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 $H^n(G/N,Z(N))\not=0$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2009
- DOI:
- 10.48550/arXiv.0901.3182
- arXiv:
- arXiv:0901.3182
- Bibcode:
- 2009arXiv0901.3182A
- Keywords:
-
- Mathematics - Group Theory;
- Mathematics - Commutative Algebra;
- 20D45;
- 20E36
- E-Print:
- to appear in Journal of Algebra