On finite $p$-groups whose central automorphisms are all class preserving

TL;DR

This paper investigates finite p-groups with the property that all central automorphisms are class preserving, revealing structural constraints such as even minimal generating set size and bounds on group and automorphism group orders.

Contribution

It establishes that for such groups, the minimal generating set size is even and provides lower bounds on the order of the group and its automorphism group.

Findings

d(G) is even for groups with all central automorphisms class preserving

Order of G is at least p^8 when all automorphisms are class preserving

Automorphism group order is at least p^12 for such groups

Abstract

We obtain certain results on a finite $p$-group whose central automorphisms are all class preserving. In particular, we prove that if $G$ is a finite $p$-group whose central automorphisms are all class preserving, then $d(G)$ is even, where $d(G)$ denotes the number of elements in any minimal generating set for $G$. As an application of these results, we obtain some results regarding finite $p$-groups whose automorphisms are all class preserving. In particular, we prove that if $G$ is a finite $p$-groups whose automorphisms are all class preserving, then order of $G$ is at least $p^8$ and the order of the automorphism group of $G$ is at least $p^12$.