Maximal cocliques in the generating graphs of the alternating and symmetric groups

The generating graph \(\Gamma(G)\) of a finite group \(G\) has vertex set the non-identity elements \(G\), with two adjacent exactly when they generate \(G\). A coclique in is an empty induced subgraph, so subset such that no pair maximal if it contained larger coclique. It easy to see subgroup form \(\Gamma(G)\), but this need not be maximal. In paper we determine intransitive subgroups \(\mathrm{S}_n\) and \(\mathrm{A}_n\) are cocliques graph. addition, prove conjecture Cameron, Lucchini, Roney-Dougal case \(G = \mathrm{A}_n\) \(\mathrm{S}_n\), \(n\) prime \({n \neq \frac{q^d-1}{q-1}}\) for all powers \(q\) \(d \geq 2\). Namely, show have identical sets neighbours only belong same subgroups.Mathematics Subject Classifications: 20D06, 05C25, 20B35