On the autocentralizer subgroups of finite $p$--groups

Authors

PARISA SEIFIZADEH
ALI GHOLAMIAN
AMIRALI FAROKHNIAEE
MOHAMMAD MEHDI NASRABADI

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.

Keywords

Automorphism, centralizer, finite $p$--group, inner automorphism

First Page

1802

Last Page

1812

Recommended Citation

SEIFIZADEH, PARISA; GHOLAMIAN, ALI; FAROKHNIAEE, AMIRALI; and NASRABADI, MOHAMMAD MEHDI (2020) "On the autocentralizer subgroups of finite $p$--groups," Turkish Journal of Mathematics: Vol. 44: No. 5, Article 22. https://doi.org/10.3906/mat-1911-89
Available at: https://journals.tubitak.gov.tr/math/vol44/iss5/22