Some results on p-nilpotence and supersolvability of finite groups

摘要

Let G be a finite group. A subgroup K of a group G is called an H-subgroup of G if N-G(K) boolean AND K-x <= K for all x epsilon 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 not less than or equal to P-1 not less than or equal to(...)not less than or equal to P-n = P is called a maximal chain of P provided that vertical bar P-i : Pi-1 vertical bar = 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 not less than or equal to P-1 not less than or equal to(...)not less than or equal to P-i not less than or equal to(...)not less than or equal to P-n = P such that P-i epsilon 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.