On noninner automorphisms of finite nonabelian $p$-groups

S. M. Ghoraishi

On noninner automorphisms of finite nonabelian $p$ -groups

Authors

S. M. Ghoraishi University of Isfahan

Keywords:

Finite

$p$ -groups, automorphisms, noninner automorphisms.

Abstract

A longstanding conjecture asserts every finite nonabelian

$p$ -group has a noninner automorphism of order

$p$ . In this paper the verification of the conjecture is reduced to the case of

$p$ -groups

$G$ satisfying

$Z_2^\star(G)\leq C_G(Z_2^\star(G))=\Phi(G)$ , where

$Z_2^\star(G)$ is the preimage of

$\Omega_1(Z_2(G)/Z(G))$ in

$G$ . This improves Deaconescu and Silberberg's reduction of the conjecture: If

$C_G(Z(\Phi(G)))\not=\Phi(G)$ , then

$G$ has a noninner automorphism of order

$p$ leaving Frattini subgroup of

$G$ elementwise fixed [â€˜Noninner automorphisms of order

$p$ of finite

$p$ -groupsâ€™, J. Algebra

$250$ (2002), 283â€“287]. 10.1017/S0004972713000403

Published

2014-01-27

Issue

Vol. 89 No. 2 (2014)

Section

Articles

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On noninner automorphisms of finite nonabelian pp-groups