Finite p-Nilpotent Groups with Some Subgroups Weakly ${\cal M}$-Supplemented

Abstract

Suppose that G is a finite group and H is a subgroup of G. Subgroup H is said to be weakly ${\cal M}$-supplemented in G if there exists a subgroup B of G such that (1) G = HB, and (2) if H₁/H_G is a maximal subgroup of H/H_G, then H₁B = BH₁ < G, where H_G is the largest normal subgroup of G contained in H. We fix in every noncyclic Sylow subgroup P of G a subgroup D satisfying 1 < ∣D∣ < ∣P∣ and study the p-nilpotency of G under the assumption that every subgroup H of P with ∣H∣ = ∣D∣ is weakly ${\cal M}$-supplemented in G. Some recent results are generalized.