Abstract :
Let M be a maximal subgroup of a finite group G; then a subgroup C of G is said to be a completion of M in G if C is not contained in M while every proper subgroup of C which is normal in G is contained in M. The set, I(M), of all completions of M is called the index complex of M in G. Set P(M) = C ε I(M) vertical, dash C is maximal in I(M) and G = CM. The purpose of this note is to prove: If a finite group G is S4-free, then G is supersolvable if and only if, for each maximal subgroup M of G, P(M) contains an element C with C/K(C) cyclic.
