有限群整群环的正规化子性质

Hertweck的反例说明,一个有限群即使它的一个正规子群和它对应的商群具有正规化子性质,该有限群也未必有正规化子性质.本文证明如下主要结果:设G是一个有限群,N是G的一个正规子群且Z(G/N)的中心单位是平凡单位.如果N的Sylow 2-子群是N的一个直因子,则G有正规化子性质.     

有限ATI-群的类保持Coleman自同构

设G是一个有限群,对G的任意阿贝尔子群A及任意g∈G,若A∩A~g=1或A,则称G为一个ATI-群.本文证明了,对任意p∈τ(G),如果ATI-群G的一个p-方幂阶类保持自同构在G的任意Sylow子群上的限制等于G的某个内自同构的限制,则它必定是一个内自同构.作为该结果的一个直接推论,我们也证明了有限ATI-群G有正规化性质.     

具有一个T．I．Sylow 2-子群的有限群的类保持Coleman自同构

设G是一个有限群,它的Sylow 2-子群是T.I.集,证明了如果G的2的方幂阶类保持自同构在G任意的Sylow子群上的限制等于G的某个内自同构的限制,则它一定是一个内自同构.对这样的自同构的研究是由整群环的同构问题所引起的.     

关于Glauberman-Isaacs特征标对映的注记

假设群A经自同构互素地作用在G上.设χ是G的一个A-不变不可约特征标,π(G,A)表示Glauberman-Isaacs特征标对映.对于B≤A,T.R.Wolf曾猜想χπ(G,A)是χπ(G,B)a的一个不可约成份,此处C=CG(A).设G=N(X)H且(|N|,|H|)=1,假定H是A-不变的且N是一个Sylow塔群,N的Sylow-子群是交换的.在本文中,我们证明了如果这个猜想对所有H的A-不变子群成立,则猜想对G也成立.     

有限π-拟幂零群

本文把有限幂零群的概念推广为有限π-拟幂零群,证明了这类群的一系列性质。文中定理9推广了[1]的两个主要结果。 本文涉及的群G均指有限群。 定义1 设π为某素数集合。N≤G。若(?)P_i∈π,G的Sylow p_i-子群均与N可换,则称N为G的一个π-拟正规子群。(当p_i(?)°(G)时,G的Sylow p_i-子群理解为G的单位元群1;若π为空集,则1是G的唯一的π-子群。此时G的每个子群是π-拟正规的)[2]。     

所有非平凡自同构均无固定点的有限群

设G是一个有限群,证明了G的每个非平凡自同构均为无固定点自同构当且仅当G是阿贝尔单群.作为应用,证明了群G的全形是以G为Frobenius核的Frobenius群当且仅当G是奇素数阶群.     

关于广义超特殊p-群的自同构群

用如下的方式确定了广义超特殊p-群G的自同构群.设|G|=p2n+m,|ζG|=pm,|N|=pl并且G'≤N≤ζG,其中n≥1且m≥2.AutnG表示AutG中平凡地作用在N上的所有自同构形成的正规子群.则(1)当p是奇素数时,AutG/AunG≌Z(p-1)pl-1.进一步地,(i)如果G的幂指数是pm,则Autn...     

有限群的最大子群的性质对群结构的影响

有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤HG=CoreG(H).本文证明了:设F是一个包含超可解群系u的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F*(H)的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.     

F-S-可补的子群对有限群结构的影响

设F是一个群系.群G的一个子群H在G中F-S-可补,如果存在G的子群K,使得G=HK且K/K∩HG∈F,其中HG表示G包含在H中的最大的正规子群.本文利用群系理论研究子群的F-S-可补性对有限群结构的影响,得到如下结论:设F是子群闭的局部群系,G是有限群且GF是可解的.则G∈F的充要条件是下列条件之一:(1)G存在正规子群N使得G/N∈F且N的极小子群及4阶循环子群(p=2)均在G中F-S-可补.(2)G存在正规子群N使得G/N∈F,N的4阶循环子群在G中有F-S-补且N的极小子群皆包含在Z∞F(G)中.应用这些结论,可以得到一些推论,其中包括已知的相关结果.     

Norm商群的Sylow子群皆循环的有限群

设$G$是有限群, $N(G)$为$G$的norm, 则$N(G)$是$G$的正规化G的每个子群的特征子群. 我们在下列条件之一下,研究了$G$的结构：1) Norm商群$G/N(G)$是循环群;2) Norm商群$G/N(G)$的所有Sylow子群都是循环群,特别地当$G/N(G)$的阶是无平方因子数时.     

关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.     

Involutions in groups of finite Morley rank of degenerate type

In a connected group of finite Morley rank, if the Sylow 2-subgroups are finite then they are trivial. The proof involves a combination of model-theoretic ideas with a device originating in black box group theory.     

On a Fitting length conjecture without the coprimeness condition

Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p ⁿ q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.     

Coleman automorphisms of finite groups

An automorphism of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer of G in the units of the integral group . Let Out be the image of these automorphisms in Out. We prove that Out is always an abelian group (based on previous work of E. C. Dade, who showed that Out is always nilpotent). We prove that if no composition factor of G has order p (a fixed prime), then Out is a -group. If O, it suffices to assume that no chief factor of G has order p. If G is solvable and no chief factor of has order 2, then , where is the center of . This improves an earlier result of S. Jackowski and Z. Marciniak. Received: 26 May 2000; in final form: 5 October 2000 / Published online: 19 October 2001     

Class-Preserving Coleman Automorphisms of Finite Groups

 Let G be a finite group whose Sylow 2-subgroups are either cyclic, dihedral, or generalized quaternion. It is shown that a class-preserving automorphism of G of order a power of 2 whose restriction to any Sylow subgroup of G equals the restriction of some inner automorphism of G is necessarily an inner automorphism. Interest in such automorphisms arose from the study of the isomorphism problem for integral group rings, see [6, 7, 13, 14].     

Finite Groups with a Pre-Fixed-Point-Free Power Automorphism

A power automorphism θ of a group G is said to be pre-fixed-point-free if C_G(θ) is an elementary abelian 2-group. G is called an E-group if G has a pre-fixed-point-free power automorphism. In this paper, finite E-groups, together with all their pre-fixed-point-free power automorphisms, are completely determined. Moreover, a characteristic of finite abelian groups is given, which explains some known facts concerning power automorphisms.     

On Preservation of Stability for Finite Extensions of Abelian Groups

We characterize preservation of superstability and ω-stability for finite extensions of abelian groups and reduce the general case to the case of p-groups. In particular we study finite extensions of divisible abelian groups. We prove that superstable abelian-by-finite groups have only finitely many conjugacy classes of Sylow p-subgroups. Mathematics Subject Classification: 03C60, 20C05.     

Finite groups with trivial Frattini subgroup

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.