On the clique number of the generating graph of a finite group

Abstract:

The generating graph $\Gamma (G)$ of a finite group $G$ is the graph defined on the elements of $G$ with an edge connecting two distinct vertices if and only if they generate $G$. The maximum size of a complete subgraph in $\Gamma (G)$ is denoted by $\omega (G)$. We prove that if $G$ is a non-cyclic finite group of Fitting height at most $2$ that can be generated by $2$ elements, then $\omega (G) = q+1$, where $q$ is the size of a smallest chief factor of $G$ which has more than one complement. We also show that if $S$ is a non-abelian finite simple group and $G$ is the largest direct power of $S$ that can be generated by $2$ elements, then $\omega (G) \leq (1+o(1))m(S)$, where $m(S)$ denotes the minimal index of a proper subgroup in $S$.