The Commuting Graph of a Solvable A-Group

TL;DR

This paper studies the maximum possible diameter of the commuting graph of solvable A-groups, establishing upper bounds of 4 and 6 depending on the group's derived length, with examples confirming these bounds are tight.

Contribution

It provides new upper bounds on the diameter of the commuting graph for solvable A-groups based on their derived length, with proofs and examples showing these bounds are optimal.

Findings

Diameter at most 4 when derived length is 2

Diameter at most 6 in the general case

Examples show bounds are sharp

Abstract

Let $G$ be a finite group. Recall that an $A$-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable $A$-group. Assuming that the commuting graph is connected, we show when the derived length of $G$ is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.