Finite groups with the same Power graph

TL;DR

This paper investigates when groups of the same order with isomorphic power graphs are necessarily isomorphic, identifying specific order conditions and properties preserved under power graph isomorphism.

Contribution

It characterizes orders for which isomorphic power graphs imply group isomorphism and explores property preservation under power graph isomorphism.

Findings

All such orders are cube-free and not multiples of 16.

Isomorphic power graphs preserve nilpotency and the existence of normal Hall subgroups.

Identifies orders where power graph isomorphism guarantees group isomorphism.

Abstract

The power graph P(G) of a group G is a graph with vertex set G, where two vertices u and v are adjacent if and only if one is the power of the other. In this paper, we raise and study the following question: For which natural numbers n every two groups of order n with isomorphic power graphs are isomorphic? In particular, we determine prove that all such n are cube-free and are not multiples of 16. Moreover, we show that if two finite groups have isomorphic power graphs and one of them is nilpotent or has a normal Hall subgroup, the same is true with the other one.