I will report on recent joint work with Dave Benson and Ellen Henke. We show that if an inclusion of finite groups H < G of index prime to p induces an F-isomorphism of mod p cohomology rings, or equivalently a homeomorphism of cohomology varieties, then H controls p-fusion in G, if p is odd -- in particular the map on cohomology is in fact an isomorphism. This generalizes classical results of Quillen, who proved this when H is the Sylow p-subgroup, and also 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, and using Hopkins-Kuhn-Ravenel character theory. The results follow from of a general algebraic theorem we prove about p-fusion systems, that roughly says that isomorphisms between p-fusion systems are detected on elementary abelian p-groups. My talk will be a gentle introduction to these results, and the surrounding mathematics.