ON THE CLIQUE NUMBER OF THE GENERATING GRAPH OF A FINITE GROUP

The generating graph Gamma(C) of a finite group G is the graph defined on the elements of C 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) <= (1 + o(1))m(S), where m(S) denotes the minimal index of a proper subgroup in S.