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 [‘Noninner automorphisms of order \(p\) of finite \(p\)-groups’, J. Algebra \(250\) (2002), 283–287]. 10.1017/S0004972713000403

Published

2014-01-27

Issue

Vol. 89 No. 2 (2014)

Section

Articles
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