VARIETY ISOMORPHISM IN GROUP COHOMOLOGY AND CONTROL OF p–FUSION

DAVID BENSON, JESPER GRODAL, AND ELLEN HENKE

Abstract. We show that if an inclusion of finite groups H ≤ G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F –isomorphism in mod p cohomology, then H controls p–fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p = 2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories.

The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p–fusion systems over the same finite p–group are detected on elementary abelian p–groups if p odd and abelian 2–groups of exponent at most 4 if p = 2.

1. Introduction

The variety of the mod p cohomology ring of a finite group was first studied by Quillen in his fundamental 1971 paper [41], and has been a central tool in group cohomology since then. The variety describes mod p group cohomology rings up to F –isomorphism, i.e., ring homomorphisms with nilpotent kernel and where every element in the target raised to a p^{k}th power lies in the image; see [41, App. B].

Quillen’s first application of the theory was to show in [40] that if the Sylow p–subgroup inclusion S ≤ G induces an F –isomorphism on mod p cohomology then S controls p–fusion in G, if p is odd, which in this case means that G is p–nilpotent. Quillen’s result has subsequently been revisited in a number of contexts [23, 12, 19, 28, 13], however all retaining the hypothesis that S is a Sylow p–subgroup in G.

The main goal of this paper is to considerably strengthen Quillen’s result by replacing S by an arbitrary subgroup H of G containing S, thereby moving from p–nilpotent groups to all finite groups. We recall that for S ≤ H ≤ G, H is said to control p–fusion in G, if pairs of tuples of elements of S are conjugate in H if they are conjugate in G, or equivalently if for all p–subgroups P, Q ≤ S, N_{H}(P, Q)/C_{H}(P) equals N_{G}(P, Q)/C_{G}(P) as homomorphisms from P to Q.

Theorem A (F –isomorphism implies control of p–fusion, p odd). Let ι: H ≤ G be an inclusion of finite groups of index prime to p, p an odd prime. If for each x ∈ H^{∗}(H; 𝔽_{p}), x^{p^{k}} ∈ im(ι^{∗}) for some k ≥ 0, then H controls p–fusion in G, and ι^{∗} is an isomorphism.

Theorem A contains the celebrated Mislin’s theorem [37] for p odd as a special case, i.e., the statement that if ι^{∗} is an isomorphism, then H controls p–fusion in G. In fact our proof gives a new, and comparatively simple, algebraic proof of this theorem for p odd as well—see Remark 5.4 for a summary of the extensive history surrounding this result. Recall that the converse, that ι^{∗} is an isomorphism if H controls p–fusion in G, is the classical 1956 Cartan–Eilenberg formula for stable elements [14, XII.10.1]. The assumption in Theorem A that H and G share a common Sylow p–subgroup is necessary as the inclusion C_{p} → C_{p^{2}} shows—we will however return to this issue below in Theorem C and D.

2000 Mathematics Subject Classification. 20J06 (20D20, 20J05, 55P60).

All three authors were supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. The second author was also supported by an ESF EURYI grant.

We deduce Theorem A from the following purely group theoretic statement, which is of independent interest—for p odd it says that if H controls p–fusion in G on elementary abelian p–subgroups then it in fact controls p–fusion. We formulate and prove the statement in terms of fusion systems, and refer the reader for example to [2] for definitions and information about these—we also recap the essential definitions in Section 2.

Theorem B (Small exponent abelian p–subgroups control p–fusion). Let G ≤ F be two saturated fusion systems on the same finite p–group S. Suppose that Hom_{G}(A, B) = Hom_{F}(A, B) for all A, B ≤ S with A, B elementary abelian if p is odd, and abelian of exponent at most 4 if p = 2. Then G = F.

Our proof of this theorem is rather short and only relies essentially on classical results of J. G. Thompson on p′–automorphisms of p–groups [20, Ch. 5.3], but nevertheless the only previous results in this direction, even for groups, appear to be in the rather limited p–nilpotent case [19, 28].

For p = 2, Theorem A does not hold. The standard counterexample, going back to Quillen, is the Sylow inclusion Q_{8} < 2A_{4} = Q_{8} ⋊C_{3}. In fact it fails more severely: Let Nil_{n} be the class of n– nilpotent unstable modules over the Steenrod algebra, defined for example in [46, Ch. 6], and recall that F –isomorphism can be rephrased as isomorphism modulo Nil_{1}. We show in Example 4.3 that for any n there exists inclusions H ≤ G which induce an Nil_{n}–equivalence but where H does not control 2–fusion in G—see Section 4 for much more information.

Switching from mod p cohomology to generalized cohomology theories, we can however combine the methods of Theorem B with Hopkins–Kuhn–Ravenel (HKR) generalized character theory [26, 27] to obtain a statement that holds for all primes, and that also avoids the assumption that H and G share a common Sylow p–subgroup.

Theorem C (Chromatic group cohomology isomorphism implies control of p–fusion). Let φ: H → G be a homomorphism of finite groups, and let E(n) denote height n Morava E–theory at an arbitrary prime p (i.e., with E(n)^{∗} = W (𝔽_{p^{n}})⟦w_{1}, …, w_{n−1}⟧[u, u^{−1}]), or any other height n cohomology theory from [27, Thm. C]. Suppose that φ induces an isomorphism

φ^{∗}: E(n)^{∗}(BG)[\frac{1}{p}] ⟶^{≃} E(n)^{∗}(BH)[\frac{1}{p}]

for some positive n ≥ rk_{p}(G). Then | ker(φ)| and |G : φ(H)| are prime to p, and φ(H) controls p–fusion in G.

As usual, the notation [\frac{1}{p}] means that we invert p after taking cohomology, producing a ℚ– algebra. Since a mod p homology isomorphism of spaces induce an E(n)^{∗}–isomorphism, Theorem C again provide a new proof of Mislin’s theorem, now valid at all primes, but at the expense of replacing the reliance on Quillen’s variety theory by the (currently) less algebraic HKR character theory. Indeed our result can be seen as an analog of Atiyah’s 1961 p–nilpotence criterion [40, 3] replacing ordinary p-adic K–theory (corresponding to n = 1) by higher chromatic K–theories; see Remark 5.1.

A p–rank restriction in Theorem C is indeed necessary: 𝔽_{p^{2}} ⋊ 𝔽_{p^{2}}^{×} < 𝔽_{p^{2}} ⋊ 𝔽_{p^{2}}^{×} ⋊ Aut(𝔽_{p^{2}}) is an example of an inclusion of groups of index prime to p, for p odd, which is an E(1)^{∗}[\frac{1}{p}]–equivalence, by HKR-character theory (3.3), but with different p–fusion; the same example with 𝔽_{p^{2}} replaced by 𝔽_{2^{3}} works for p = 2. We speculate that the bound n ≥ rk_{p}(G) we give may be close to optimal, but we currently do not know an example to this effect.

Finally, we remark that isomorphism on E(n)^{∗} is equivalent to having isomorphism on Morava K–theory K(n)^{∗}, whereas an E(n)^{∗}[\frac{1}{p}]–isomorphism is a priori much weaker. It corresponds to having homeomorphism only on a certain “zeroth pure stratum” on the variety of E(n)–cohomology in the sense of Greenlees–Strickland [22]—see Remark 3.5 for this geometric interpretation and Remark 3.6 for the relationship to general localization results of Bousfield and others.

VARIETY ISOMORPHISM AND CONTROL OF p–FUSION 3

To prove Theorem C we need the following variant of Theorem B, which again appears to be new, even in special cases.

Theorem D (Abelian p–subgroups control fusion). Assume that a finite group homomorphism φ: H → G induces a bijection

Rep(A, H) ⟶^{≃} Rep(A, G)

for all finite abelian p–groups A with rk_{p}(A) ≤ rk_{p}(G). Then | ker(φ)| and |G : φ(H)| are prime to p, and φ(H) controls p–fusion in G. More generally, suppose that (T, G) and (S, F) are saturated fusion systems and T → S is a fusion preserving homomorphism that induces a bijection Rep(A, G) ⟶^{≃} Rep(A, F) for any finite abelian p–group A, then (T, G) ⟶^{≃} (S, F).

Here Rep(A, G) denotes the quotient of Hom(A, G) where we identify φ with c_{g} ∘φ for all g ∈ G, and likewise Rep(A, F) is the quotient of Hom(A, S), identifying two morphisms if they differ by a morphism in F; we spell out what the assumption of the theorem means in Remark 2.6.

Organization. In Section 2 we give a proof of the algebraic Theorems B and D, which we use in Section 3 to prove Theorems A and C. In Section 4 we briefly examine how far Theorem A extends to p = 2 using Lannes’ T –functor techniques, and in the final Section 5 we make some concluding remarks, and further clarify the relationship with the existing literature.

Acknowledgement. Our interest in Theorem A was piqued by a discussion during the problem session at the August 2011 workshop on Homotopical Approaches to Group Actions in Copenhagen with Peter Symonds and others, and also stimulated by [13]. The project started as a collaboration between the first two authors at the Centre for Symmetry and Deformation, Copenhagen in early spring 2012, and was joined by the third author when she completed the proof of Theorem B, following a lunchtime conversation. We thank Mike Hopkins for pointers concerning the relationship between Theorem C and classical stable homotopy theory, explained in Remark 3.6, and Lucho Avramov, Nick Kuhn, and Neil Strickland for other literature references.

2. p′–automorphisms of p–groups and proofs of Theorems B and D

The goal of this section is to prove Theorems B and D by group theoretic methods, combining manipulations with fusion systems with now textbook results of J. G. Thompson on p′– automorphisms of p–groups.

Thompson’s so-called critical subgroup theorem [18, Lem. 2.8.2] (see also [20, Thm. 5.3.11]) says that for any finite p–group P there exists a characteristic subgroup C of P such that C/Z(C) is elementary abelian, [P, C] ≤ Z(C), C_{P} (C) = Z(C), and every nontrivial p′–automorphism of P induces non-trivial p′–automorphisms of C. Our main classical group theoretic tool in this paper is a variant of that theorem, where instead of a critical subgroup we use a certain characteristic subgroups of P of small exponent and consider its maximal abelian subgroups.

Theorem 2.1 (Small exponent abelian subgroups detect p′–automorphisms). Let P be a finite p–group. There exists a characteristic subgroup D of P, of exponent p when p is odd and exponent 4 if p = 2, such that [D, P] ≤ Z(D), and such that every non-trivial p′–automorphism of P induces a non-trivial automorphism of D. Furthermore, for any such subgroup D and any maximal abelian subgroup A of D (with respect to inclusion) A ⊴ P and C_{Aut(P)}(A) is a p–group.

Note that for example the extra-special group p_{+}^{1+2} shows that an abelian characteristic subgroup that detects p′–automorphisms may not exist.

Proof of Theorem 2.1. Taking D = Ω_{1}(C), the subgroup generated by elements of order p of a critical subgroup C, produces such a subgroup D as in the theorem, for p odd, as proved in [20, Thm. 5.3.13]. For p = 2 the claim holds for D = Ω_{2}(C), the subgroup of C generated by elements of order at most 2^{2}; we establish this fact in Lemma 2.2 below, as we have not found this statement previously in the literature.

For the last part, let A be a maximal abelian subgroup of D with respect to inclusion. Since [A, P] ≤ Z(D) ≤ A it follows that A ⊴ P. Furthermore, if B ≤ C_{Aut(P)}(A) is a p′–group, then A × B acts on P and thus on D. Since A is maximal abelian it follows that C_{D}(A) = A, and in particular B acts trivially on C_{D}(A); Thompson’s A × B–lemma [20, Thm. 5.3.4] now says that [D, B] = 1 and so B = 1, and we conclude that C_{Aut(P)}(A) is a p–group as wanted. □

We now provide a proof of the postponed lemma for p = 2.

Lemma 2.2. Let P be a 2–group such that P/Z(P) is elementary abelian. Then for all x, y ∈ P, (xy)^{4} = x^{4}y^{4} and in particular Ω_{2}(P) is of exponent at most 4. Furthermore if B is a p′–group of automorphisms of P with [Ω_{2}(P), B] = 1, then then B = 1.

Proof. The proof is a variant of the p odd statements in [20, Ch. 5.3]. The first part is similar to [20, Lem. 5.3.9]: Let x, y ∈ P and set z = [y, x]. A small calculation shows that (xy)^{4} = z^{6}x^{4}y^{4}, and z^{2} = 1 so (xy)^{4} = x^{4}y^{4}. Hence the first part follows. (For general commutator formulas, see [20, Lem. 2.2.2].)

For the last part we follow [20, Thm. 5.3.10]: Let P be a minimal counterexample to the last statement. If Q is a proper B–invariant subgroup of P then Q/(Q ∩ Z(P)) is elementary abelian and Q ∩ Z(P) ≤ Z(Q), so Q/Z(Q) is elementary abelian. Moreover, Ω_{2}(Q) ≤ Ω_{2}(P) and thus [Ω_{2}(Q), B] = 1. So, as P is a minimal counterexample, [Q, B] = 1. In particular, [P′, B] = 1, since P′ is a proper B–invariant subgroup of P. Furthermore, by [20, Thm. 5.3.7(ii)], P/P′ is elementary abelian. We now show that [P, B] ≤ Ω_{2}(P). Let x ∈ P and a ∈ B. Then by the first part of the lemma ([x, a])^{4} = (x^{−1}x^{a})^{4} = x^{−4}(x^{4})^{a}. As P/P′ is elementary abelian, x^{4} ∈ P′ and thus, as [P′, B] = 1, (x^{4})^{a} = x^{4}. We conclude that ([x, a])^{4} = 1 and so [x, a] ≤ Ω_{2}(P). This shows that [P, B] ≤ Ω_{2}(P). By assumption, [Ω_{2}(P), B] = 1, so by [20, Thm. 5.3.6], [P, B] = 1. □

In the case where F is the fusion system of a semidirect product G = S ⋊ K, with p ∤ |K|, Theorem B follows directly from Theorem 2.1, as the action of elements of K on S is detected by small abelian subgroups of S. The general case requires more work, for which we use fusion systems; we however do not rely on any non-trivial theorems about fusion systems, and our arguments can easily be translated into the setup of ordinary finite groups. However the group case is not essentially easier and we indeed feel that the fusion system setup makes the arguments more transparent, once the initial language barrier is overcome.

Recall that a saturated fusion system F on a finite p–group S [10, Def. 1.2][2, Prop. I.2.5] is a category whose objects are the subgroups of S, and morphisms are group monomorphims satisfying axioms which mimic those satisfied by morphisms induced by conjugation in some ambient group G: conjugation by elements in S need to be in the category, every map needs to factor as an isomorphism followed by an inclusion, and furthermore two non-trivial conditions need to be satisfied, called the Sylow and extension axiom, which we recall below together with some terminology. We refer to the book [2] and the original references [39, 10] for detailed information. A subgroup Q ≤ S is called fully F–normalized if |N_{S}(Q)| is maximal among F–conjugates of Q, it is called fully F–centralized if the corresponding property holds for the centralizer, and it is called F–centric if C_{S}(Q′) = Z(Q′) for all F-conjugates Q′ of Q. The Sylow axiom says that if Q is fully F-normalized then it is fully F-centralized and Aut_{S}(Q) is a Sylow p-subgroup of Aut_{F}(Q). (Here Aut_{S}(Q) means the automorphisms of Q induced by elements in S.) The extension axiom says that any morphism φ: Q → S with φ(Q) fully F-centralized extends to

N_{φ} = {g ∈ N_{S}(Q)|^{φ}(c_{g}|_{Q}) ∈ Aut_{S}(φ(Q))}.

The first tool we need is the following variant of the extension axiom.

Lemma 2.3. Fix a saturated fusion system F on S and let φ: P → S be any monomorphism (not necessarily in F). For Q ⊴ P and ψ = φ|_{Q} the following hold.

(1) For all g ∈ P, ^{ψ}(c_{g}|_{Q}) = c_{φ(g)}|_{φ(Q)}, and hence P ≤ N_{ψ} and ^{ψ}Aut_{P} (Q) = Aut_{φ(P)}(φ(Q)).

(2) If ψ ∈ F and φ(Q) is fully F–centralized then ψ extends to ψ̂ ∈ Hom_{F}(P, φ(P)C_{S}(φ(Q))).

Proof. For (1) calculate (^{ψ}c_{g})(x) = ψ ∘ c_{g} ∘ ψ^{−1}(x) = ψ(gψ^{−1}(x)g^{−1}) = φ(g)xφ(g^{−1}) = c_{φ(g)}(x) for any x ∈ φ(Q), from which it is clear that P ≤ N_{ψ} and ^{ψ} Aut_{P} (Q) = Aut_{φ(P)}(φ(Q)).

For (2) note that the extension axioms imply that ψ extends to ψ̂ ∈ Hom_{F}(P, S). And, since Aut_{φ(P)}(φ(Q)) = ^{ψ} Aut_{P} (Q) = Aut_{ψ̂(P)}(φ(Q)), where the last equality is by applying (1) with ψ̂

in place of φ, we conclude that ψ̂(P) ≤ φ(P)C_{S}(φ(Q)) as wanted. □

For the purpose of the next proof, recall that a proper subgroup H of a finite group G is called strongly p-embedded if p divides the order of H and, for all g ∈ G\H, H ∩ ^{g}H has order prime to p. Provided p divides |G|, one easily shows that H is strongly p–embedded in G if and only if H contains a Sylow p–subgroup S of G such that N_{G}(R) ≤ H for every 1 ≠ R ≤ S (see for example [21, Lem. 17.10] or [42, Prop. 5.2]); in particular an overgroup of a strongly p–embedded subgroup is again strongly p–embedded, if it is a proper subgroup. (Groups with strongly embedded subgroups play a central role in many aspects of local group theory, and in particular they show up in connection with Alperin’s fusion theorem [2, Thm. I.3.6], though we shall only indirectly need them in that capacity here.)

We now give the key step in deducing Theorem B from Theorem 2.1.

Main Lemma 2.4. Let G ≤ F be two saturated fusion systems on the same finite p–group S, and P an F–centric and fully F–normalized subgroup, with Aut_{F}(R) = Aut_{G}(R) for every P < R ≤ N_{S}(P).

Suppose that there exists a subgroup Q ⊴ P with Hom_{F}(Q, Q′) = Hom_{G}(Q, Q′) for all F– conjugates Q′ of Q. Then Aut_{F}(P) = ⟨Aut_{G}(P), C_{Aut_{F} (P)}(Q)⟩.

Proof. To ease the notation set G = Aut_{F}(P), H = Aut_{G}(P), and \overline{G} = G/ Inn(P), and denote by \overline{U} the image of any subgroup U ≤ G in \overline{G}. We want to show that G = ⟨H, C_{G}(Q)⟩. Step 1: We first assume in addition that

C_{S}(ξ(Q)) ≤ P for all ξ ∈ H  (∗)

and show that G = HC_{G}(Q). Let γ ∈ G be arbitrary; set ψ = (γ^{−1})|_{γ(Q)} ∈ Hom_{F}(γ(Q), Q). Then ψ ∈ Hom_{G}(γ(Q), Q) by assumption. We claim that Q is fully centralized in G, and postpone the proof to Lemma 2.5 below, since it is a general statement. Granted this, Lemma 2.3(2) implies that we can extend ψ: γ(Q) → Q to ψ̂: P → P C_{S}(Q) in G, and, as C_{S}(Q) ≤ P by assumption (∗), we conclude that ψ̂ ∈ H. Since γ ∘ψ̂ ∈ C_{G}(Q), we have γ ∈ C_{G}(Q)H, and, as γ was arbitrary, this yields G = HC_{G}(Q) as required. Step 2: If P = S assumption (∗) is automatically satisfied and the lemma follows from Step 1; likewise we are done if H = G. In this step we show that if P < S and H < G then H̄ is strongly p–embedded in Ḡ. Consider P < R ≤ N_{S}(P). For φ ∈ N_{G}(Aut_{R}(P)) it follows from the extension axiom that φ extends to φ̂ ∈ Hom_{F}(R, S), since P is fully F–normalized and R ≤ N_{φ}, cf. Lemma 2.3. Furthermore, by Lemma 2.3(1), Aut_{R}(P) = ^{φ} Aut_{R}(P) = Aut_{φ̂(R)}(P), so since C_{S}(P) ≤ P by F–centricity of P, we have φ̂(R) = R. It then follows from our hypothesis that φ̂ ∈ Aut_{G}(R) and thus φ ∈ H. We conclude that \overline{H} is strongly p–embedded in \overline{G}. Step 3: Finally set H_{0} = ⟨H, C_{G}(Q)⟩, and suppose for contradiction that there exists χ ∈ G \ H_{0}. Then by Step 2, \overline{H}_{0} is a strongly p–embedded subgroup of \overline{G}, so in particular Aut_{S}(P) ∩^{χ}H_{0} = Inn(P) as Aut_{S}(P) ≤ H_{0}. Note that C_{G}(χ(Q)) = ^{χ}C_{G}(Q) ≤ ^{χ}H_{0}, so C_{Aut_{S}(P)}(χ(Q)) ≤ Aut_{S}(P)∩ χH_{0} = Inn(P). Hence C_{S}(χ(Q)) ≤ P, using that P is centric. Note that ξ ∘ χ ∈ G\H_{0} for any ξ ∈ H; so as χ was arbitrary the argument actually shows that C_{S}(ξ(χ((Q)))) ≤ P for any ξ ∈ H. But now (∗) holds with χ(Q) in place of Q, and Step 1 reveals that G = HC_{G}(χ(Q)). As C_{G}(χ(Q)) is conjugate to C_{G}(Q) in G, it follows that C_{G}(χ(Q)) is conjugate to C_{G}(Q) under an element of H and thus G = HC_{G}(Q). This is a contradiction, and we conclude that G = H_{0} as wanted. □

We next prove the postponed lemma using the following well-known consequence of the extension axiom: If Q ≤ S and R is a fully F–normalized F–conjugate of Q, then there exists α ∈ Hom_{F}(N_{S}(Q), N_{S}(R)) with α(Q) = R; see [33, Lem. 2.6].

Lemma 2.5. Let F be a saturated fusion system on S and suppose that Q ⊴ P ≤ S, with P fully F–normalized, and C_{S}(ξ(Q)) ≤ P for all ξ ∈ Aut_{F}(P). Then Q is fully F–centralized.

Proof. As stated above we may choose α: N_{S}(Q) → S in F such that α(Q) is fully normalized. Furthermore, as P is fully normalized, there is β ∈ Hom_{F}(N_{S}(α(P)), N_{S}(P)) such that β(α(P)) = P, again using the fact above. Then

β(C_{S}(α(Q)) ∩ N_{S}(α(P))) ≤ C_{S}(β(α(Q))) ≤ P = β(α(P))

where the second inclusion follows by assumption as β ∘α restricts to an element of Aut_{F}(P). This yields C_{S}(α(Q)) ∩ N_{S}(α(P)) ≤ α(P), so by a standard fact about p–groups, C_{S}(α(Q)) ≤ α(P). Hence, C_{S}(α(Q)) = C_{α(P)}(α(Q)) = α(C_{P} (Q)) = α(C_{S}(Q)) where the last equality holds since our assumption gives C_{S}(Q) ≤ P. It follows |C_{S}(α(Q))| = |α(C_{S}(Q))| = |C_{S}(Q)|; so Q is fully F–centralized as α(Q) is fully F–centralized. □

Proof of Theorem B. By Alperin’s fusion theorem, F is generated by the F–automorphism of fully F–normalized and F–centric subgroups; see [2, Thm. I.3.6] (in fact we only need “F–essential” subgroups and S). We want to show that Aut_{G}(P) = Aut_{F}(P) for any P ≤ S; by downward induction on the order we can assume that Aut_{G}(R) = Aut_{F}(R) for all subgroups R ≤ S with |R| > |P |, and by the fusion theorem we can furthermore assume that P is F–centric and fully F–normalized. Now choose a characteristic subgroup D of P as described in Theorem 2.1, and a maximal abelian subgroup A of D. Recall that C_{Aut_{F} (P)}(A) is a p–group and, as P is fully F–normalized, Aut_{S}(P) is a Sylow p–subgroup of Aut_{F}(P). So if we replace A by a conjugate of A under Aut_{F}(P), we can arrange that C_{Aut_{F} (P)}(A) ≤ Aut_{S}(P) ≤ Aut_{G}(P). But A also satisfies the assumptions on Q in Lemma 2.4, so Aut_{F}(P) = ⟨Aut_{G}(P), C_{Aut_{F} (P)}(A)⟩, and we conclude that Aut_{G}(P) = Aut_{F}(P) as wanted. □

We now head towards a proof of Theorem D. Recall that for Q a group and F a fusion system on S we define Rep(Q, F) = Hom(Q, S)/F as the quotient of Hom(Q, S) under F–conjugation, i.e., where we identify φ ∈ Hom(Q, S) with α ∘ φ, for all α ∈ Hom_{F}(φ(Q), S). The proof of Theorem D reduces quickly to the case that G is a subsystem of F. We first make explicit what the assumption in Theorem D then means.

Remark 2.6. Let (T, G) be a sub-fusion system of a fusion system (S, F), and Q a group. The induced map Rep(Q, G) → Rep(Q, F) is surjective if and only if every epimorphic image of Q in S is F–conjugate to a subgroup of T. It is injective, if and only if G controls fusion on the epimorphic images of Q in T, i.e., for any epimorphic image Q′ ≤ T of Q we have Hom_{F}(Q′, T) = Hom_{G}(Q′, T).

The next lemma, together with Theorem B, will easily imply Theorem D.

Lemma 2.7. Let (T, G) ≤ (S, F) be two saturated fusion systems, and suppose that there exists an F–centric subgroup Q of T with Hom_{F}(Q, T) = Hom_{G}(Q, T). Then Aut_{F}(T) = Aut_{G}(T) and T = S.

Proof. As T is a finite p–group, there is a finite chain

Q = T_{0} ◁ T_{1} ◁ ⋯ ◁ T_{n} = T

with T_{i+1} = N_{T} (T_{i}) for 0 ≤ i < n. Note that, as Q is F–centric, every T_{i} is F–centric and thus also G–centric. We want to show that Aut_{F}(T) = Aut_{G}(T) by proving that

Hom_{F}(T_{i}, T) = Hom_{G}(T_{i}, T), for all 0 ≤ i ≤ n,  (∗∗)

by induction on i. For i = 0 the claim is true by assumption. Let now 0 ≤ i < n such that Hom_{F}(T_{i}, T) = Hom_{G}(T_{i}, T). Let γ ∈ Hom_{F}(T_{i+1}, T). Then ψ = γ|_{T_{i}} ∈ Hom_{F}(T_{i}, T) = Hom_{G}(T_{i}, T). As T_{i} is G–centric, γ(T_{i}) is fully G–centralized and C_{T} (γ(T_{i})) ≤ γ(T_{i}); so by Lemma 2.3(2), ψ extends to ψ̂ ∈ Hom_{G}(T_{i+1}, γ(T_{i+1})). Then ψ̂^{−1} ∘γ ∈ C_{Aut_{F} (T_{i+1})}(T_{i}). By [10, Prop. A.8] C_{Aut_{F} (T_{i+1})}(T_{i}) = Aut_{Z(T_{i})}(T_{i+1}) (a different reference is [2, Lem. 5.6]; one can also give a direct proof). We conclude that ψ̂^{−1} ∘ γ ∈ C_{Aut_{F} (T_{i+1})}(T_{i}) = Aut_{Z(T_{i})}(T_{i+1}) ≤ Aut_{G}(T_{i+1}) and thus γ ∈ Hom_{G}(T_{i+1}, T), i.e., (∗∗) holds. So Aut_{G}(T) = Hom_{G}(T, T) = Hom_{F}(T, T) = Aut_{F}(T). If Aut_{F}(T) = Aut_{G}(T) then, in particular, Aut_{F}(T)/ Inn(T) has order prime to p, by the Sylow axiom for (G, T), and so Aut_{S}(T) = Inn(T). Since Q ≤ T is F–centric, this implies that N_{S}(T) = T, and thus S = T. □

Proof of Theorem D. We only prove the claim about fusion systems, since the claim about groups is a special case of that. First, it is obvious that T → S has to be a monomorphism, since if an element is conjugate to the trivial element, then it is indeed trivial. Hence, we may consider G as a subsystem of F. Choose a subgroup A ≤ T such that A is of maximal order among the abelian subgroups of T. As explained in Remark 2.6 the assumption in Theorem D implies that every abelian subgroup of S is F–conjugate to a subgroup of T, and thus A is of maximal order among the abelian subgroups of S. This implies that A is F–centric. Furthermore, by assumption, Hom_{G}(A, T) = Hom_{F}(A, T), see Remark 2.6, so Lemma 2.7 shows that T = S. This reduces us to a special case of the setup of Theorem B, and the result follows. □

3. Proofs of Theorems A and C using Theorems B and D

Proof of Theorem A. By our main Theorem B we just need to verify that an F –isomorphism on cohomology rings implies that H controls G–fusion on elementary abelian p–groups. However this is the statement of [40, Prop. 10.9] (see also [1, 51]). Finally, for the last part, the Cartan– Eilenberg stable elements formula [14, XII.10.1] shows that if H controls p–fusion in G then

H^{∗}(G; 𝔽_{p}) ⟶^{≃} H^{∗}(H; 𝔽_{p}). □

Before proving Theorem C we state a lemma explaining the condition on n, whose proof is elementary and seems best left to the reader. Below ℤ_{p} denotes the p–adic integers.

Lemma 3.1. For φ: H → G a morphism of finite groups and a fixed natural number n, Rep(ℤ_{p}^{n}, H) ⟶^{≃} Rep(ℤ_{p}^{n}, G) if and only if Rep(A, H) ⟶^{≃} Rep(A, G) for all finite abelian p–groups A with rk_{p}(A) = n. Furthermore, the isomorphism for a fixed positive n ≥ min{rk_{p}(G), rk_{p}(H) + 1} implies rk_{p}(G) = rk_{p}(H) and isomorphism for all n. □

In further preparation for the proof of Theorem C, we briefly recall the HKR character theorem [27, Thm C]: For any multiplicative cohomology theory E and finite group G, taking E^{∗}– cohomology induces a map

Rep(ℤ_{p}^{n}, G) → Hom_{E^{∗}-alg}(E^{∗}(BG), E_{cont}^{∗}(Bℤ_{p}^{n}))

with E_{cont}^{∗}(Bℤ_{p}^{n}) = colim_{r} E^{∗}(B(ℤ/p^{r})^{n}), since any α: ℤ_{p}^{n} → G factors canonically through (ℤ/p^{r})^{n} for r large. By adjunction we can view this as an E^{∗}–algebra homomorphism

E^{∗}(BG) → ∏_{Rep(ℤ_{p}^{n},G)} E_{cont}^{∗}(Bℤ_{p}^{n})  (3.2)

where the right-hand side is E_{cont}^{∗}(Bℤ_{p}^{n})–valued functions on the finite set Rep(ℤ_{p}^{n}, G), with pointwise multiplication.

The map (3.2) is the n–character map and the HKR character theorem [27, Thm. C] says that, for particular E, this becomes an isomorphism after suitable localization. More precisely, assume that E = E(n), in particular E^{∗}(BS^{1}) ≅ E^{∗}⟦x⟧, |x| = 2, and define L(E^{∗}) to be the ring of fractions of E_{cont}^{∗}(Bℤ_{p}^{n}) obtained by inverting α^{∗}(x) for all for all non-zero α ∈ Hom_{cont}(ℤ_{p}^{n}, S^{1}) ≅ (ℤ/p^{∞})^{n}. Then, by [27, Thm. C], L(E^{∗}) is faithfully flat over E(n)^{∗}[\frac{1}{p}] = ℚ_{p^{n}}⟦w_{1}, …, w_{n−1}⟧[u, u^{−1}] (and in particular non-zero) and (3.2) induces an isomorphism

L(E^{∗}) ⊗_{E^{∗}} E^{∗}(BG) ⟶^{≃} ∏_{Rep(ℤ_{p}^{n},G)} L(E^{∗})  (3.3)

Proof of Theorem C. By the assumption of the theorem and the HKR character isomorphism (3.3) we have an isomorphism

∏_{Rep(ℤ_{p}^{n},G)} L(E^{∗}) → ∏_{Rep(ℤ_{p}^{n},H)} L(E^{∗})  (3.4)

by precomposing with the natural map Rep(ℤ_{p}^{n}, H) → Rep(ℤ_{p}^{n}, G). Since L(E^{∗}) ≠ 0 we conclude that Rep(ℤ_{p}^{n}, H) → Rep(ℤ_{p}^{n}, G) is an isomorphism. By the assumption on n and Lemma 3.1 this implies that Rep(A, H) → Rep(A, G) is an isomorphism for all finite abelian groups. Theorem C now follows from Theorem B. □

Remark 3.5 (A variety version of Theorem C and the role of inverting p). One easily sees, using some ring theory, that φ^{∗}: E(n)^{∗}(BG)[\frac{1}{p}] → E(n)^{∗}(BH)[\frac{1}{p}] is an isomorphism if and only if it induces a variety isomorphism (i.e., a bijection of K–points for all fields K, see [41, App. B] and [22, App. A]); this provides an alternative version of Theorem C, in terms of varieties, structurally similar to that of Theorem A. We quickly explain the ring theoretic argument: Since the rings are reduced, φ^{∗} cannot induce a variety isomorphism unless it is injective, so we assume this. We now argue that φ is seminormal in the sense of Traverso and Swan [49, §2], which implies that it induces a variety isomorphism (i.e., it is subintegral in their terminology) only if it is an isomorphism. By [49, Rem. p.215] it is enough to verify this after extending scalars along the faithfully flat map E(n)[\frac{1}{p}] → L(E^{∗}), which in turns by (3.3) reduces to showing that the map (3.4) of HKR character rings is seminormal. However that this is seminormal is easily seen e.g., from [49, Thm. 2.5].

In the paper [22] Greenlees–Strickland explain how Spec(E(n)^{∗}(BG)[\frac{1}{p}]) constitute a certain “zeroth pure stratum” of a chromatic stratification of the formal spectrum Spf(E(n)^{∗}(BG)), cf. the discussion before [22, Thm. 3.7]. Hence having an isomorphism on E(n)^{∗}(−)[\frac{1}{p}] is a priori significantly weaker information than having isomorphism on E(n)^{∗}(−) or Spf(E(n)^{∗}(−)). We note in passing that a variety version of Theorem C can also be obtained by in our proof, instead of [27], referencing the stratification theorem [22, Thm. 3.8, k = 0] (in a strengthen form that also says that W̄_{G}(A) acts freely on Level′_{k}(A^{∗}, 𝔾), as defined there).

Remark 3.6 (Relationship to the K(n)–Whitehead theorem). Recall that a map induces an isomorphism on E(n) if and only if it is an isomorphism on the corresponding uncompleted Johnson–Wilson theory, or an isomorphism on K(i) for all i ≤ n (see [43, Thm. 2.1] and [34, Lec. 23]). This in turn happens if and only if it is an isomorphism on K(n), by a result of Bousfield [9, Thm. 1.1]. Furthermore Bousfield proved in 1982 a “K(n)–Whitehead theorem” relating K(n)–isomorphisms to mod p–homology equivalences. More precisely he proved that a map between spaces that induces an isomorphism on K(n)^{∗} induces an isomorphism on H^{i}(−; 𝔽_{p}) for i ≤ n (stated in [8, Ex. 8.4] for p odd, and in [9, Thm. 1.4] in general). Furthermore, as explained in Remark 5.2, it is possible (but non-trivial) to give a value n (depending on the Sylow p–subgroup S) such that isomorphism in H^{i}(−; 𝔽_{p}) for i ≤ n implies isomorphism on H^{∗}(−; 𝔽_{p}). Hence these results, combined with the original Mislin’s theorem [37], implies a weaker version of Theorem C, that E(n)^{∗}–isomorphism for n suitably large implies that H controls p–fusion in G.

Note that Theorem C in particular says that E(n)^{∗}[\frac{1}{p}]–isomorphism implies E(n)^{∗}–isomorphism, in the setup of that theorem; we do not know an alternative, a priori, reason for this to be true, cf. Remark 3.5. However for so-called good groups in sense of Hopkins–Kuhn–Ravenel [27, §7] it follows from their work that the conditions are equivalent. (Non-good groups do exist, the first discovered example being UT_{4}(𝔽_{3}), upper triangular 4 × 4 matrices over 𝔽_{3} with ones on the diagonal [31].)

Remark 3.7 (Theorems A and C for fusion systems). Naturally, Theorems A and C should also hold in the setup of fusion systems. Indeed, as is clear from our proofs the only missing piece is a published reference for Quillen stratification and the Hopkins–Kuhn–Ravenel character theorem in the context of fusion systems, and once these become available Theorem A and C will also hold for those as well—however, we will not pursue this here.

4. Cohomology isomorphism modulo higher nilpotence and control of 2–fusion

In this section we briefly give a much weakened version of Theorem A also valid for p = 2 using Lannes T –functor techniques, specifically work of Henn–Lannes-Schwartz [32, 24]. We replace the assumption of F –isomorphism by the stronger assumption of isomorphism modulo n–nilpotence \overline{Nil}_{n}, recalled below, were n is a constant depending on the Sylow p–subgroup S. We also show by example that no fixed finite n works for all finite groups when p = 2, in contrast to p odd where n = 1 always works by Theorem A.

Recall that \overline{Nil}_{n} is the smallest subcategory of the category of unstable modules over the Steenrod algebra generated by locally finite modules and n–fold suspensions, under taking extensions and filtered colimits. Concretely, for p = 2, an unstable module M lies in \overline{Nil}_{n} if and only if for every element x of degree at least n and all 0 ≤ s < n, (Sq_{s})^{n}x = 0 for some n, where Sq_{s} y = Sq^{|x|−s} y, and Sq^{k} is the kth Steenrod square; see [46, Ch. 6] [45] [47, §6]. One sees that a map of connected algebras is an isomorphism modulo \overline{Nil}_{1} if and only if it is an F –isomorphism. Note that membership of \overline{Nil}_{n} only depends on the structure of M in high degrees. Removing ‘locally finite modules’ in the above definition one gets the definition of the smaller subcategory Nil_{n} more often used elsewhere in the literature; a module lies in Nil_{n} if and only if M^{<n} = 0 and M ∈ \overline{Nil}_{n}.

Theorem 4.1. Let ι: H < G be an inclusion of finite groups with common Sylow p–subgroup S. If ι^{∗}: H^{∗}(G; 𝔽_{p}) → H^{∗}(H; 𝔽_{p}) is an isomorphism modulo \overline{Nil}_{2r^{2}}, for r the dimension of a faithful complex representation of S, then ι^{∗} is an isomorphism and hence, by Mislin’s theorem, H controls p–fusion in G.

For any n ∈ ℕ there exists finite groups H_{n} < G_{n} with |G_{n} : H_{n}| = 3 and H^{∗}(G_{n}; 𝔽_{2}) → H^{∗}(H_{n}; 𝔽_{2}) an isomorphism modulo \overline{Nil}_{n} but not modulo \overline{Nil}_{n+1}.

We give the proof below, after two lemmas. The first part with \overline{Nil}_{n} replaced by Nil_{n} will follow directly from work of Henn–Lannes–Schwartz [24]; our addition concerning \overline{Nil}_{n} is obtained by combining an earlier argument of Mislin [37] with a small but useful simplification of the Henn– Lannes–Schwartz Nil_{n}–localization formula in the case of group cohomology rings obtained via results of Jackowski–McClure [30] and Dwyer–Wilkerson [15].

Lemma 4.2. Let A = H^{∗}(G; 𝔽_{p}) for a finite group G, or let A be any other unstable algebra satisfying the Dwyer–Wilkerson assumptions stated in [15, Thm. 1.2].

(1) The localization of A away from Nil_{n} can be described as an equalizer

Eq(  ⨁_{(V,f)∈Ob(ℛ^{∗}(A))} H^{∗}(V) ⊗ T_{V,f}(A)^{<n} ⇉ ⨁_{((V′,f′)→(V′′,f′′))∈Mor(ℛ^{∗}(A))} H^{∗}(V′) ⊗ (H^{∗}(V′) ⊗ T_{V′′,f′′}(A))^{<n})

induced by comultiplication of H^{∗}(V) and the H^{∗}(V)–comodule structure on T_{V,f}(A); here H^{∗}(V) means H^{∗}(V; 𝔽_{p}), ℛ^{∗}(A) denotes the full subcategory of the opposite Rector category ℛ(A) with objects finite maps of unstable algebras A → H^{∗}(V), for V a non-trivial elementary abelian p-group, and T_{V,f} is the component of Lannes T_{V} –functor corresponding to f: A → H^{∗}(V).

(2) Suppose that φ: A → B is a map of unstable algebras making B finitely generated over A, and assume that B also satisfies the Dwyer–Wilkerson assumptions. If φ is an isomorphism modulo \overline{Nil}_{n}, then it is also an isomorphism modulo Nil_{n}.

Sketch of proof. The formula in (1) is the formula for the localization L_{n}(A) of a noetherian unstable algebra A away from Nil_{n} given in [24, Thm. 4.9], except that we replace the opposite Rector category ℛ(A) [24, 4.6] with the full subcategory without the “trivial subgroup” i.e., the augmentation map A → H^{∗}(1; 𝔽_{p}) = 𝔽_{p}. Note that A → 𝔽_{p} only influences L_{n}(A)^{<n}, by the equalizer formula above. Following the discussion after [24, (5.5.1)] one sees that the equalizer above can be manipulated to produce

L_{n}(A)^{<n} ≅ lim_{(V,f)∈ℛ(A)} T_{V,f}(A)^{<n},

i.e., the standard “centralizer decomposition formula”. However by [15, Thm. 1.2] (extending work of Jackowski–McClure [30])

A^{<n} = lim_{(V,f)∈ℛ(A)} T_{V,f}(A)^{<n} = lim_{(V,f)∈ℛ^{∗}(A)} T_{V,f}(A)^{<n}

i.e., we can remove A → 𝔽_{p}. Doing the equalizer manipulations in reverse shows (1).

For the second part, note that obviously φ induces an isomorphism ℛ(A) ⟶^{≃} ℛ(B). Furthermore the argument presented in [37, Pf. of Prop. 2.3] shows that for (V, f) ∈ ℛ^{∗}(A) we have φ: T_{V,f}(A) ⟶^{≃} T_{V,φf}(B). We conclude that L_{n}(A) ⟶^{≃} L_{n}(B), and hence φ: A → B is an isomorphism modulo Nil_{n} as wanted. □

Lemma 4.3. Let G_{n} = (2A_{4})^{n}, S_{n} = (Q_{8})^{n} and S_{n} = ker(ψ), where ψ: G_{n} → G_{n}/S_{n} ≅ (C_{3})^{3} → C_{3} is given by (g_{1}, …, g_{n}) ↦ g_{1} … g_{n}. Then the restriction map H^{∗}(G_{n}, 𝔽_{2}) → H^{∗}(H_{n}, 𝔽_{2}) is an isomorphism modulo Nil_{n} but not modulo Nil_{n+1}.

Proof. Recall that H^{∗}(Q_{8}; 𝔽_{2}) ≅ H^{<4}(Q_{8}, 𝔽_{2}) ⊗ k[z] with |z| = 4, where the action of C_{3} on k[z] is trivial while the action on H^{<4}(Q_{8}; 𝔽_{2}) consists of 𝔽_{2} in degrees zero and three, and the two dimensional irreducible 𝔽_{2}C_{3}–module V in degrees one and two. Hence H^{∗}(S_{n}; 𝔽_{2}) is a tensor product of k[z_{1}, …, z_{n}], |z_{i}| = 4, and n copies of H^{<4}(Q_{8}; 𝔽_{2}).

Since G_{n} and H_{n} share the common Sylow 2–subgroup S_{n}, the restriction map H^{∗}(G_{n}, 𝔽_{2}) → H^{∗}(H_{n}, 𝔽_{2}) is injective, and the cokernel is a tensor product of k[z_{1}, …, z_{n}] with a certain finite module M, described explicitly as the sum of the non-trivial irreducible G_{n}–representations on H^{<4}(Q_{8}; 𝔽_{2})^{⊗n} which restrict trivially to H_{n}. Using the definition of Nil_{m}, the largest m for which the restriction map is an isomorphism modulo Nil_{m} therefore is the first degree where M is non-zero. This can again be described the first degree in which H_{n}/S_{n} has an invariant on (H^{<4}(Q_{8}; 𝔽_{2}))^{⊗n} while G_{n}/S_{n} does not.

To determine the largest such m we use Frobenius reciprocity

Hom_{𝔽_{2}(H_{n}/S_{n})}(𝔽_{2}, −) ≅ Hom_{𝔽_{2}(G_{n}/S_{n})}((𝔽_{2})_{H_{n}}↑^{G_{n}}, −).

We can without restriction extend coefficients to 𝔽_{4} where

(𝔽_{4})_{H_{n}}↑^{G_{n}} ≅ (𝔽_{4} ⊗ ⋯ ⊗ 𝔽_{4}) ⊕ (ω ⊗ ⋯ ⊗ ω) ⊕ (ω̄ ⊗ ⋯ ⊗ ω̄),

for 𝔽_{4}, ω and ω̄ the three one-dimensional 𝔽_{4}C_{3}–modules. Note that with this notation 𝔽_{4}⊗_{𝔽_{2}} V ≅ ω ⊕ ω̄. In order to have an H_{n}–invariant element in (H^{<4}(Q_{8}; 𝔽_{4}))^{⊗n} that is not G_{n}–invariant, it follows that we need to be in a degree where there is a copy of ω ⊗ ⋯ ⊗ ω or ω̄ ⊗ ⋯ ⊗ ω̄. This happens for the first time in degree n, where there is a summand V ⊗ ⋯ ⊗ V, which extended over 𝔽_{4} becomes (ω ⊕ ω̄) ⊗ ⋯ ⊗ (ω ⊕ ω̄) completing the proof. □

Proof of Theorem 4.1. By Lemma 4.2(2) the map ι^{∗} in the theorem is in fact an isomorphism modulo Nil_{2r^{2}}. By [24, Thm. 0.5], H^{∗}(S; 𝔽_{p}) is Nil_{2r^{2}}–closed, and therefore this holds for H^{∗}(G; 𝔽_{p}) and H^{∗}(H; 𝔽_{p}) as well, since they are both retracts of the cohomology of the Sylow p-subgroup H^{∗}(S; 𝔽_{p}) as modules over the Steenrod algebra. Hence ι^{∗} is an actual isomorphism, and Mislin’s theorem (for example via Theorem C) gives the statement about fusion. The counterexample in the last part of the theorem was given in Lemma 4.3. □

Remark 4.4 (Bounds in Theorem 4.1). For S_{n} = (Q_{8})^{n}, considered in Lemma 4.3 the bound stated in Theorem 4.1 gives 2 · (2n)^{2}; it follows from [11, Prop. 1.17] and [24, II.4.6] that the cohomology is Nil_{m}–closed if and only if m ≥ 3n + 2.

Remark 4.5 (Additional history of Theorem 4.1). Mislin proved Lemma 4.2(2) for \overline{Nil}_{∞} (i.e., locally finite modules) [37, Prop. 2.3], and used this to prove that p cohomology isomorphism in high degrees implies isomorphism in all degrees [37, Main Thm], i.e., a weaker version of Theorem 4.1. That conclusion can also be reached purely algebraically by looking at the relative cohomology, and appealing to a general vanishing result [5, Thm. 1.1].

5. Additional remarks

We can now diagrammatically display, in shorthand form, the way we have established the relations on an inclusion of finite groups H < G sharing a common Sylow p–subgroup, with dotted arrows indicating trivial implications, dashed arrows indicating previous theorems, and solid arrows indicating current contributions.

[figure: p-fusion iso; Mislin; Cartan– Eilenberg; ab p-fusion iso; p^{2}-ab p-fusion iso; Thm B; elt ab p-fusion iso; Quillen; Thm B, p odd; E(n)^{∗}-iso; E(n)^{∗}[\frac{1}{p}]-iso; HKR, n = rk_{p}(G); 𝔽_{p}-iso; high deg 𝔽_{p}-iso; 𝔽_{p}-iso mod \overline{Nil}_{k}; Thm. 4.1, k ≫ 0; F -iso]

Here for example “p^{2}–ab p–fusion iso” is short for isomorphism on p–fusion on abelian subgroups of exponent at most p^{2}, as made precise in Theorem B, and 𝔽_{p}–iso is short for isomorphism on mod p cohomology. Theorem A is the implication from the bottom right corner to the top left, making all conditions equivalent for p odd. For p = 2 only the conditions not all the way to the right in the diagram are equivalent, and Theorem C is the implication from the center-right entry to the top left—we again refer to the earlier sections for the detailed statements.

We take the opportunity to discuss some related conditions, in light of this work.

Remark 5.1 (High degree cohomology isomorphism implies control of p–fusion). We already gave various ways of proving cohomology isomorphism and control of p–fusion from high degree cohomology isomorphism in Theorem A and Theorem 4.1, as well as in Remark 4.5. In the special case of a Sylow inclusion S < G the result goes all the way back to Atiyah in 1961 [3] and was a precursor to Quillen’s work. Atiyah’s original argument is in fact interesting from the current perspective: As explained in [40, p. 362] Atiyah argues that if G and S have the same cohomology in high degrees, then their E(1)–cohomology (= p–adic K–theory) have the same rank, by the Atiyah–Hirzebruch spectral sequence, and hence G and S have the same fusion on cyclic p– subgroups, which in this case implies that G is p–nilpotent. We can now generalize that argument to an arbitrary inclusion H < G, using the E(n)–based Atiyah–Hirzebruch sequence instead: Given a high degree mod p cohomology isomorphism the Atiyah–Hirzebruch spectral sequence shows that we obtain an isomorphism on E(n)–cohomology, after inverting p. HKR character theory and Theorem B now shows that H controls p–fusion in G, as explained in Theorem C.

Remark 5.2 (Low degree cohomology isomorphism implies control of p–fusion). Orthogonally to Theorem A and Theorem 4.1, one can ask when knowing H^{∗}(BG; 𝔽_{p}) → H^{∗}(BH; 𝔽_{p}) is an isomorphism in low degrees implies that it is an isomorphism. The first result in this direction was Tate’s p–nilpotence criterion from 1964 [53]: If a Sylow inclusion S ≤ G induces isomorphism on H^{1}(−; 𝔽_{p}), then it controls fusion. Examples such as Lemma 4.3 shows that there in general does not exist any n such that inducing isomorphism on the bottom n cohomology groups implies fusion isomorphism. Jackowski in [29, Cor. 1.5] however produced a bound n depending on the large ambient group G. And, since there are only finitely fusion systems on a given p–group, there also exists a constant n(S) only depending on the Sylow p–subgroup. Using Symond’s recent solution of Benson’s regularity conjecture one can get another bound [52, §10] [4, §10].

Remark 5.3 (Integral cohomology isomorphism implies group isomorphism). Mislin’s theorem includes the statement that a homomorphism between finite groups inducing a p cohomology isomorphism has kernel and index of order prime to p, which implies as a corollary that a homomorphism between finite groups inducing an isomorphism on integral cohomology is an isomorphism. That statement was originally made by Evens [17, p. 237–8], proved by topological means by Jackowski [29], and finally proved algebraically by Benson–Evens [6], relying on earlier work by Benson–Carlson. HKR character theory together with Lemma 2.7 of course also implies this result, though this is an overkill. We also note that Bojanowska-Jackowska proved that a homomorphism between finite groups that induces isomorphism of integral K-theory varieties is an isomorphism; see [7, Thm. 6.2] and also [36, Thm. D].

Remark 5.4 (History of Mislin’s theorem on cohomology isomorphism). As noted in the introduction our proof of Theorem A provides a new simple algebraic proof of Mislin’s theorem [37] at odd primes, and Theorem C provides another proof for all primes, with similar algebraic input, but replacing the theory of cohomology varieties with the HKR character theory. (Note that Mislin’s theorem is most generally formulated in the setup of an arbitrary map as in Theorem C; but, as explained in Remark 5.3, it is easily reduced to the case of the inclusion of a subgroup of index prime to p, as in Theorem A.) We here summarize the history behind Mislin’s theorem and alternative proofs. Mislin’s original proof [37] uses the Dwyer–Zabrodsky theorem [16] in algebraic topology, whose proof again relies on Lannes’ theory [32] extending Miller’s proof of the Sullivan conjecture [35]. In the early 1990s, for example in his lecture series at the 1994 Banff conference on representation theory, Jon Alperin made the highly publicized challenge to find a purely algebraic proof of this theorem, and this was pursued by several workers. Snaith gave a somewhat different topological argument [48] and Symonds [50], following an idea of Robinson [44, §7], gave an algebraic reduction of the problem to a statement about cohomology of trivial source modules, and gave a topological proof of the latter statement. Algebraic proofs were finally completed independently by Hida [25] and Okuyama [38], who gave algebraic proofs of Symonds’ statement about cohomology of trivial source modules, through some quite delicate arguments and a good amount of modular representation theory. In a separate paper, Symonds [51] also showed how to prove Mislin’s theorem following ideas of Alperin [1], using the algebraic machinery of Lannes’ T –functor.

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Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE

Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark

Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark