A characterization of Gaussian measures on locally compact Abelian groups

Let and be independent random variables having equal variance. In order that + and – be independent, it is necessary and sufficient that and have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when and take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of + and –, functions of and are considered which satisfy the associativity equation.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 759–762, November, 1977.     

Abelian groups

The present, fourth survey of review articles on Abelian groups includes works reviewed in the years 1979–1984. Also, as in the preceding surveys, no attention has been given to questions involving finite Abelian groups, topological groups, ordered groups, group algebras, modules, or the structure of subgroups, or to questions connected with logic. The word group throughout is understood to mean Abelian group (except in the case of the group of automorphisms of an Abelian group). Concepts and notation not defined in this survey can be found in the books [117, 118]. The letterZ throughout denotes the group (or ring) of integers,Q the group (or field) of rational numbers,Q _p the group (ring) of all rational numbers whose denominator is not divisible by the primep, I_p the group of allp-adic integers. The torsion part of an Abelian groupG is throughout denoted bytG. Translated from Itogi Nauki i Tekhniki. Seriya Algebra, Topologiya, Geometriya, Vol. 23, pp. 51–118.     

Abelian groups

In the present, third survey of reviews of articles on Abelian groups are included papers reviewed during 1972–1978. Here, as in the previous surveys, the questions concerning finite Abelian groups, topological groups, ordered groups, group algebras, modules, structures of subgroups, as well as the questions connected with logic, are not touched on.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 17, pp. 3–63, 1979.     

A theorem on direct expansions of Abelian groups

The following assertion is proved: if the basis subgroups of the periodic part t(G) of a non-denumerable Abelian group G have the same cardinality as G, then each of the subgroups contains, as a subgroup of the same cardinality as G, a direct component of G. The restriction on the cardinality of G is essential.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 879–884, December, 1973.     

Periodic groups acting freely on Abelian groups

We describe {2, 3}-groups without elements of order 27, acting freely on an Abelian group. In particular, it is proved that such groups are locally finite.     

On the Shunkov groups acting freely on Abelian groups

We prove the existence of a locally finite periodic part in every rank 1 Shunkov group with solvable finite subgroups, in every Shunkov group acting freely on an abelian group, and in the groups of affine transformations of neardomains with finite elements.     

A result about Abelian automorphism groups

Some new techniques are given to prove that two sorts of Abelian groups cannot function as the full automorphism groups of the finite groups. With generality these techniques made a breakthrough in MacHale’s problem. Project supported by the National Natural Science Foundation of China.     

Robinson's conjecture on Abelian groups

Let x₁,…,x_n be elements of a finite abelian group G, having respective orders k₁,…,k_n such that (x₁−1)(x₂−1)⋯(x_n−1)=0 in $\text{Z}\text{(G)}$, where n⩽1. We prove that min k_i≤n−1 with equalityossible if only if n−1 is prime. If all k_i are equal, and not divisible by the cube of a prime, we prove $\text{n⩾k}{}_1\text{(1+}\text{1}\text{r)}$ where r is the least prime dividing k₁. We also establish an inequality concerning coverings of a set by subsets.     

Abelian groups as autocommutator groups

In this article, we establish that every finite abelian group is isomorphic to the autocommutator subgroup of some finite abelian group. Received: 21 September 2007