Article contents
ON NONINNER AUTOMORPHISMS OF FINITE NONABELIAN
$p$-GROUPS
Published online by Cambridge University Press: 07 June 2013
Abstract
A long-standing conjecture asserts that 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 the Frattini subgroup of
$G$ elementwise fixed [‘Noninner automorphisms of order
$p$ of finite
$p$-groups’, J. Algebra 250 (2002), 283–287].
MSC classification
- Type
- Research Article
- Information
- Copyright
- Copyright ©2013 Australian Mathematical Publishing Association Inc.
References















- 8
- Cited by