剩余有限minimax可解群的自同构

设G是剩余有限minimax可解群,α是G的自同构且φ:G→G(g→[g,α])是满射,则有以下结果:(1)当α~p=1时,G是幂零类不超过h(p)的幂零群的有限扩张,其中h(p)是只与p有关的函数;(2)当α~4=1时,G存在一个指数有限的特征子群H,使得H″≤Z(H)和C_H(α~2)是Abel群.并且C_G(α~2)和G/[G,α~2]都是Abel群的有限扩张.     

有限秩的可解群的正则自同构

设G是有限秩的剩余有限可解群或是有限秩的剩余有限可解群的有限扩张,α是G的一个索数p阶正则自同构且φ:G→G(g→[g,α])是满射,则G是幂零类不超过h(p)的幂零群,其中h(p)是只与p有关的函数.     

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.     

关于允许一个无不动点自同构群的有限群的可解性

关于允许一个无不动点自同构(群)的有限群的可解性的猜想是有限群研究中的一个重要问题。结果比较丰富的是限制该自同构群为一个p-群的情形。Thompson于1959年证明了p阶群的情形。Martineail于1971年证明了初等Abel p-群的情形。Rickman于1979年证明了p²阶群的情形。本文借助Glauberman的一个定理,对p=2或3的一般情形给出了肯定的回答。实际上是用较初等的方法证明了更为广泛一些的结论。     

有限线性空间的可解线-传递自同构群

设S是一个有限线性空间,G是S的自同构群的一个可解线-传递子群,则对于给定的线长k,除了有限对(S,G)外,S有v=pn个点,且G≤AΓL(1,pn).     

有限秩的可解群的剩余有限性质

本文讨论了有限秩的可解群的剩余有限性质,推广了Smelkin等人关于多重循环群的几个同类结果．     

剩余有限Minimax可解群的4阶正则自同构

设G是剩余有限minimax可解群,α是G的4阶正则自同构,则下面结果成立:(1)如果映射φ:G→G (g→[g,α])是满射,那么G是中心子群被亚Abel群的扩张.(2)C_G(α~2)和[G,n-1α~2]/[G,nα~2](n∈Z~+)都是Abel群的有限扩张.     

有限线性空间的可解线-传递自同构群

设S是一个有限线性空间 ,G是S的自同构群的一个可解线 传递子群 ,则对于给定的线长k ,除了有限对 (S ,G)外 ,S有v =pⁿ 个点 ,且G≤AΓL( 1 ,pⁿ ) .     

无限的可解SD2—群（II）—有限剩余情形

如果对多重循环群Ｇ的每个有限剩余Ｇ，Ｇ的真子群都具有可以由二元生成的，那么我们就把Ｇ叫做ＲＤ２－群，在本文里，我们确定了无限的ＲＤ２－群的结构，证明了ＲＤ２－群是可以由二元生成的，这些结构推广了作者已经得到了关于无限的可解ＳＤ２群的全部结果，见（４）。     

Polycyclic groups admitting a regular automorphism of order four

Let G be a polycyclic group and α a regular automorphism of order four of G. If the map φ: G→ G defined by g~φ= [g, α] is surjective, then the second derived group of G is contained in the centre of G. Abandoning the condition on surjectivity, we prove that C_G(α~2) and G/[G, α~2] are both abelian-by-finite.     

Regular automorphisms of order p2

Let G be a group, and let α be a regular automorphism of order p² of G, where p is a prime. If G is polycyclic-by-finite and the map ϕ : G →G defined by g^ϕ= [g,α] is surjective, then G is soluble. If G is polycyclic, then C_G(α^p) and G/[G,α^p] are both nilpotent-by-finite.     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

Carter subgroups of finite almost simple groups

In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups of every finite group are conjugate. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) for Support of Young Russian Scientists via projects MK-1455.2005.1 and MK-3036.2007.1; by SB RAS Young Researchers Support grant No. 29; via Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007.     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

On the prefrattini subgroups of finite soluble groups

We study the partially prefrattini groups of a finite soluble group. We prove that the set of all partially prefrattini subgroups associated with the Gaschütz system of complements to crowns is a Boolean lattice.     

On Coleman automorphisms of finite nilpotent groups by cyclic groups

Let G be a finite group and let N be a nilpotent normal subgroup of G such that G/N is cyclic. It is shown that under some conditions all Coleman automorphisms of G are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.