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.