On equality of absolute central and IA-automorphism groups of finite p-groups

Abstract

Let G be a finite p-group. Let $\text{Var}(G)$ and $\text{IA}(G)$ respectively denote the group of all absolute central and $\text{IA}$-automorphisms of G. We give necessary and sufficient conditions for a finite non-abelian p-group G such that $\text{Var}(G)=\text{IA}(G),$ and as a consequence, we prove that for a finite non-abelian p-group G of order up to p⁴, $\text{Var}(G)=\text{IA}(G)$ if and only if $G^{\prime }=L(G),$ where L(G) denotes the absolute center of G.

Keywords:

• Absolute central automorphism
• finite p-group
• $\text{IA}$-automorphism

2020 Mathematics Subject Classification:

• 20D45
• 20D15