An Analog of the Baer-Suzuki Theorem for Infinite Groups

We consider a group without infinite ascending chains of nilpotent subgroups and prove that if every two elements of some conjugacy class generates a nilpotent subgroup then the whole class also generates a nilpotent subgroup.     

On Ordering the Groups with Nilpotent Commutant

We prove that every group with nilpotent commutant, having an abelian normal subgroup such that the factor by this subgroup is nilpotent, is preorderable if and only if the group is -torsion-free. An example is exhibited of a nonorderable -torsion-free group with two-step nilpotent radical. This example demonstrates that for the variety of groups with nilpotent commutant the absence of -torsion in a group is not a sufficient condition for orderability.     

Maximal and Sylow Subgroups of Solvable Finite Groups

The structure of finite solvable groups in which any Sylow subgroup is the product of two cyclic subgroups is studied. In particular, it is proved that the nilpotent length of such a group is no greater than 4. It is also proved that the nilpotent length of a finite solvable group in which the index of any maximal subgroup is either a prime or the square of a prime or the cube of a prime does not exceed 5.     

Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.     

On the lower central series of the free nilpotent groups of finite rank

In their article, “On the derived subgroup of the free nilpotent groups of finite rank” R. D. Blyth, P. Moravec, and R. F. Morse describe the structure of the derived subgroup of a free nilpotent group of finite rank n as a direct product of a nonabelian group and a free abelian group, each with a minimal generating set of cardinality that is a given function of n. They apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank. We generalize their main result to investigate the structure of the other terms of the lower central series of a free nilpotent group of finite rank, each again described as a direct product of a nonabelian group and a free abelian group. In order to compute the ranks of the free abelian components and the size of minimal generating sets for the nonabelian components we introduce what we call weight partitions.     

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.     

On the fundamental group of manifolds with almost nonnegative Ricci curvature

Gromov conjectured that the fundamental group of a manifold with almost nonnegative Ricci curvature is almost nilpotent. This conjecture is proved under the additional assumption on the conjugate radius. We show that there exists a nilpotent subgroup of finite index depending on a lower bound of the conjugate radius.

     

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.     

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     

Structure of a group with elements of order at most 4

We prove that every group in which the order of each element is at most 4 either possesses a nontrivial class 2 nilpotent normal Sylow subgroup or includes a normal elementary abelian 2-subgroup the quotient by which is isomorphic to the nonabelian group of order 6.     

幂零群的若干充分条件

本文利用Ｃ－正规子群的性质，得到了若干有限群为幂零群的充分条件，推广了一些著名定理，如Ｔｈｏｍｐｓｏｎ定理及Ｉｔ＾ｏ定理。     

有限群的c-半置换子群

研究了有限群的结构问题,利用子群c-半置换和完全c-半置换的定义和性质,通过对有限群sylow子群的2-极大子群的研究,获得了有限群幂零、p-幂零的充分条件和另外两个决定群结构的充要条件.     

On D. I. Moldavanskii's Question About p-Separable Subgroups of a Free Group

Siberian Mathematical Journal - We prove that every free nonabelian group has a finitely generated isolated subgroup not separable in the class of nilpotent groups. This enables us to give a...     

On Hall Normally Embedded Subgroups of Finite Groups

A subgroup H of a finite group G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure H ^G . Groups which contain a Hall normally embedded subgroup of order d for every factor d of | G | are characterized. Such groups are supersolvable with a cyclic nilpotent residual of square-free order.     

Finite groups with hall subnormally embedded Schmidt subgroups

Abstract

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.     

Finite groups with Hall subnormally embedded Schmidt subgroups

Abstract

A subgroup H of a finite group G is said to be Hall subnormally embedded in G if there is a subnormal subgroup N of G such that H is a Hall subgroup of N. A Schmidt group is a finite non-nilpotent group whose all proper subgroups are nilpotent. We prove the nilpotency of the second derived subgroup of a finite group in which each Schmidt subgroup is Hall subnormally embedded.     

Algebraic Entropy of Elementary Amenable Groups

We prove that any finitely generated elementary amenable group of zero (algebraic) entropy contains a nilpotent subgroup of finite index or, equivalently, any finitely generated elementary amenable group of exponential growth is of uniformly exponential growth. We also show that 0 is an accumulation point of the set of entropies of elementary amenable groups.     

Dominions in finitely generated nilpotent groups

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.     

Pseudocomplementation in (Normal) Subgroup Lattices

The goal of this article is to study finite groups admitting a pseudocomplemented subgroup lattice (PK-groups) or a pseudocomplemented normal subgroup lattice (PKN-groups). In particular, we obtain a complete classification of finite PK-groups and of finite nilpotent PKN-groups. We also study groups with a Stone normal subgroup lattice, and we classify finite groups for which every subgroup has a Stone normal subgroup lattice. Finally, we obtain a complete classification of finite groups for which every subgroup is monolithic.     

有限生成的幂零群的共轭分离性质

研究了有限生成的幂零群中元素的共轭分离问题.设ω表示全部素数组成的集合,π是ω的非空真子集,G是有限生成的幂零群,则下述三条等价:(i)如果x和y是G中的任意两个不共轭的元素,则x和y在G的某个有限p-商群中不共轭,其中p∈π;(ii)如果x和y是G中的任意两个不共轭的元素,则x和y在G的某个有限π-商群中不共轭;(iii)G的挠子群T(G)是π-群且G/T(G)是Abel群.同时举例说明:设G是有限生成的无挠幂零群,对于任意素数p,x和y都在G的有限p-商群G/G~p中共轭,但x和y在G中不共轭.