SΦ-SUPPLEMENTED AND COPRIME ACTION IN FINITE GROUPS

An A-invariant subgroup H of G is said to be ASΦ-supplemented in G if there exists an A-invariant normal subgroup T of G such that G = HT and H ∩ T ≤ Φ(H), where Φ(H) is the Frattini subgroup of H. In this paper, we study SΦ-supplement in the coprime action setting so as to obtain several p-nilpotency criteria in terms of certain subsets of maximal invariant subgroups of its Sylow subgroups.