Difference graphs of finite abelian groups with two Sylow subgroups

TL;DR

This paper proves that for finite abelian groups with at most two prime factors, isomorphic difference graphs imply the groups themselves are isomorphic, revealing a graph-theoretic characterization of such groups.

Contribution

It establishes that the difference graph uniquely determines finite abelian groups with up to two prime factors, a novel result linking graph isomorphism to group isomorphism.

Findings

Difference graphs distinguish finite abelian groups with at most two prime factors.

Isomorphic difference graphs imply group isomorphism in this class.

Provides a new graph-theoretic approach to classify certain finite abelian groups.

Abstract

The power graph and the enhanced power graph of a group $\mathbf G$ are simple graphs with vertex set $G$; two elements of $G$ are adjacent in the power graph if one of them is a power of the other, and they are adjacent in the enhanced power graph if they generate a cyclic subgroup. The difference graph of a group $\mathbf G$, denoted by $\mathcal D(\mathbf G)$, is the difference of the enhanced power graph and the power graph of group $\mathbf G$ with all the isolated vertices removed. In this paper, we prove that, if a pair of finite abelian groups of order divisible by at most two primes have isomorphic difference graphs, then they are isomorphic.