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- Title
ON NONINNER AUTOMORPHISMS OF FINITE NONABELIAN $p$-GROUPS.
- Authors
GHORAISHI, S. M.
- 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].
- Subjects
AUTOMORPHISMS; GROUP theory; FINITE groups; NONABELIAN groups; FRATTINI subgroups
- Publication
Bulletin of the Australian Mathematical Society, 2014, Vol 89, Issue 2, p202
- ISSN
0004-9727
- Publication type
Article
- DOI
10.1017/S0004972713000403