Non-Isomorphic Groups with Isomorphic Power and Commuting Graphs
Surbhi, Geetha Venkataraman

TL;DR
This paper constructs examples of non-isomorphic groups whose power, commuting, and enhanced power graphs are isomorphic, answering a question about the relationships between these graph structures in group theory.
Contribution
It demonstrates the existence of non-isomorphic groups with isomorphic power and commuting graphs, specifically involving quaternion, dihedral, and dicyclic groups.
Findings
Power graph of quaternion group is isomorphic to commuting graph of dihedral group.
Enhanced power graph of dicyclic group is isomorphic to commuting graph of dihedral group.
Examples show non-isomorphic groups can have identical associated graphs.
Abstract
The power graph of a group is a graph with vertex set , in which two vertices are adjacent if one is some power of the other. In the commuting graph, with as the vertex set, two vertices are joined by an edge if they commute in . The enhanced power graph of a group is a graph with vertex set and an edge joining two vertices and if is cyclic. In this paper, we answer a question posed by P. J. Cameron, namely, if there exist groups and such that the power graph of is isomorphic to the commuting graph of . We show that the answer is yes if is the generalised quaternion group and is the dihedral group. We also show that the enhanced power graph of the dicyclic group is isomorphic to the commuting graph of the dihedral group.
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Taxonomy
TopicsFinite Group Theory Research · Advanced Topics in Algebra · Geometric and Algebraic Topology
