On the soluble graph of a finite group

TL;DR

This paper studies the soluble graph of finite insoluble groups, proving it is always connected with a diameter at most 5, and provides specific bounds for various group families, including almost simple groups.

Contribution

It introduces the soluble graph for finite groups, establishes its connectivity and diameter bounds, and analyzes these bounds for different classes of groups, including almost simple groups.

Findings

The soluble graph is always connected with diameter ≤ 5.

For non-almost simple groups, the diameter is ≤ 3.

Certain simple groups have diameter ≥ 4, close to the upper bound.

Abstract

Let $G$ be a finite insoluble group with soluble radical $R(G)$. In this paper we investigate the soluble graph of $G$, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in $G \setminus R(G)$, with $x$ adjacent to $y$ if they generate a soluble subgroup of $G$. Our main result states that this graph is always connected and its diameter, denoted $\delta_{\mathcal{S}}(G)$, is at most $5$. More precisely, we show that $\delta_{\mathcal{S}}(G) \leqslant 3$ if $G$ is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that $\delta_{\mathcal{S}}(S_n) = 3$ for all $n \geqslant 6$. We also establish the existence of simple groups with $\delta_{\mathcal{S}}(G) \geqslant 4$. For instance, we prove that $\delta_{\mathcal{S}}(A_{2p+1}) \geqslant 4$ for every Sophie Germain prime $p \geqslant 5$, which demonstrates that our general upper bound of $5$ is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.