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

By Alireza Abdollahi. 2009

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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 $H^n(G/N,Z(N))\not=0$ for all $n\geq 0$, 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 $N\lhd G$ with $G/N$ non-cyclic.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f6c3760ce9e5bac888bbfe12eeac78bc5537c95d