Group cohomology and control of p-fusion

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.

A cohomological characterization of nilpotent fusion systems

“…Already several authors have provided fusion system counterparts to characterizations of p-nilpotency for finite groups, see [2], [3], [7], [9], [10], [11], [13] and [14]. In this work, we prove the fusion system version of a p-nilpotency criterion from the late 60's due to Wong [16] and Hoechsmann, Roquette and Zassenhaus [12].…”

Mislin’s theorem for fusion systems through Mackey functors

“…Recently Todea [11] proved the above theorem when p is odd following Benson, Grodal and Henke's new algebraic proof [2] of Mislin's theorem for finite groups for p odd. We prove the above theorem for all primes p. Consequently Todea's results in [11] on block algebras now hold for all primes p. Our approach follows closely Symonds's proof [10] of Mislin's theorem for finite groups, which uses Mackey functors.…”

Control of fusion by Abelian subgroups of the hyperfocal subgroup

“…1 More precisely, Quillen studied the variety of the commutative subring of H * (G; F p ) of elements of even degree. However, his stratification theorem holds similarly for the variety of H * (G; F p ); see Remark 3.2. proved a result that holds more generally for any subgroup H of index prime to p; see [5]. More precisely, it is shown that H controls fusion in G (and thus the inclusion map from H to G induces an isomorphism in mod p group cohomology), if the inclusion map induces an isomorphism between the corresponding varieties, i.e.…”

“…More precisely, it is shown that H controls fusion in G (and thus the inclusion map from H to G induces an isomorphism in mod p group cohomology), if the inclusion map induces an isomorphism between the corresponding varieties, i.e. if H controls fusion of elementary abelian subgroups of G. This is obtained as a consequence of a theorem that is stated and proved for saturated fusion systems; see [5,Theorem B]. In this short note, we point out that actually a slightly stronger version of this theorem holds.…”

“…If p is an odd prime and H is a subgroup of G of index prime to p, then the results in [5] say basically that H controls fusion in G if res * G,H : V H → V G is an isomorphism of varieties. Theorem A implies a slightly stronger statement which is stated in the next theorem.…”