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Turkish Journal of Mathematics

DOI

10.3906/mat-1911-89

Abstract

Let $G$ be a finite group and ${\rm Aut}(G)$ be the group of automorphisms of $G.$ Then, the autocentralizer of an automorphism $\alpha\in {\rm Aut}(G)$ in $G$ is defined as $C_{G}(\alpha)= \lbrace g \in G\vert \alpha(g)=g\rbrace.$ Let $Acent(G)=\lbrace C_{G}(\alpha)\vert \alpha \in {\rm Aut}(G) \rbrace.$ If $\vert Acent(G)\vert= n,$ then $G$ is an $n$--autocentralizer group. In this paper, we classify all $n$--autocentralizer abelian groups for $n=$ 6, 7 and 8. We also obtain a lower bound on the number of autocentralizer subgroups for $p$--groups, where $p$ is a prime number. We show that if $p\neq 2,$ there is no $n$--autocentralizer $p$--group for $n=6,7.$ Moreover, if $p=2,$ then there is no $6$--autocentralizer $p$--group.

First Page

1802

Last Page

1812

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Mathematics Commons

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