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

Abstract

The generating graph G(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 G(G) is denoted by ω(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 ω(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 ω(G) ≥ (1 + o(1))m(S), where m(S) denotes the minimal index of a proper subgroup in S. © 2009 American Mathematical Society.