Group cohomology and control of p-fusion

TY - JOUR

T1 - Group cohomology and control of p-fusion

AU - Benson, David John

AU - Grodal, Jesper

AU - Henke, Ellen

PY - 2014/9/1

Y1 - 2014/9/1

N2 - 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.

AB - 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.

KW - math.AT

KW - math.GR

KW - 20J06 (20D20, 20J05, 55P60)

U2 - 10.1007/s00222-013-0489-5

DO - 10.1007/s00222-013-0489-5

M3 - Journal article

SN - 0020-9910

VL - 197

SP - 491

EP - 507

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -