Control of fusion by Abelian subgroups of the hyperfocal subgroup

Abstract

We prove that an isomorphism between saturated fusion systems over the same finite [Formula: see text]-group is detected on the elementary abelian subgroups of the hyperfocal subgroup if [Formula: see text] is odd, and on the abelian subgroups of the hyperfocal subgroup of exponent at most [Formula: see text] if [Formula: see text]. For odd [Formula: see text], this has implications for mod [Formula: see text] group cohomology.