局部环上正交群中一类子群的扩群

定出了局部环上正交群中一类子群的扩群,得到了如下结果:设R是局部环,M是R的唯一极大理想,O(2m,R)为R上正交群.对R的任意理想S,G(2m,S)表示子群{A BC D∈O(2m,R)|B∈Sm×m}.如果char(R)≠2,m≥3,G(2m,0)≤X≤G(2m,M),那么存在R的理想S,使得X=G(2m,S).     

欧氏环上特殊正交群的一类极大子群

定出欧氏环上特殊正交群的一类极大子群,得到如下结果:设R是带有欧氏映射σ的特征不为2的欧氏环且不是域,SO(2m,R)为R上的特殊正交群,R*=R\{0},l=min{σ(x)|x∈R*\U(R)},任取a∈R*\U(R)使σ(a)=l,记a在R中生成的主理想为M.那么当m≥3时,AB CD∈SO(2m,R)|B∈Mm×m是SO(2m,R)的一个极大子群.     

关于半完全环的K1群

本文研究了半完全环的K1群. 利用半局部环的K1群的已知结果和半完全环的构造, 证明了半完全环R的K1群是R的一些子环的K1群的直和与R的单位群的一个子群的商群.     

x_0-Φ满射环上酉群的构造

设 R 为 X_0-φ满射环,则在 Witt 指数 i(H)≥3时,R 上酉群 U_n(R,H)的满阶正规子群包含酉群的换位子群 E_n(R);在 Witt 指数 i(H)≥1及2,3为单位时,U_n(R,H)的子群 G 为E_n(R)-正规子群的充要条件为 E_n(R,A)(R,A),其中 A 由 G 唯一确定.特别当 R为交换环时,A 为 G 的阶理想.     

关于CAP-子群的一点注记（英文）

对于有限群G的每一主因子H/K来说,若G的子群L满足LH=LK或者L∩H=L∩K,则称L是G的CAP-子群.本文通过假设G的每个非循环Sylow子群P有一个子群D使得1〈｜D｜〈｜P｜,且P的所有阶为｜D｜和2｜D｜（若P是非交换2-群且｜P∶D｜〉2）的子群H是G的CAP-子群,得到G为p-幂零群的一个结果.     

有限局部环上酉群阶的计算

设K=F_(q^2),其特征为p, q=p^α,K有对合自同构ω:a→a^q. G是一个p 群，其阶为p^β, 群代数R=KG为一局部环. K的2阶自同构ω可延拓为R的一个2阶自同构，记为ω'，为方便，对任意a∈R, 记ω‘(a)为~a. R上2n级酉群定义为U_(2n)R={A∈GL_(2n)R|A(0,I^n,I^n,0)~A^t=(0,I^n,I^n,0)} 该文计算了U_(2n)R的阶.      

τ-拟置换子群对有限群结构的影响

群G的一个子群H称为τ-拟置换的,如果G有一个子群B满足G=N_G(H)B且HB=BH,同时对于B的Sylow q-子群Q,只要满足(|H|,q)=1但(|H|,|Q~G|)≠1,便有HQ=QH,其中q是|B|的任一素因子.研究了τ-拟置换子群对有限群结构的影响.应用极小阶反例的方法得到了群G是p-超可解群的一个新的判定,又利用群G的F-剩余子群G~F的性质以及群G的准素数子群的τ-拟置换性得到了群G的半直积结构     

有限可解(q)群与(s—q)群的一个注记

文[l].[2]分别研究了每个次正规子群为拟正规的有限群(即(q)群)以及每个次正规子群为s—q拟正规的有限群(即(s—q)群).本文利用广幂零群的概念对(q)群与(s—q)群给出了一个新的刻划,并得到内(s—q)群的完全分类。     

∑＊-嵌入子群对有限群的可解性的影响

设H是有限群G的子群, K/L是G的任一非Frattini主因子.如果对每一满足L≤A＜B≤K且A是B的极大子群的子群对（A,B）,都有HA=HB或者H∩A=H∩B,则称H是G的∑＊-嵌入子群.通过有限群G的某些子群的∑＊-嵌入性,给出了一些有限群G的正规子群为可解群的一些判别条件,推广了已有的一些结论.     

与群PSL2(R)相关的交叉积R(A,α)的一点注记

在上半复平面H上给定双曲测度dxdy/y2,群G=PSL2(R)在H上的分式线性作用导出了G在Hilbert空间L2(H,dxdy/y2)上的酉表示α.证明了交叉积R(A,α)是Ⅰ型von Neumann代数,其中A={Mf:f∈L∞(H,dxdy/y2)}.具体地,交叉积代数R(A,α)与von Neumann代数B(L2(P,v))-(×)LK是*-同构的,其中LK是G中子群K的左正则表示生成的群von Neumann代数.     

有限交换环上典型群的Sylow子群

令Ｒ为有限交换局部环，Ｍ表其唯一的极大理想，ｋ表其剩余类域．本文定出了Ｒ上的一般线性群ＧＬｎＲ，辛群Ｓｐ２ｎＲ及双曲正交群Ｏ２ｎＲ的Ｓｙｌｏｗ子群．一般讲，若ｃｈａｒｘ＝ｐ，上述三类典型群的Ｓｙｌｏｗ ｐ－子群分别同构于由某些特殊形式的矩阵生成的子群；若ｃｈａｒｋ≠ｐ，上述三类典型群的Ｓｙｌｏｗｐ－子群分别同构于一循环群或半二面体群与若干Ｚｐ型循环群的圈积。     

有限交换环上线性群的Carter子群

令Ｒ为有限交换局部环，Ｋ为其剩余类域．本文研究了Ｒ上一般线性群ＧＬｎＲ的Ｃａｒｔｅｒ子群的存在性及结构．得到的结果是：若ｃｈａｒＫ为奇数或Ｋ＝Ｆ２，ＧＬｎＲ中存在唯一的Ｃａｒｔｅｒ子群的共轭类，即Ｓｙｌｏｗ－２子群的正规化子；若ｃｈａｒＫ＝２且｜Ｋ｜＞２，ＧＬｎＲ中不含Ｃａｒｔｅｒ子群．     

Tychonoff property for linear groups

A criterion for a wide class of topological groups which includes linear discrete groups and Lie groups to be Tychonoff groups is established. The main result provides a criterion for an almost polycyclic group to have the Tychonoff property. By the well-known Tits alternative, this yields the required criterion for linear discrete groups. In conclusion it is pointed out that a particular case of the presented proof yields a Tychonoff property criterion for Lie groups. In addition, an example of a polycyclic group without Tychonoff subgroups of finite index is constructed. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 269–279, February, 1998. The author wishes to express his gratitude to R. I. Grigorchuk for setting the problem and his interest in the work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00182 and by the American Mathematical Society Fund.     

矩阵环的有限乘法子群

将Minkowski关于有限整数矩阵群的著名结果推广到一般的环上,主要结果是证明了:对任意环R,如果R的加法群为有限生成的自由Abel群,则R的所有乘法可逆元构成的群U(R)中的有限子群精确到同构只有有限多个.     

Completion of Linearly Ordered Groups

We give examples of linearly ordered groups that are not embeddable in divisible orderable. In the first example, the group does not embed in any divisible group with strictly isolated unity. In the second example, the group in question is an O*-group, and in the third, it is a group with a central system of convex subgroups. To my teacher A. I. Kokorin Supported by RFBR grant Nos. 96-01-00358, 99-01-00335, and 03-01-00320. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 664–681, November–December, 2005.     

子群的θ-偶和群的结构

研究极大子群和2-极大子群的θ-偶对群结构的影响.设G是有限群,本文得到了:如果G的每一个极大子群M都有极大θ-偶(C,D),使MC=G且C/D是2-闭的,那么G可解;如果G的每一个2-极大子群H都有θ-偶(C,D),使C/D幂零且G=HC,那么G是幂零.     

Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

A criterion for subnormality of subgroups in finite groups

Based on Wielandt’s criterion for subnormality of subgroups in finite groups, we study 2-maximal subgroups of finite groups and present another subnormality criterion in finite solvable groups.     

Constructions for Terraces and R‐Sequencings,Including a Proof That Bailey's Conjecture Holds for Abelian Groups

Every abelian group of even order with a noncyclic Sylow 2‐subgroup is known to be R‐sequenceable except possibly when the Sylow 2‐subgroup has order 8. We construct an R‐sequencing for many groups with elementary abelian Sylow 2‐subgroups of order 8 and use this to show that all such groups of order other than 8 also have terraces. This completes the proof of Bailey's Conjecture in the abelian case: all abelian groups other than the noncyclic elementary abelian 2‐groups have terraces. For odd orders it is known that abelian groups are R‐sequenceable except possibly those with noncyclic Sylow 3‐subgroups. We show how the theory of narcissistic terraces can be exploited to find R‐sequencings for many such groups, including infinitely many groups with each possible of Sylow 3‐subgroup type of exponent at most 3¹² and all groups whose Sylow 3‐subgroups are of the form or .