The Power Graph of a Torsion-Free Group of Nilpotency Class $2$

TL;DR

This paper investigates the structure of power graphs in torsion-free groups of nilpotency class 2, proving their equivalence under different definitions and solving an open problem about their isomorphism properties.

Contribution

It establishes the isomorphism invariance of various power graph definitions and proves that power graph isomorphism implies directed power graph isomorphism for these groups.

Findings

Power graphs under three definitions are mutually isomorphic.

Power graph isomorphism implies directed power graph isomorphism in torsion-free nilpotent groups.

Solved an open problem on the relationship between power graph and directed power graph.

Abstract

The directed power graph $\mathcal G(\mathbf G)$ of a group $\mathbf G$ is the simple digraph with vertex set $G$ in which $x\rightarrow y$ if $y$ is a power of $x$, the power graph is the underlying simple graph, and the enhanced power graph of $\mathbf G$ is the simple graph with the same vertex set such that two vertices are adjacent if they are powers of some element of $\mathbf G$. In this paper three versions of the definition of the power graphs are discussed, and it is proved that the power graph by any of the three versions of the definitions determines the other two up to isomorphism. It is also proved that, if $\mathbf G$ is a torsion-free group of nilpotency class $2$ and if $\mathbf H$ is a group such that $\mathcal G(\mathbf H)\cong\mathcal G(\mathbf G)$, then $\mathbf G$ and $\mathbf H$ have isomorphic directed power graphs, which was an open problem proposed by Cameron, Guerra and Jurina.