COMMUTING AUTOMORPHISM OF p-GROUPS WITH CYCLIC MAXIMAL SUBGROUPS

Title & Authors
COMMUTING AUTOMORPHISM OF p-GROUPS WITH CYCLIC MAXIMAL SUBGROUPS
Vosooghpour, Fatemeh; Kargarian, Zeinab; Akhavan-Malayeri, Mehri;

Abstract
Let G be a group and let $\small{p}$ be a prime number. If the set $\small{\mathcal{A}(G)}$ of all commuting automorphisms of G forms a subgroup of Aut(G), then G is called $\small{\mathcal{A}(G)}$-group. In this paper we show that any $\small{p}$-group with cyclic maximal subgroup is an $\small{\mathcal{A}(G)}$-group. We also find the structure of the group $\small{\mathcal{A}(G)}$ and we show that $\small{\mathcal{A}(G)=Aut_c(G)}$. Moreover, we prove that for any prime $\small{p}$ and all integers $\small{n{\geq}3}$, there exists a non-abelian $\small{\mathcal{A}(G)}$-group of order $\small{p^n}$ in which $\small{\mathcal{A}(G)=Aut_c(G)}$. If $\small{p}$ > 2, then $\small{\mathcal{A}(G)={\cong}\mathbb{Z}_p{\times}\mathbb{Z}_{p^{n-2}}}$ and if $\small{p=2}$, then $\small{\mathcal{A}(G)={\cong}\mathbb{Z}_2{\times}\mathbb{Z}_2{\times}\mathbb{Z}_{2^{n-3}}}$ or $\small{\mathbb{Z}_2{\times}\mathbb{Z}_2}$.
Keywords
commuting automorphism;cyclic maximal subgroup;
Language
English
Cited by
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