A theorem of Tate and p-solvability

10 June 2014
Jon Gonzalez Sanchez
Let G be a finite group, p a prime and S a Sylow p-subgroup. The group G is called p-nilpotent if S has a normal complement N in G, that is, G is the semidirect product between S and N. The notion of p-nilpotency plays an important role in finite group theory. For instance, Thompson's criterion for p-nilpotency leads to the important structural result that finite groups with fixed-point-free automorphisms are nilpotent. By a classical result of Tate one can detect p-nilpotency using mod p cohomology in dimension 1: the group G is p-nilpotent if and only if the restriction map in cohomology from G to S is an isomorphism in dimension 1. In this talk we will discuss cohomological criteria for p-nilpotency by Tate, and Atiyah/Quillen (using high-dimensional cohomology) from the 1960s and 1970s. Finally, we will discuss how one can extend Tate's result to study p-solvable and more general finite groups.