ F, the group m(F) is contained in the p-part of H^{2}(G, ℂ^{×}), hence zero by the assumptions. The last statement follows then from 4.1. □

We refer to the [1, 1.5] for the definition of the direct product of two fusion systems occurring in the next statement, which is a reformulation of 4.3 making use of 2.1:

Corollary 4.4. Let G be a perfect finite group such that H^{2}(G, ℂ^{×}) is a p′-group, let P be a Sylow-p-subgroup of G and let F be a fusion system on P such that F_{P} (G) ⊆ F. If there is a central extension of finite p-groups

1 ⟶ Z ⟶ P̂ ⟶ P ⟶ 1

such that there exists a fusion system F̂ on P̂ satisfying F̂ = CF̂(Z) and F̂/Z = F then P̂ ≅ Z × P and F̂ ≅ F_{Z}(Z) × F.

The Solomon fusion system arising in [5] is a fusion system on a Sylow-2-subgroup P of Spin(7, q) which properly contains the fusion system F_{P} (Spin(7, q)) of the group Spin(7, q) on P. Since Spin(7, q) is a perfect group whose Schur multiplier is odd, Corollary 4.4 applies in this case. This fact is one of the ingredients used in Kessar [3] to show that the Solomon fusion system cannot occur as the category of Brauer pairs of a p-block of a finite group; this application has been the main motivation to develop the material in this note.

Appendix: fusion systems

Let p be a prime. Given a finite p-group P, a category on P is a category F whose objects are the subgroups of P and whose morphism sets Hom_{F}(Q, R) are sets of injective group homomorphisms from Q to R, where Q, R are subgroups of P, with the following properties:

• the composition of morphisms in F is given by the usual composition of group homomorphisms;

• if Q ⊆ R then the inclusion map from Q to R is a morphism in F;

• if φ : Q → R is a morphism in F then so is the induced isomorphism Q ≅ φ(Q) and its inverse.

If Q, R are subgroups of P we denote by Hom_{P} (Q, R) the set of group homomorphisms from Q to R induced by conjugation with elements in P, and we write Aut_{P} (Q) = Hom_{P} (Q, Q).

Given a category F on a finite p-group P and a subgroup Q of P we say that

- Q is fully F-normalised if |N_{P} (Q)| ≥ |N_{P} (Q′)| for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F;

- Q is fully F-centralised if |C_{P} (Q)| ≥ |C_{P} (Q′)| for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F;

- Q is F-centric if C_{P} (Q′) = Z(Q′) for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F;

- Q is F-radical if O_{p}(Aut_{F}(Q)) ⊆ Aut_{Q}(Q);

- Q is weakly F-closed if for every morphism φ : Q → P we have φ(Q) = Q;

- Q is strongly F-closed if for every morphism φ : R → P we have φ(R∩Q) = φ(R)∩Q;

Furthermore, if φ : Q → P is a morphism in F we denote by N_{φ} the subgroup of all elements y in N_{Q}(P) for which there exists an element z ∈ N_{P} (φ(Q)) such that φ(yuy^{−1}) = zφ(u)z^{−1} for all u ∈ Q. Note that QC_{P} (Q) ⊆ N_{φ} ⊆ N_{P} (Q).

A category F on a finite p-group P is called a fusion system if in addition the following properties hold:

• Hom_{P} (Q, R) ⊆ Hom_{F}(Q, R) for all subgroups Q, R of P;

• Aut_{P} (P) is a Sylow-p-subgroup of Aut_{F}(P);

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

The concept of a fusion system on a finite p-group is due to L. Puig; the above list of axioms uses simplifications of the original definition due to Broto, Levi, Oliver [1] and Stancu [6]. As observed in [6], the above definition implies that any fully F-normalised subgroup is fully F-centralised, from which follows in turn that the last axiom holds more generally whenever φ(Q) is fully F-centralised (which is the formulation in [1]). Another easy consequence of the axioms is that Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q) for any fully F-normalised subgroup Q of P; see e.g. [4, §1] for proofs.

If G is a finite group and P a Sylow-p-subgroup of G we denote by F_{P} (G) the category on P whose morphism sets are the group homomorphisms induced by conjugation with elements in G; that is, Hom_{F_{P} (G)}(Q, R) = Hom_{G}(Q, R) for any two subgroups Q, R of P. It is well-known and easy to verify that F_{P} (G) is a fusion system on P. In particular, F_{P} (P) is the fusion system on P consisting precisely of all morphisms given by conjugation with elements in P. For any fusion system F on P we have F_{P} (P) ⊆ F and F_{1}(1) ⊴ F.

One of the most fundamental properties of fusion systems is Alperin’s fusion theorem, which says that any ismorphism in a fusion system F on a finite p-group P can be written as the composition of isomorphisms φ : Q → R for which there is a radical F-centric subgroup S of P containing both Q, R and an automorphism α of S in F such that φ is the restriction to Q of α. See e.g. [1] or [7] for proofs and more precise statements.

Given a category F on a finite p-group P and a subgroup Q of P we define the category N_{F}(Q) on N_{P} (Q) by Hom_{N_{F} (Q)}(R, R′) = {φ : R → R′| φ extends to a morphism ψ : QR → QR′ in F such that ψ(Q) = Q}, for any two subgroups R, R′ of N_{P} (Q). Similarly, we define the category C_{F}(Q) on C_{P} (Q) by Hom_{C_{F} (Q)}(R, R′) = {φ : R → R′| φ extends to a morphism ψ : QR → QR′ in F such that ψ|_{Q} = Id_{Q}}.

We have clearly inclusions of categories C_{F}(Q) ⊆ N_{F}(Q) ⊆ F. If F = N_{F}(Q) for some subgroup Q of P, then clearly Q is strongly F-closed. The converse of this statement is not true, in general.

It has been observed by Puig that if Q is fully F-centralised, then C_{F}(Q) is a fusion system on C_{P} (Q); if Q is fully normalised, then N_{F}(Q) is a fusion system on N_{P} (Q).

In particular, if F is a fusion system on P such that F = C_{F}(Z) for some subgroup Z of Z(P) then the category on P/Z induced by F is a fusion system on P/Z, denoted by F/Z. In that case, if F′ is a fusion system on P contained in F we have F′ = F if and only if F′/Z = F/Z; this follows from Alperin’s fusion theorem together with the fact that if Q is a subgroup of P then the canonical map Aut_{F}(Q) → Aut_{F/Z}(Q/Z) has a p-group as kernel. Thus if Z ⊆ Q ⊆ P, then Q is F-centric radical if and only if Q/Z is F/Z-centric radical. In that case we have an isomorphism Aut_{F}(Q)/Aut_{Q}(Q) ≅ Aut_{F/Z}(Q/Z)/(Q/Z). Such a statement would not be true in general for all centric subgroups: if Q/Z is F/Z-centric, then Q is F-centric, but the converse need not hold.

References

1. C. Broto, R. Levi, R. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856.

2. D. Gorenstein, Finite Groups, Chelsea Publishing Co., New York, 1980.

3. R. Kessar, The Solomon fusion system is not the Brauer category of a block, preprint (2003).

4. M. Linckelmann, Simple fusion systems and the Solomon 2-local group, preprint (2002).

5. R. Solomon, Finite groups with Sylow-2-subgroups of type .3, J. Algebra 28 (1974), 182–198.

6. R. Stancu, Equivalent definitions of fusion systems.

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

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