A combinatorial characterization of finite groups of prime exponent

TL;DR

This paper characterizes non-cyclic finite groups of prime exponent and elementary abelian 2-groups (rank ≥ 2) using their power graphs, providing a graph-theoretic perspective on their structure.

Contribution

It offers a novel characterization of specific finite groups of prime exponent and elementary abelian 2-groups through their power graphs.

Findings

Characterization of non-cyclic finite groups of prime exponent via power graphs

Identification of elementary abelian 2-groups of rank ≥ 2 using power graphs

Establishment of graph-theoretic criteria for group classification

Abstract

The power graph of a group $G$ is a simple and undirected graph with vertex set $G$ and two distinct vertices are adjacent if one is a power of the other. In this article, we characterize (non-cyclic) finite groups of prime exponent and finite elementary abelian $2$-groups (of rank at least $2$) in terms of their power graphs.