On ℋC-Subgroups and the Structure of Finite Groups

Abstract

A subgroup H of a group G is said to be an ℋC-subgroup of G if there exists a normal subgroup T of G such that G = HT and H ^g ∩ N _T (H) ≤ H for all g ∈ G. In this article, some new characterizations for p-nilpotency and supersolvability of finite groups are presented. In addition, we discuss the structure of ℋC*-group (a group G is called an ℋC*-group if every subgroup of G is an ℋC-subgroup of G) and prove that a group G is an ℋC*-group if and only if there is a nilpotent normal subgroup H of G such that G/H is abelian and each element of G induces a power automorphism on H by conjugation.

Key Words:

• p-Nilpotent group
• c-Normal subgroup
• ℋ-Subgroup
• ℋC-Subgroup
• Supersolvable group

2000 Mathematics Subject Classification:

• Primary 20D10
• 20D20

ACKNOWLEDGMENTS

The authors are very grateful to the referee for her/his valuable suggestions and useful comments.

The research of the work was partially supported by the National Natural Science Foundation of China (11071155), Shanghai Leading Academic Discipline Project (J50101) and NSF of Anhui Provence (KJ2008A030).

Notes

Communicated by T. Lenagan.