Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000

The Thompson-Lyons transfer lemma for fusion systems

Justin Lynd

Abstract

A generalization of the Thompson transfer lemma and its various extensions, most recently due to Lyons, is proven in the context of saturated fusion systems. A strengthening of Alperin’s fusion theorem is also given in this setting, following Alperin’s own “up and down” fusion.

The classical Thompson transfer lemma appeared as Lemma 5.38 in [12]; for a 2-perfect group G with S ∈ Syl_{2}(G), it says that if T is a maximal subgroup of S and u is an involution in S − T, then the element u has a G-conjugate in T. Thompson’s lemma has since been generalized in a number of ways. Harada showed [9, Lemma 16] that the same conclusion holds provided one takes u to be of least order in S − T. Unpublished notes of Goldschmidt [6] extended this to show that one may find an G-conjugate of u in T which is extremal under the same conditions. An element t ∈ S is said to be extremal in S with respect to G if C_{S}(t) is a Sylow subgroup of C_{G}(t). In other words, ⟨t⟩ is fully F_{S}(G)-centralized, where F_{S}(G) is the fusion system of G. Later, Thompson’s result and its extensions were generalized to all primes via an argument of Lyons [7, Proposition 15.15]. We prove here a common generalization of Lyons’ extension and his similar transfer result [8, Chapter 2, Lemma 3.1] (which relaxes the requirement that S/T be cyclic) in the context of saturated fusion systems.

Theorem TL. Suppose that F is a saturated fusion system on the p-group S, and that T is a proper normal subgroup of S with S/T abelian. Let u ∈ S − T and let I be the set of fully F-centralized F-conjugates of u in S − T. Assume

(1) u is of least order in S − T and

(2) the set of cosets IT = {vT | v ∈ I} is linearly independent in Ω_{1}(S/T). If u ∈ 𝔣𝔬𝔠(F), then u has a fully F-centralized F-conjugate in T.

Since 𝔥𝔶𝔭(F) ⩽ 𝔣𝔬𝔠(F), an immediate consequence of the Theorem is that the conclusion holds in any fusion system with F = O^{p}(F). We refer to [2] and [4] for background on fusion systems. Of course, both conditions in Theorem TL are necessary. Take p = 2, F = F_{S}(G), S ∈ Syl_{2}(G), and for (1): G = SL_{3}(2), T a four group, and u ∈ S − T of order 4; while for (2): G = SL_{2}(3), T = Z(S) and u ∈ S − T.

In the classification of finite simple groups, direct applications of the transfer map are primarily those arising from extensions of Thompson’s lemma, as well as Yoshida’s Theorem on control of transfer. The latter has an analogue in fusion systems [5]. To the extent that history is a guide, Theorem TL should serve as a key ingredient in the analysis of simple fusion systems at the prime 2, especially in the component-type portion of a program for the classification of simple 2-fusion systems outlined recently by M. Aschbacher; see [2, Sections I.13-15]. Indeed, we postpone applications of Theorem TL to the companion [11], where we classify certain 2-fusion systems with an involution centralizer having a component based on a dihedral 2-group.

2000 Mathematics Subject Classification 20D20 (primary).

1. Bisets and notation

The proof we present is modeled on Lyons’ argument mentioned above, and relies on the transfer in saturated fusion systems. Transfer in the fusion system setting is defined by way of a characteristic biset associated to a saturated fusion system, which is an S-S biset Ω satisfying certain properties first outlined by Linckelmann and Webb [10]. We motivate these properties now while fixing notation; more details can be found in [4, Section 7.6].

We compose maps left-to-right, sometimes writing applications of injective homomorphisms in the exponent, as in s^{φ} when s ∈ S and φ ∈ Hom_{F}(⟨s⟩, S). For groups H and K, an H-K biset is a set X = _{H}X_{K} together with an action of H on the left and K on the right, such that (hx)k = h(xk) for each x ∈ X, h ∈ H, and k ∈ K. An H-K biset X may be regarded as a right (H × K)-set via x · (h, k) = h^{−1}xk. A transitive biset is a biset with a single orbit under the action of H × K.

Let G be a finite group and S a Sylow p-subgroup of G. Then G = _{S}G_{S} is an S-S biset with left and right action given by multiplication in G. The transitive subbisets of G are the S-S double cosets of G. For g ∈ G, the stabilizer in S × S of g is the graph subgroup

Δ_{^{g}S∩S}^{c_{g}} = { (s, s^{g}) | s ∈ ^{g}S ∩ S}

of S × S. Thus, the double coset

SgS ≅ (S × S)/Δ_{^{g}S∩S}^{c_{g}}

as a right (S × S)-set. In general, for P ⩽ S and an injective group homomorphism φ: P → S, we write Δ_{P}^{φ} for the graph subgroup { (t, t^{φ}) | t ∈ P }.

Let ψ: S → A be a map to an abelian group A, and T the kernel of ψ. The transfer map tr_{S,ψ}^{G}: G → A relative to ψ is the group homomorphism given by

u tr_{S,ψ}^{G} = ∏_{h∈[G/S]}([uh]^{−1}uh)ψ for u ∈ G,

where [G/S] is a set of representatives for the left cosets of S in G and [uh] ∈ [G/S] is the chosen representative for the coset uhS. Restricting tr_{S,ψ}^{G} to S and decomposing this product by the left action of S on the set of cosets G/S gives rise to the Mackey decomposition (cf. [7, Lemma 15.13]) of the transfer map:

u tr_{S,ψ}^{G} = ∏_{g∈[S\G/S]} u tr_{^{g}S∩S, c_{g}ψ}^{S} for u ∈ S,

where [S\G/S] is a set of representatives for the S-S double cosets in G and tr_{^{g}S∩S, c_{g}ψ}^{S} is the transfer of the composite

gS ∩ S ⟶^{c_{g}} S ∩ S^{g} ⟶^{ψ|_{S∩S^{g}}} (S ∩ S^{g})T/T.

Thus, tr_{S, ψ}^{G}|_{S} is determined by the collection of morphisms {c_{g}: ^{g}S ∩ S → S ∩ S^{g} | g ∈ [S\G/S]} in F_{S}(G).

Let P ⩽ S. For an S-S biset X, denote by _{S}X_{P} the set X considered as an S-P biset upon restriction on the right to P. More generally, for an injective group homomorphism φ: P → S, denote by _{S}X_{φ} the S-P biset with action s · x · t = sxt^{φ} for x ∈ X, s ∈ S, and t ∈ P. In the case of X = _{S}G_{S} and x ∈ G with φ = c_{x}, the map g ↦ gx gives an isomorphism of S-P bisets SG_{P} ≅ _{S}G_{c_{x}}. This property determines from fusion data the information that

u tr_{S,ψ}^{G} = (x^{−1}ux) tr_{S,ψ}^{G} for [u, x] ∈ S,

i.e. that S ∩ [G, G] ⩽ ker(tr_{S,ψ}^{G} |_{S}). So tr_{S,ψ}^{G} induces a group homomorphism S/𝔣𝔬𝔠(F_{S}(G)) → A.

Definition 1.1. Let F be a saturated fusion system on the p-group S. An S-S biset Ω is said to be a characteristic biset for F if

(a) for each transitive subbiset of Ω isomorphic to (S × S)/Δ_{P}^{φ} as a right (S × S)-set for some φ ∈ Hom_{F}(P, S),

(b) for each P ⩽ S and each φ ∈ Hom_{F}(P, S), the S-P bisets _{S}Ω_{P} and _{S}Ω_{φ} are isomorphic, and

(c) |Ω|/|S| is prime to p.

Theorem 1.2 [3, Proposition 5.5]. Each saturated fusion system has a characteristic biset.

When F = F_{S}(G) for a finite group G, the above discussion indicates one may take Ω = G. A characteristic biset is not uniquely determined by the fusion system; for instance, a disjoint union of a p′ number of copies of Ω also satisfies the Linckelmann-Webb properties (a)–(c) if Ω does. But for what follows, any characteristic biset will do.

Fix a saturated fusion system F over the p-group S, and let Ω = Ω_{F} be a characteristic biset for F. From 1.1(a), fix a decomposition of Ω into transitive S-S bisets:

Ω = ∐_{i∈I}(S × S)/Δ_{S_{i}}^{φ_{i}}.

where S_{i} ⩽ S and φ_{i} ∈ Hom_{F}(S_{i}, S) for each i ∈ I. For a an abelian group A and a group homomorphism ψ: S → A, define the transfer map relative to Ω to be the homomorphism tr_{Ω,ψ} : S → A given by

u tr_{Ω,ψ} = ∏_{i∈I} u tr_{S_{i},φ_{i}ψ}^{S} .

for u ∈ S, where tr_{S_{i},φ_{i}ψ}^{S} is the ordinary transfer. From 1.1(b),

ker(tr_{Ω,ψ}) ⩾ 𝔣𝔬𝔠(F).  (1.3)

2. Proof of Theorem TL

Proof of Theorem TL. Let ψ: S → S/T be the quotient map. Suppose that u has no fully F-centralized F-conjugate in T. We shall show that u lies outside the kernel of the transfer map tr_{Ω,ψ}. Once this is done, the Theorem follows from (1.3).

Without loss of generality we may assume that u itself is fully F-centralized. Let P = C_{S}(u). Applying the Mackey formula for, and the definition of, ordinary transfer, we have

u tr_{Ω,ψ} = ∏_{i∈I} u tr_{S_{i},φ_{i}ψ}^{S}

= ∏_{i∈I} ∏_{t∈[P \S/S_{i}]} u tr_{^{t}S_{i}∩P, c_{t}φ_{i}ψ}^{P}

= ∏_{i∈I} ∏_{t∈[P \S/S_{i}]} ∏_{r∈[P/^{t}S_{i}∩P]}(([ur]^{−1}ur)^{c_{t}φ_{i}})ψ

where [ur] is the representative in [P/^{t}S_{i} ∩ P] corresponding to the coset ur(S_{i}^{t} ∩ P). As P commutes with u, we have for an orbit O of ⟨u⟩ on the cosets P/^{t}S_{i} ∩ P, that Π_{r∈[O]}[ur]^{−1}ur = u^{|O|} mod T. Taking the product over these orbits,

u tr_{Ω,ψ} = ∏_{i∈I} ∏_{t∈[P \S/S_{i}]}(u^{c_{t}φ_{i}})^{|P:^{t}S_{i}∩P |} mod T.

Suppose i and t are such that the index |P : ^{t}S_{i} ∩ P | is divisible by p. Then the corresponding factor (u^{c_{t}φ_{i}})^{|P:S_{i}^{t}∩P |} has order less than that of u, so lies in T by assumption and contributes nothing to the transfer. On the other hand, |P : ^{t}S_{i} ∩ P | = 1 if and only if P ⩽ ^{t}S_{i}. In this case, φ_{i} is defined on P^{t} = C_{S}(u^{t}), and so the corresponding factor u^{c_{t}φ_{i}} is fully F-centralized. By assumption, u^{c_{t}φ_{i}} ∉ T. Write IT = {u_{j}T }_{j∈J} with u_{j} a fully F-centralized F-conjugate of u for each j ∈ J. Let

T = {(i, t) | i ∈ I, t ∈ [P \S/S_{i}], P ⩽ ^{t}S_{i}}.

Then by the above remarks,

u tr_{Ω,ψ} = ∏_{j∈J} u_{j}^{k_{j}}T

with ∑_{j∈J} k_{j} = |T |.

We finish by showing that the cardinality of the set T is prime to p. Now P ⩽ ^{t}S_{i} if and only if P fixes the left coset tS_{i} in its action from the left. Furthermore, Ω decomposes as a disjoint union of orbits of the form S/S_{i} as a right S-set. Therefore |T | = |(Ω/S)^{P} |, the number of P -fixed points in its left action on this set of orbits. Since |Ω/S| is prime to p by 1.1(c), it follows that |T | is also prime to p. As ∑_{j∈J} k_{j} = |T |, there exists j_{0} ∈ J with p ∤ k_{j_{0}}. By linear independence of {u_{j}T }_{j∈J}, we have u tr_{Ω,ψ} ≠ T and so u ∉ ker(tr_{Ω,ψ}), completing the proof.

When F = F_{S}(G) for some finite group G with Sylow p-subgroup S, we can specialize Theorem TL to obtain a generalization of Lyons’ results [7, Proposition 15.15] and [8, Lemma 2.3.1].

Corollary 2.1 (cf. [7, Proposition 15.15]). Let F be a saturated fusion system on a pgroup S with F = O^{p}(F). Suppose T is a proper normal subgroup of S with S/T cyclic, and let u be an element of least order in S − T. Assume that every fully F-centralized F-conjugate of u lies in the coset T or in uT. Then u has a fully F-centralized F-conjugate in T.

Note that if p = 2 in Corollary 2.1, then uT is the unique involution in the quotient S/T, and so the condition that each F-conjugate of u lies in T ∪ uT is automatically satsified.

Corollary 2.2 (cf. [8, Lemma 2.3.1]). Let F be a saturated fusion system on a finite 2-group S with F = O^{2}(F). Suppose T is a proper normal subgroup of S with S/T abelian. Let I be the set of fully F-centralized involutions in S − T, and suppose that the set IT = {vT | v ∈ I} is linearly independent in Ω_{1}(S/T). Then each involution u ∈ S − T has a fully F-centralized F-conjugate in T.

3. Up and down fusion

We close by pointing out a strengthening of Alperin’s fusion theorem for saturated fusion systems which may be useful when applying Theorem TL in the case where one has some knowledge of the structure of F-automorphism groups of subgroups of S in a given conjugation family C for F. This is motivated by Alperin’s observations in [1], and the proof is his.

In practice, one applies Theorem TL in a fusion system with F = O^{p}(F) to obtain conjugacy information (usually of involutions). The result of applying the transfer lemma is to get a morphism φ: C_{S}(u) → C_{S}(u^{φ}) with u^{φ} ∈ T. Sometimes it is advantageous to decompose this morphism into a composition of restrictions of automorphism groups of subgroups of S. In particular, by Alperin’s fusion theorem there will be some conjugate u_{1} of u, an F-centric subgroup P of S, and α ∈ Aut_{F}(P) such that u_{1} ∈ P, u_{1} ∉ T, and u_{1}^{α} ∈ T. This increased control comes at the cost that u_{1}^{α} may no longer be fully F-centralized. However, Proposition 3.1 below shows that a decomposition may be chosen so that the centralizers of intermediate conjugates always “go up”. In particular, the above α can always be chosen so that it extends to α: C_{S}(u_{1}) → C_{S}(u_{1}^{α}).

A conjugation family C for F is a set of subgroups Q of S such that every morphism in F is a composition of restrictions of elements in Aut_{F}(Q) as Q ranges over C. Examples are the set of fully F-normalized, F-centric, and F-radical subgroups of S, or the set {P ⩽ S | P is F-essential or P = S}. Given an isomorphism φ ∈ Hom_{F}(P, P′), say for short that a sequence (Q_{i}, α_{i})_{1⩽i⩽n} is an up (resp. down) C-decomposition of φ if Q_{i} ∈ C with α_{i} ∈ Aut_{F}(Q_{i}) and φ = α_{1} ⋯ α_{n} (after restriction of the α_{i}’s) and with |C_{S}(P_{i−1})| ⩽ |C_{S}(P_{i})| (resp. |C_{S}(P_{i−1})| ⩾ |C_{S}(P_{i})|) for all 1 ⩽ i ⩽ n. (Here, we set P_{0} = P and P_{i} = P^{α_{1}···α_{i}}.)

Proposition 3.1. Let F be a saturated fusion system on the p-group S, and suppose that C is a conjugation family for F. Let P, P′ ⩽ S, let φ: P → P′ be an isomorphism in F. Then there exists a C-decomposition (Q_{i}, α_{i})_{1⩽i⩽n} of φ together with an integer 0 ⩽ k ⩽ n such that if we set P_{i} = P^{α_{1}···α_{i}}, then

|C_{S}(P)| ⩽ |C_{S}(P_{1})| ⩽ ⋯ ⩽ |C_{S}(P_{k})| ⩾ ⋯ ⩾ |C_{S}(P_{n−1})| ⩾ |C_{S}(P′)|.

Proof. Let φ: P → P′ be an isomorphism in F. Is suffices to consider the case when P′ is fully F-centralized. Indeed, if ψ: P′ → P′′ is a map in F with |C_{S}(P′′)| maximal and there are up C-decompositions for ψ and φψ, then there is an up-down decomposition for φ = φψψ^{−1}.

Assume P′ is fully F-centralized. Fix an extension φ̃: P C_{S}(P) → P′C_{S}(P′) of φ and denote also by φ̃ the induced isomorphism P C_{S}(P) → (P C_{S}(P))φ̃. Let (Q_{i}, α_{i})_{1⩽i⩽n} be any Cdecomposition of φ̃. Set P_{0} = P, P_{i} = P_{i−1}^{α_{i}}, and C_{0} = C_{S}(P), C_{i} = C_{i−1}^{α_{i}}, for 1 ⩽ i ⩽ n. The proof is by induction on the index of C_{S}(P) in S.

Assume first that |C_{S}(P)| = |C_{S}(P′)|. We claim that

C_{i} = C_{S}(P_{i}) for each i

in this case, so that (Q_{i}, α_{i}) is already an up C-decomposition of φ. The case i = 0 holds by definition. If i ⩽ n and C_{i−1} = C_{S}(P_{i−1}), then as Q_{i} ⩾ P_{i−1}C_{S}(P_{i−1}), we have

C_{i} = C_{S}(P_{i−1})^{α_{i}} ⩽ C_{S}(P_{i}).

By induction then

|C_{S}(P)| = |C_{i−1}| ⩽ |C_{S}(P_{i})| ⩽ |C_{S}(P′)| = |C_{S}(P)|.

Therefore inequalities are equalities, and C_{i} = C_{S}(P_{i}) as claimed.

Now suppose |C_{S}(P)| < |C_{S}(P′)| and the proposition holds for all F-conjugates R of P′ with |S : C_{S}(R)| < |S : C_{S}(P)|. With notation as before, let 1 ⩽ l ⩽ n be the smallest integer such that |C_{S}(P_{l−1})| < |C_{S}(P_{l})|. Then |C_{S}(P)| = |C_{S}(P_{1})| = ⋯ = |C_{S}(P_{l−1}|, since C_{S}(P_{i−1})^{α_{i}} ⩽ C_{S}(P_{i}) for each i. By the inductive hypothesis, there is an up C-decomposition (Q′_{i}, α′_{i})_{l⩽i⩽n} for α_{l} ⋯ α_{n} : P_{l} → P′, and this completes the proof of the proposition.

Remark. Proposition 3.1 does not hold if centralizers are replaced by the normalizers in S of the p-subgroups. The point is that there need not exist a decomposition (Q_{i}, α_{i})_{1⩽i⩽n} of

Page 6 of 6 THOMPSON-LYONS TRANSFER LEMMA

an automorphism φ ∈ Aut(P) with the property P_{i} = P^{α_{1}···α_{i}} are all fully F-normalized for 1 ⩽ i ⩽ n.

As an example, let H_{i} ≅ S_{4} for i = 1, 2 and set G = H_{1} × H_{2}. Let S_{i} ∈ Syl_{2}(H_{i}), S = S_{1} × S_{2}, and V = O_{2}(G). The essential subgroups of F_{S}(G) are V S_{1} and V S_{2}. Let h_{i} ∈ N_{H_{i}}(V S_{3−i}) of order 3, and let P be the four subgroup of V which is normalized by h := h_{1}h_{2} and with N_{S}(P) of index 2 in S. Then |P^{F}| = 3 and N_{S}(R) = V for R ∈ P^{F} − {P }. Thus, (V S_{3−i}, c_{h_{i}})_{1⩽i⩽2} is the only essential decomposition of φ = c_{h} ∈ Aut_{F_{S}(G)}(P) and whereas P is fully F-normalized, P^{h_{1}} is not.

Acknowledgments

This work forms part of the author’s Ph.D. thesis at Ohio State under the direction of Ron Solomon, and was supported in part by a fellowship from the Ohio State University Graduate School. The author would also like to thank Richard Lyons for his comments and suggestions.

References

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6. David M. Goldschmidt. A transfer lemma. unpublished notes.

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10. Markus Linckelmann and Peter Webb. Unpublished notes.

11. Justin Lynd. A characterization of the 2-fusion system of L_{4}(q). 2013. 37pp., submitted.

12. John G. Thompson. Nonsolvable finite groups all of whose local subgroups are solvable. Bull. Amer. Math. Soc., 74:383–437, 1968.

Justin Lynd Department of Mathematics Rutgers University 110 Frelinghuysen Rd Piscataway, NJ 08854 USA

jlynd@math.rutgers.edu