Universal G-spaces for proper actions of locally compact groups

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w(G)?τ, can act properly on Rτ?{0} such that Rτ?{0} contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ?τ. The metric cones Cone(G/H) with H⊂G a compact subgroup such that G/H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H⊂G does not exceed the weight of G.     

Partitions of groups into absolutely dense subsets

It is proved that any infinite Abelian group with finitely many elements of order two can be partitioned into two subsets that are dense in any nondiscrete group topology, and hence contain no cosets of infinite subgroups. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 706–711, May, 2000.     

Groups with the weak minimal condition for non-subnormal subgroups

Let G be a group with the property that there are no infinite descending chains of non-subnormal subgroups of G for which all successive indices are infinite. The main results are as follows. If G is locally nilpotent then either G is minimax or G has all subgroups subnormal; if G is a Baer group then all subgroups of G are subnormal. It is also proved that a generalised radical group with this property is soluble-by-finite and either is minimax or has all subgroups subnormal.     

Large soluble groups and the control of embedding properties

The aim of this survey article is to show that the behaviour of subgroups of finite rank in a (generalized soluble) group of infinite rank can be neglected. Some new results on this subject are also presented.     

Strongly indecomposable finite groups

Motivated by examples in infinite group theory, we classify the finite groups whose subgroups can never be decomposed as direct products.     

有限循环群的Fuzzy子群的等价类数

有限循环群G的F子群可以有无数个．但是．若当两个F子群的水平集构成的集合相等就称其等价的话，那么其等价类数是有限的。通过研究群的合成群列、商群列以及数的因数列和极大因数列找出了有限循环群的极大F子群和F子群的等价类数的求解公式．并给出二者之间的关系式．     

On a class of metahamiltonian groups

A group is metahamiltonian if all its non-abelian subgroups are normal. It is known that any infinite (generalized) soluble group whose proper subgroups are metahamiltonian is itself metahamiltonian. Moreover, it turns out that the study of soluble groups whose infinite proper subgroups are metahamiltonian can be reduced to the case of a finite extension of a central subgroup of type $p^\infty $ for some prime $p$ . A classification of metahamiltonian groups in this latter class is given.     

Zur Kohomologie von Gruppen mit abelschen Sylowuntergruppen

In 1962 Atiyah showed, with, the help of topological methods, the existence of a spectral sequence joining the cohomology ring of a finite group (with values in the rational integers) and the ring of generalized complex characters of the group. This paper undertakes to get that connection (at least in part) in a purely algebraic way. To that effect such ring filtrations of the character ring of a group are described that are functorially compatible with group morphisms and compatible with the induction of representations belonging to subgroups. The graded ring formed canonically by a filtration of that kind is to be compared with the cohomology ring of the same group, and it is shown, the group assumed to have abelian Sylow subgroups or to be a periodic group in a cohomological sense, that by taking a certain filtration its graded ring can be identified with a homomorphic part of the cohomology ring. (That corresponds to the infinite term of Atiyah's spectral sequence). The generalization of this result with respect to all finite groups has not yet been achieved; there is, however, a reduction theorem saying that it is sufficient to deal with groups of prime power order only. - As an application one gets lower bounds for the orders of the cohomology groups belonging to a finite group, especially, as a corollary, the theorem of its infinite cohomological dimension.     

Description of periodic p-harmonic functions on Cayley tree

We show that any periodic (with respect to normal subgroups of finite index of the group representation of the Cayley tree) p-harmonic function on a Cayley tree is a constant. For some normal subgroups of infinite index we describe a class of (non-constant) periodic p-harmonic functions. We also prove that linear combinations of the p-harmonic functions described for normal subgroups of infinite index are also p-harmonic.     

On Quasiconvex Subgroups of Word Hyperbolic Groups

We prove that a quasiconvex subgroup H of infinite index of a torsion free word hyperbolic group can be embedded in a larger quasiconvex subgroup which is the free product of H and an infinite cyclic group. Some properties of quasiconvex subgroups of word hyperbolic group are also discussed.     

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so‐called word‐based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k ‐tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so‐called graph‐based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph‐based distribution, while they are exponentially generic in the word‐based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

On one property of hypercyclic groups

We prove that the condition of periodicity of all normal Abelian subgroups of a group with increasing normal series with cyclic factors such that infinite factors are central is preserved on passing to subgroups of finite index. We present an example which demonstrates that the requirement that all infinite cyclic factors must be central cannot be omitted.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 436–438, March, 1995.     

The Distance-Regular Graphs of Valency Four

Let W be a Coxeter group acting as a matrix group by way of the dual of the geometric representation. Let L be the lattice of intersections of all reflecting hyperplanes associated with the reflections in this representation. We show that L is isomorphic to the lattice consisting of all parabolic subgroups of W. We use this correspondence to find all W for which L is supersolvable. In particular, we show that the only infinite Coxeter group for which L is supersolvable is the infinite dihedral group. Also, we show how this isomorphism gives an embedding of L into the partition lattice whenever W is of type An, Bn or Dn. In addition, we give several results concerning non-broken circuit bases (NBC bases) when W is finite. We show that L is supersolvable if and only if all NBC bases are obtainable by a certain specific combinatorial procedure, and we use the lattice of parabolic subgroups to identify a natural subcollection of the collection of all NBC bases.     

Homological stability for classical groups

Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.     

Intersections of Magnus subgroups and embedding theorems for cyclically presented groups

It is now known that the intersection of two Magnus subgroups M_i=〈Y_i〉 (1≤i≤2) in a one-relator group is either the free group F on Y₁∩Y₂ or the free product of F together with an infinite cyclic group (so-called exceptional intersection). Using this, we give conditions under which two embedding theorems for cyclically presented groups can be obtained. This provides a new method for proving such groups infinite. We also give a combinatorial method for checking the presence of exceptional intersections.     

On fixed subgroups of maximal rank

We show that, in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fix^{^} for some 4> L Aut(F). We further show that, in the rank two case, if the intersection of an arbitrary family of proper maximal rank fixed subgroups has rank two then all those subgroups are equal. In particular, every G < Aut(F) with r(FixG) = 2 is either trivial or infinite cyclic.     

Groups all of whose infinite abelian pd-subgroups are normal

The author studies groups in which any infinite Abelian pd-subgroup (p is a prime) is normal, on the assumption that the group indeed contains such subgroups (IH_p-groups). Necessary and sufficient conditions are established for a group to be an IH_p-group. Relationships are established between the class of IH_p-groups and the class of groups in which all infinite Abelian subgroups are normal, and the class of groups in which all pd-subgroups are normal.Translated from Ukrainskii Matematicheskii Zhurnal, Vo. 44, No. 6, pp. 796–800, June, 1992.     

On the Direct Indecomposability of Infinite Irreducible Coxeter Groups and the Isomorphism Problem of Coxeter Groups

In this article, we prove that any irreducible Coxeter group of infinite order, which is possibly of infinite rank, is directly indecomposable as an abstract group. The key ingredient of the proof is that we can determine, for an irreducible Coxeter group W, the centralizers in W of the normal subgroups of W that are generated by involu-tions. As a consequence, the problem of deciding whether two general Coxeter groups are isomorphic is reduced to the case of irreducible ones. We also describe the automorphism group of a general Coxeter group in terms of those of its irreducible components.     

On Influence of Contranormal Subgroups on the Structure of Infinite Groups

Following Rose, a subgroup H of a group G is called contranormal, if G = H ^G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.