Open Access
May 2015 On commuting automorphisms of finite $p$-groups
Pradeep Kumar Rai
Proc. Japan Acad. Ser. A Math. Sci. 91(5): 57-60 (May 2015). DOI: 10.3792/pjaa.91.57


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 $p^{5}$.


Download Citation

Pradeep Kumar Rai. "On commuting automorphisms of finite $p$-groups." Proc. Japan Acad. Ser. A Math. Sci. 91 (5) 57 - 60, May 2015.


Published: May 2015
First available in Project Euclid: 30 April 2015

zbMATH: 1325.20024
MathSciNet: MR3342027
Digital Object Identifier: 10.3792/pjaa.91.57

Primary: 20F28

Keywords: coclass 2 group , Commuting automorphism

Rights: Copyright © 2015 The Japan Academy

Vol.91 • No. 5 • May 2015
Back to Top