Infinite groups with isomorphic power graph and commuting graph

TL;DR

This paper explores conditions under which the power, enhanced power, and commuting graphs of infinite groups are equal, extending finite group results and revealing new isomorphisms between specific infinite groups' graphs.

Contribution

It provides a necessary and sufficient condition for graph equality among these group graphs and constructs new isomorphisms for infinite groups, answering a question by P. J. Cameron.

Findings

Characterizes when two of the graphs are equal for infinite groups.

Shows power graph of locally quaternion group is isomorphic to commuting graph of locally dihedral group.

Answers Cameron's question about isomorphic power and commuting graphs across different groups.

Abstract

In this paper, we investigate certain graphs defined on groups, with a focus on infinite groups. The graphs discussed are the power graph, the enhanced power graph, and the commuting graph whose vertex set is a group $G$. The power graph is a graph in which two vertices are adjacent if one is some power of the other. In the enhanced power graph, an edge joins two vertices if they generate a cyclic subgroup of $G$. In the commuting graph, two vertices are adjacent if they commute in $G$. We prove a necessary and sufficient condition for any two of these graphs to be equal. This extends existing results for finite groups. In addition, we show that the power graph of the locally quaternion group is isomorphic to the commuting graph of the locally dihedral group. Lastly, we also answer a question posed by P. J. Cameron about the existence of groups $G_1$ and $G_2$ both of whom have power graph not equal to commuting graph but the power graph of $G_1$ and the commuting graph of $G_2$ are isomorphic.