Finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index

We determine the composition factors of the finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index.     

二极大子群为TI-子群的有限群的类保持Coleman自同构

设$H$是有限群$G$的一个子群,若对任意$g\in G$, $H\cap H^g=1$或者$H$,则称$H$为TI-子群. 设$G$是一个所有二极大子群为TI-子群的有限群,本文证明了$G$的每个类保持Coleman自同构是内自同构. 作为本结果的一个直接推论,得到了这样的群$G$有正规化子性质.     

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N?H = N^|Ω| ? H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.     

The normalizer property of integral group rings of semidirect products of finite 2-closed groups by rational groups

Let G = N?Q be a semidirect product of a finite 2-closed group N by a rational group Q. It is shown that under some conditions the normalizer property holds for G.     

Finite Groups in Which Sylow Normalizers Admit Supplements

For a group G and a subgroup M of G, we say that a subgroup A of G is a supplement to M in G, if G = MA. We prove that a finite group in which every Sylow normalizer admits a nilpotent supplement is solvable. In particular we confirm a conjecture of Buchthal. Moreover we investigate finite groups in which Sylow normalizers admit solvable supplements. Supported by SRFDP(20010001046)     

On the group with the same orders of Sylow normalizers as the finite projective special unitary group

In this paper we prove that a finite group G is isomorphic to the finite projective special unitary group Un(q) if and only if they have the same order of Sylow r-normalizer for every prime r.     

具有阿贝尔Sylow 2-子群的有限群的整群环的正规化子性质

Mazur猜想:具有阿贝尔Sylow 2-子群的有限群有正规化子性质.设G是一个有限群,N是G的一个正规子群且Z(G/N)仅有平凡单位,本文建立了由Z(G/N)中单位诱导的G的自同构与N的Coleman自同构之间的联系,在此基础上证明了若G是一个具有阿贝尔Sylow 2-子群的有限群且Z(G/F*(G))仅有平凡单位,则Mazur猜想对G成立.     

Coleman automorphisms of holomorphs of finite abelian groups

Let K be a finite abelian group and let H be the holomorph of K. It is shown that every Coleman automorphism of H is an inner automorphism. As an immediate consequence of this result, it is obtained that the normalizer property holds for H.