On generalization of minimal non-nilpotent groups

Abstract

A finite group is said to be p-closed, if its Sylow p-subgroup is normal. A minimal non-nilpotent group means that all of its proper subgroups are nilpotent but the group itself is not. O.J. Schmidt showed that this group is soluble and p-nilpotent for some prime p. The present paper generalizes minimal non-nilpotent groups and Schmidt’s result. Consider a group which has s (s ≤ 2) proper non-p-closed subgroups H _i , i = 1, …, s so that for any proper subgroup R of it, if R ≰ H _i , then R is p-closed. Obtain that if such a group G satisfies the condition H ₁ ≰ H ₂ and H ₂ ≰ H ₁, then (1) H ₁, H ₂ are maximal normal subgroups of G and H ₁ ∩ H ₂ ≠ 1. Let N be the minimal normal subgroup of G contained in H ₁ ∩ H ₂. Then (2) If N is insoluble, then G/N is a cyclic group and N/Φ(N) is a non-abelian simple group. (3) If N is soluble, then G/N (sometimes G itself) is a q-nilpotent group for some prime q.