On connectedness of power graphs of finite groups

TL;DR

This paper studies the connectivity and separating sets of power graphs of finite groups, providing bounds and characterizations for cyclic and p-groups, advancing understanding of their structural properties.

Contribution

It introduces new bounds for the connectivity of power graphs and characterizes components of proper power graphs of p-groups, including abelian cases.

Findings

Minimal separating sets for cyclic groups identified

Sharp upper bounds for connectivity established

Number of components in abelian p-groups determined

Abstract

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of $p$-groups are studied. In particular, the number of components of that of abelian $p$-groups are determined.