On equality of central and class preserving automorphisms of finite p-groups

TL;DR

This paper investigates conditions under which the group of class preserving automorphisms equals the group of central automorphisms in finite non-abelian p-groups, providing classifications for groups up to order p^7.

Contribution

It establishes necessary and sufficient conditions for the equality of these automorphism groups in specific p-groups, extending classifications up to order p^7.

Findings

Necessary condition for automorphism groups equality

Characterization for groups with elementary abelian or cyclic center

Complete classification for groups of order ≤ p^5

Abstract

Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. Let $\mathrm{Aut}_c(G)$ and $\mathrm{Aut}_z(G)$ respectively denote the group of all class preserving and central automorphisms of $G$. We give a necessary condition for $G$ such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G)$ and give necessary and sufficient conditions for $G$ with elementary abelian or cyclic center such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G).$ We also characterize all finite $p$-groups $G$ of order $\leq p^7$ such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G)$ and complete the classification of all finite $p$-groups of order $\le p^5$ for which there exist non-inner class preserving automorphisms.