The commuting graphs of certain cyclic-by-abelian groups

TL;DR

This paper investigates the structure of commuting graphs in certain finite groups, showing they are either connected with small diameter or a union of complete graphs, with implications for metacyclic and square-free order groups.

Contribution

It characterizes the commuting graphs of cyclic-by-abelian groups, establishing their connectivity and diameter properties, extending to metacyclic and square-free order groups.

Findings

Commuting graph is either connected with diameter ≤ 4 or a union of |G'|+1 complete graphs.

Results apply to all finite metacyclic groups.

Specific structure identified for groups of square-free order.

Abstract

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or the disjoint union of $|G'| + 1$ complete graphs. These results apply to all finite metacyclic groups, and to groups of square-free order in particular.