Groups with small deviation for non-subnormal subgroups

We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent, while a Baer group with deviation at most 1 has all of its subgroups subnormal.      

Factorization of Periodic Locally Solvable Groups by Locally Nilpotent and Nilpotent Subgroups

We establish a series of new results concerning periodic locally solvable and finite solvable groups G = AB with locally nilpotent or nilpotent subgroups A and B.     

Infinite groups with Sylow permutable subgroups

If a subgroup H of a periodic group G satisfies HP = PH for all Sylow subgroups P of G, then we call H a Sylow-permutable, or S-permutable, subgroup of G. It is well known that S-permutability is not a transitive relation. In this paper, we study infinite periodic groups in which the relation to be S -permutable is transitive (PST-groups) and infinite periodic groups whose ascendant subgroups are S-permutable (ASP-groups).     

On Groups with Finite Abelian Section Rank Factorized by Mutually Permutable Subgroups

It is proved that if G = AB is a soluble group with finite abelian section rank which is factorized by two mutually permutable finite-by-nilpotent subgroups A and B such that A′ and B′ are locally nilpotent, then also the normal closure ? A′, B′ ?^G is locally nilpotent and the subgroups A′ and B′ are ascendant in G.     

Kazhdan's property (T), L^2-spectrum and isoperimetric inequalities for locally symmetric spaces

Let V = G\G/KV =\Gamma\backslash G/K be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups G ì G\Gamma\subset G in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).     

On One Class of Separable Dedekind Groups

We describe locally solvable groups G such that all their infinite subgroups that do not belong to a certain proper subgroup of the group under consideration are normal.     

Stochastic processes on geometric loop groups,diffeomorphism groups of connected manifolds,and associated unitary representations

This paper is devoted to the investigation of semidirect products of loop groups and homeomorphism or diffeomorphism groups of finite-and infinite-dimensional real, complex, and quaternion manifolds. Necessary statements about quaternion manifolds with quaternion holomorphic transition mappings between charts of atlases are proved. It is shown that these groups exist and have the structure of infinite-dimensional Lie groups, i.e., they are continuous or differentiable manifolds and the composition (f, g) ↦ f ⁻¹ g is continuous or differentiable depending on the smoothness class of groups. Moreover, it is proved that in the cases of complex and quaternion manifolds, these groups have the structures of complex and quaternion manifolds, respectively. Nevertheless, it is proved that these groups do not necessarily satisfy the Campbell-Hausdorff formula even locally outside of the exceptional case of a group of holomorphic diffeomorphisms of a compact complex manifold. Unitary representations of these groups G′, including irreducible ones, are constructed by using quasi-invariant measures on groups G relative to dense subgroups G′. It is proved that this procedure provides a family of cardinality card(ℝ) of pairwise nonequivalent, irreducible, unitary representations. The differentiabilty of such representations is studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 28, Algebra and Analysis, 2005.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

Abnormal,Pronormal, Contranormal,and Carter Subgroups in Some Generalized Minimax Groups

ABSTRACT

Some properties of abnormal and pronormal subgroups in generalized minimax groups are considered. For generalized minimax groups (not only periodic) whose locally nilpotent residual is nilpotent and satisfies Min-G the existence of Carter subgroups and their conjugations have been proven. Some generalizations of results of J. Rose on abnormal and contranormal subgroups have been also obtained.     

On totally inert simple groups

A subgroup H of a group G is inert if |H: H ∩ H ^g | is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally graded simple groups cannot be totally inert.     

On Complementability of Nonmetacyclic Subgroups in Finite A-Groups

We study groups G that satisfy the following conditions: (i) G is a finite solvable group with nonprimary metacyclic second subgroup and (ii) all Sylow subgroups of the group G are elementary Abelian subgroups. We describe the structure of groups of this type with complementable nonmetacyclic subgroups.     

Locally minimal topological groups 2

We continue in this paper the study of locally minimal groups started in Außenhofer et al. (2010) [4]. The minimality criterion for dense subgroups of compact groups is extended to local minimality. Using this criterion we characterize the compact abelian groups containing dense countable locally minimal subgroups, as well as those containing dense locally minimal subgroups of countable free-rank. We also characterize the compact abelian groups whose torsion part is dense and locally minimal. We call a topological group G almost minimal if it has a closed, minimal normal subgroup N such that the quotient group G/N is uniformly free from small subgroups. The class of almost minimal groups includes all locally compact groups, and is contained in the class of locally minimal groups. On the other hand, we provide examples of countable precompact metrizable locally minimal groups which are not almost minimal. Some other significant properties of this new class are obtained.     

On the products of ℙ-subnormal subgroups of finite groups

A subgroup H of a finite group G is called ℙ-subnormal in G whenever H either coincides with G or is connected to G by a chain of subgroups of prime indices. If every Sylow subgroup of G is ℙ-subnormal in G then G is called a w-supersoluble group. We obtain some properties of ℙ-subnormal subgroups and the groups that are products of two ℙ-subnormal subgroups, in particular, of ℙ-subnormal w-supersoluble subgroups.     

ON MINIMAL NON-<Emphasis Type="Italic">MSP</Emphasis>-GROUPS

A finite group G is called an MSP-group if all maximal subgroups of the Sylow subgroups of G are S-quasinormal in G: We give a complete classification of groups that are not MSP-groups but all their proper subgroups are MSP-groups.     

Confined subgroups in periodic simple finitary linear groups

A subgroupX of the locally finite groupG is said to beconfined, if there exists a finite subgroupF≤G such thatX ^g∩F≠1 for allg∈G. Since there seems to be a certain correspondence between proper confined subgroups inG and non-trivial ideals in the complex group algebra ℂG, we determine the confined subgroups of periodic simple finitary linear groups in this paper. Dedicated to the memory of our friend and collaborator Richard E. Phillips     

Trifactorized locally finite groups with min-p for every prime p

A group G is trifactorized if G = AB = AC = BC with three subgroups A, B and C of G. Some structural theorems about trifactorized locally finite groups with minimum condition on p-subgroups for every prime p are proved. For instance, it is shown that G is locally supersoluble (locally nilpotent) if A and B are locally nilpotent and C is locally supersoluble (locally nilpotent). The second and the third author wish to thank the Institute of Mathematics of the University of Mainz for their excellent hospitality during the preparation of this paper. The second author is grateful to the University of Stellenbosch, South Africa, and the third author to the Deutsche Forschungsgemeinschaft (DFG) for financial assistance.     

On Periodic Locally Solvable Groups Decomposable into the Product of Two Locally Nilpotent Subgroups

We establish new results concerning various properties of a periodic locally solvable group G = A B with locally nilpotent subgroups A and B one of which is hyper-Abelian.