On the Norm of Finite Groups

• 来源期刊: 代数集刊（英文版）
• 关键词: norm p-nilpotent group p-group
• 年，卷（期）: 2007,（4）
• 页码范围: 605-612
• 页数: 8页
• 语种: 英文

摘要

The norm N(G) of a group G is the intersection of normalizers of all the subgroups of G. Let G be a finite group, p a prime dividing the order of G, and P a Sylow p-subgroup of G. In this paper, it is proved that G is p-nilpotent if Ω1(P)≤N(NG(P)),and when p = 2, Ω2(P) = <Ω1(P), x | x ∈ P is quasi-central in NG(P) and o(x) = 4>.Some applications of this result are given. Finally, a class of finite p-groups in which the index of the norm is exactly p is described.