On a group with three supersolvable subgroups of pairwise relatively prime indices

摘要

In this paper, we show that if G is a finite group with three supersolvable subgroups of pairwise relatively prime indices in G and G′ is nilpotent, then G is supersolvable. Let π(G) denote the set of prime divisors of |G| and max(π(G)) denote the largest prime divisor of |G|. We also establish that if G is a finite group such that G has three supersolvable subgroups H, K, and L whose indices in G are pairwise relatively prime, where p = max(π(G)) and q = max(π(L)) with L a Hall p′-subgroup of G, then G is supersolvable.