Powerful $p$-groups have noninner automorphisms of order $p$ and some cohomology

Abstract

In this paper we study the longstanding conjecture of whether there exists a noninner automorphism of order $p$ for a finite non-abelian $p$-group. We prove that if $G$ is a finite non-abelian $p$-group such that $G/Z(G)$ is powerful then $G$ has a noninner automorphism of order $p$ leaving either $\Phi (G)$ or ${\Omega }_1(Z(G))$ elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for odd $p$, by showing that the Tate cohomology $H^n(G/N,Z(N))\ne 0$ for all $n\ge 0$, where $G$ is a finite $p$-group, $p$ is odd, $G/Z(G)$ is $p$-central (i.e., elements of order $p$ are central) and $N⊲G$ with $G/N$ non-cyclic.

to appear in Journal of Algebra

Cited by in corpus (1)

• Some automorphism groups of finite p-groups