Combinatorial Theory

The generating graph \(\Gamma(G)\) of a finite group \(G\) has vertex set the non-identity elements of \(G\), with two elements adjacent exactly when they generate \(G\). A coclique in a graph is an empty induced subgraph, so a coclique in \(\Gamma(G)\) is a subset of \(G\) such that no pair of elements generate \(G\). A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of \(G\) form a coclique in \(\Gamma(G)\), but this coclique need not be maximal. In this paper we determine when the intransitive maximal subgroups of \(\mathrm{S}_n\) and \(\mathrm{A}_n\) are maximal cocliques in the generating graph. In addition, we prove a conjecture of Cameron, Lucchini, and Roney-Dougal in the case of \(G = \mathrm{A}_n\) and \(\mathrm{S}_n\), when \(n\) is prime and \({n \neq \frac{q^d-1}{q-1}}\) for all prime powers \(q\) and \(d \geq 2\). Namely, we show that two elements of \(G\) have identical sets of neighbours in \(\Gamma(G)\) if and only if they belong to exactly the same maximal subgroups.

Mathematics Subject Classifications: 20D06, 05C25, 20B35