On compact engel groups

In 1992, Wilson and Zelmanov proved that a profinite Engel group is locally nilpotent. Here we prove the stronger result that every compact Engel group is locally nilpotent.     

On amenable and compact groups

In the sequel ofB. E. Johnson's work on amenable Banach algebras we characterize amenable and compact groups.     

On quasiresolvent periodic abelian groups

This is a continuation of [1]. We introduce the concept of a primarily quasiresolvent periodic abelian group and describe primarily quasiresolvent and 1-quasiresolvent periodic abelian groups. We construct an example of a quasiresolvent but not primarily quasiresolvent periodic abelian group. For a direct sum of primary cyclic groups we obtain criteria for a group to be quasiresolvent, 1-quasiresolvent, and resolvent, and establish relations among them. We construct a set S of primes such that the direct sum of some cyclic groups of orders p ∈ S is not a quasiresolvent group.     

On multipliers on compact Lie groups

In this note we announce L ^p multiplier theorems for invariant and noninvariant operators on compact Lie groups in the spirit of the well-known Hörmander-Mikhlin theorem on ? ⁿ and its versions on the torus $\mathbb{T}^n$ . Applications to mapping properties of pseudo-differential operators on L ^p -spaces and to a priori estimates for nonhypoelliptic operators are given.     

On periodic groups of automorphisms of extremal groups

It is proved that if a periodic group has an extremal normal divisor , determining a complete abelian factor group , then the center of the group contains a complete abelian subgroup , satisfying the relation and intersecting on a finite subgroup. It is also established with the aid of this proposition that every periodic group of automorphisms of an extremal group is a finite extension of a contained in it subgroup of inner automorphisms of the group .Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 91–96, July, 1968.     

On projective representations for compact quantum groups

We study actions of compact quantum groups on type I-factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz?s results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).     

On stable cohomotopy groups of compact spaces

The stable cohomotopy theory is a useful tool for studying global properties of compact spaces. In the stable shape theory we have infinite-dimensional versions of the Whitehead theorem. They require to use the stable cohomotopy groups.     

On periodic groups with narrow spectrum

We study groups with no elements of big orders. We prove that if the set of element orders of G is {1, 2, 3, 4, p, 9}, where p ∈ {7, 5}, then G is locally finite.     

On the structure of compact torsion groups

It is shown that every profinite torsion group has a finite series of closed characteristic subgroups in which each factor either is a pro-p-group for some primep or is isomorphic to a Cartesian product of isomorphic finite simple groups.     

On cocycle twisting of compact quantum groups

We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C^∗-algebra of the quantum group changes.     

On characterized subgroups of compact abelian groups

Let X be a compact abelian group. A subgroup H of X   is called characterized if there exists a sequence u=(_un)$\mathbf{u}=(u_n)$ of characters of X   such that H=_su(X)$H=s_{\mathbf{u}}(X)$, where _su(X):={x∈X:(_un,x)→0 in T}$s_{\mathbf{u}}(X):=\{x\in X:(u_n,x)\to 0\text{in}\mathbb{T}\}$. Every characterized subgroup is an _Fσδ$F_{\sigma \delta }$-subgroup of X  . We show that every _Gδ$G_{\delta }$-subgroup of X is characterized. On the other hand, X   has non-characterized _Fσ$F_{\sigma }$-subgroups.     

On mixed Abelian groups with periodic groups of automorphisms

In the paper, sufficient conditions for the splittability of mixed Abelian groups with periodic automorphism groups are established. Classes of mixed splittable Abelian groups with perfect holomorphs are distinguished. Translated fromMaternaticheskie Zametki, Vol. 61, No. 4, pp. 483–493, April, 1997. Translated by A. I. Shtern     

On the commutativity degree of compact groups

In any finite group G, the commutativity degree of G (denoted by d(G)) is the probability that two randomly chosen elements of G commute. More generally, for every n ≥ 2 the nth commutativity degree (denoted by d _n (G)) is the probability that a randomly chosen ordered (n + 1)-tuple of the group elements is mutually commuting. The aim of this paper is to generalize the definition of d(G) and d _n (G) to every compact group G (infinite and even uncountable). We shall state some results concerning compact groups and we will extend some results in Erfanian et al. (Comm. Algebra 35 (2007), 4183–4197) and Lescot (J. Algebra 177 (1995), 847–869).     

On Sylow subgroups of Shunkov periodic groups

We study the structure of Sylow 2-subgroups in Shunkov periodic groups with almost layer-finite normalizers of finite nontrivial subgroups. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1548–1556, November, 2005.