Some necessary and sufficient conditions on commuting automorphisms of some finite p-groups

Abstract

Let G be a finite non-abelian p-group of order greater than p⁴, p an odd prime, such that $Z({\Omega }_1(H))$ is cyclic and Z(H) is elementary abelian, where $H=C_G(\Phi (G)).$ We prove that the set $A(G)=\{\alpha \in \text{Aut}(G)|[x,\alpha (x)]=1\text{for}\text{all}x\in G\}$ of all commuting automorphisms of G forms a subgroup of $\text{Aut}(G)$ if and only if ${\Omega }_1(H)$ is abelian. Also, we find the structure of $A(G)={\text{Aut}}_z(G)$ for a finite 2-group G of almost maximal class with cyclic center Z(G), where ${\text{Aut}}_z(G)$ denotes the set of all central automorphisms of G.

Keywords:

• Commuting automorphism
• central automorphism
• almost maximal class

2020 Mathematics Subject Classification:

• 20D15
• 20D45