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.     

Locally nilpotent groups satisfying the weak minimum or maximum condition for subgroups of bounded nilpotency class

We study locally nilpotent groups containing subgroups of classc, c>1, and satisfying the weak maximum condition or the weak minimum condition on c-nilpotent subgroups. It is proved that nilpotent groups of this type are minimax and periodic locally nilpotent groups of this type are Chernikov groups. It is also proved that if a group G is either nilpotent or periodic locally nilpotent and if all of its c-nilpotent subgroups are of finite rank, then G is of finite rank. If G is a non-periodic locally nilpotent group, these results, in general, are not valid.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 384–389, March, 1992.     

“具有幂零局部子群的有限群”一文的注记

称有限群$G$为一个PN-群若 $G$非幂零群,且对$G$的每一个$p$-子群$P$, 或者$P$是$G$的正规子群, 或者$P \subseteq Z_\infty(G)$, 或者$N_G(P)$是幂零群, $\forall p \in \pi(G)$. 本文证明了PN-群是亚幂零群. 特别地, PN-群是可解的 且给出了PN-群结构定理的一个初等的、直观的、简洁的证明.     

有限幂零群通过单群扩张的整群环的正规化子性质

设G是一个有限幂零群通过单群的扩张,即G有一个幂零正规子群N,使得G/N是单群.本文证明了这样的有限群G具有正规化子性质.特别地,内可解群有正规化子性质.     

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2). Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 314–333, May–June, 1996.     

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.     

Finite Groups with Subnormal Residuals of Sylow Normalizers

Siberian Mathematical Journal - Considering a&nbsp;nonempty formation $ {\mathfrak{X}} $ of nilpotent groups, we prove that a group $ G $ is an extension of a nilpotent group by an $...     

Eine Bemerkung zur Nilpotenz lokal kompakter Gruppen

Let G be a locally compact group which is connected or finite dimensional and compact over its connected component. If every (closed) subgroup of G is nilpotent then G is nilpotent.     

恰有两个子群共轭类长的有限群

设G是有限群,Ns(G)表示G的子群共轭类长构成的集合.本文研究Ns(G)中只有两个元素时有限群G的结构,在非幂零情形时给出了G的完全分类,在幂零情形时获得了G的一些性质.     

A nilpotent group and its elliptic curve: Non-uniformity of local zeta functions of groups

A nilpotent group is defined whose local zeta functions counting subgroups and normal subgroups depend on counting points modp on the elliptic curvey ²=x ³−x. This example answers negatively a question raised in the paper of F. J. Grunewald, D. Segal and G. C. Smith where these local zeta functions were first defined. They speculated that local zeta functions of nilpotent groups might be finitely uniform asp varies. A proof is given that counting points on the elliptic curvey ²=x ³−x are not finitely uniform, and hence the same is true for the zeta function of the associated nilpotent group. This example demonstrates that nilpotent groups have a rich arithmetic beyond the connection with quadratic forms.     

Gruppenkopplungen Mit Zyklischer Ausgangsgruppe Bzv7. Zyklischer Ableitung

Two problems are treated: (A) We look for all couplings of a finite cyclic group G. (B) We try to determine all finite groups G bearing couplings with cyclic derivations. These problems will be solved completely, if G is a p-group and nilpotent, respectively. The results enable us to construct couplings with cyclic Dickson groups.     

Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)\leqq X?X^g{\rm C}_{G}(X\cap X^{{g}})\leqq X\cap X^{{g}} for all g ? G{g}\in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if N\triangleleft GN\triangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)\leqq XC_G(N\cap X)\leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.     

Locally nilpotent groups with a centralizer of finite rank

Locally nilpotent groups in which the centralizer of some finitely generated subgroup has finite rank are studied. It is shown that if G is such a group and F is a finitely generated subgroup with centralizer C_G(F) of finite rank, then the centralizer of the image of F in the factor group G/t(G) modulo the periodic part t(G) also has finite rank. It is also shown that G is hypercentral when F is cyclic and either G is torsion-free or all Sylow subgroups of the periodic part of C_G(F) are finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1511–1517, November, 1992.     

Nilpotency of the multiplicative group of a group ring

It is proven that if K is a commutative ring of characteristic p^m while group G contains p-elements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G is a finite p-group. Those group algebras KG are described for which the nilpotency classes of groups G and UKG coincide.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 191–200, February, 1972.In conclusion, the author wishes to express her gratitude to A. A. Bovdi for his scientific direction.     

On theta pairs for maximal subgroups ∗

For a maximal subgroup M of a finite group G, a θ -pair is any pair of subgroups (CD) of G such that (i) DjG, D>C, (ii) (M, C=G, <M,D> = M and (iii) C/D has no proper normal subgroup of G/D. A natural partial ordering is defined on the family of 0 -pairs. We study the further properties of the maximal 0 -pairs of M and obtain several results on 0 -pairs which imply G to be n -solvable, 7t - supersolvable and π - nilpotent     

Locally nilpotent groups with the min−∞−n condition

The study of locally nilpotent groups with the weak minimality condition for normal subgroups, the min^––n condition, is continued. The following results are obtained.THEOREM 1. A locally nilpotent group with the min^––n condition is countable.THEOREM 2. If G is a locally nilpotent group with the min^––n condition whose periodic part is nilpotent and the orders of the elements of the periodic part are bounded in the aggregate, then G=t(G)A, where the subgroup A is minimax.THEOREM 3. Suppose G is a locally nilpotent group with the min^––n condition and T is its periodic part. If T is nilpotent and G/T is Abelian, then G=TA, where the subgroup A is minimax.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 340–346, March, 1990.     

Semisimple crossed products of groups and rings

Let K?G be a crossed product of a multiplicative group G over an associative ring K with 1 and let C(G) be the center of G. If K has no C(G)-invariant ideals, then the Jacobson radical of the center of K?G is a nil ideal. In addition, if G is a ZA-group, then K?G is semisimple if and only if K?G has no central nilpotent elements.     

Finite groups in which every self-centralizing subgroup is nilpotent or subnormal or a TI-subgroup

Czechoslovak Mathematical Journal - Let G be a finite group. We prove that if every self-centralizing subgroup of G is nilpotent or subnormal or a TI-subgroup, then every subgroup of G is nilpotent...     

Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT _n (Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT _n (Z) has no proper existentially closed subgroups.     

剩余有限minimax 可解群的几乎正则自同构

设G 是一个剩余有限的minimax 可解群, α 是G 的几乎正则自同构, 则G/[G, α] 是有限群, 并且(1) 当α^p = 1 时, G 有一个指数有限的幂零群其幂零类不超过h(p), 其中h(p) 是只与素数p 有关的函数.(2) 当α² = 1 时, G 有一个指数有限的Abel 特征子群且[G, α]′ 是有限群.关键词剩余有限minimax 可解群几乎正则自同构