Some Results on p-Nilpotence and Supersolvability of Finite Groups

Abstract

Let G be a finite group. A subgroup K of a group G is called an ℋ-subgroup of G if N G (K) ∩ K x ≦ K for all x ∈ G. The set of all ℋ-subgroups of G will be denoted by ℋ(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P 0 ≨ P 1 ≨ ··· ≨ P n = P is called a maximal chain of P provided that |P i : P i−1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P 0 ≨ P 1 ≨ ··· ≨ P i ≨ ··· ≨ P n = P such that P i ∈ ℋ(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups.