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 -