Proceedings of the Japan Academy, Series A, Mathematical Sciences

On commuting automorphisms of finite $p$-groups

Pradeep Kumar Rai

Full-text: Open access

Abstract

Let $G$ be a group. An automorphism $\alpha$ of $G$ is called a commuting automorphism if $[\alpha(x), x] = 1$ for all $x \in G$. Let $A(G)$ be the set of all commuting automorphisms of $G$. A group $G$ is said to be an $A(G)$-group if $A(G)$ forms a subgroup of $\mathit{Aut}(G)$. We give some sufficient conditions on a finite $p$-group $G$ such that $G$ is an $A(G)$-group. As an application we prove that a finite $p$-group $G$ of coclass 2 for an odd prime $p$ is an $A(G)$-group. Also we classify non-$A(G)$ groups $G$ of order $p^{5}$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 5 (2015), 57-60.

Dates
First available in Project Euclid: 30 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1430397893

Digital Object Identifier
doi:10.3792/pjaa.91.57

Mathematical Reviews number (MathSciNet)
MR3342027

Zentralblatt MATH identifier
1325.20024

Subjects
Primary: 20F28: Automorphism groups of groups [See also 20E36]

Keywords
Commuting automorphism coclass 2 group

Citation

Rai, Pradeep Kumar. On commuting automorphisms of finite $p$-groups. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 5, 57--60. doi:10.3792/pjaa.91.57. https://projecteuclid.org/euclid.pja/1430397893


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