Small subsets of groups. (Q1032906)

Relations between different kinds of ``small'' subsets of infinite groups are studied (related results can be found in [\textit{Ie. Lutsenko} and \textit{I. V. Protasov}, Int. J. Algebra Comput. 19, No. 4, 491-510 (2009; Zbl 1186.20024)]). Let \(G\) be an infinite group and \(k\) an infinite cardinal. The symbol \([G]^{<k}\) stands for the set consisting of all subsets of \(G\) of cardinality \(<k\). A subset \(X\) of \(G\) is called `\(k\)-large' provided there exists \(F\in[G]^{<k}\) such that \(G=FX\). A subset \(Y\) of \(G\) is called `\(k\)-small' provided \(G\setminus FY\) is large for each \(F\in[G]^{<k}\). Given any family \(\mathcal F\) of subsets of \(G\), then \(\mathcal F^*:=\{X\subset G\mid\forall_{A\in\mathcal F}[X^{-1}A\neq G]\}\). The set \(S_k\) of all \(k\)-small subsets is the main object in the present paper. In each of the following cases holds \(S_k^*=[G]^{<k}\): (i) the cardinal \(k=|G|\) is regular; (ii) \(G\) is Abelian and \(k\) is a limit ordinal; (iii) \(G\) is a divisible Abelian group (Theorems 6.1-6.3). If the cardinal \(k=|G|\) is regular then there is no family \(\mathcal F\) of subsets of \(G\) such that \(\mathcal F^*=S_k\) (Theorem 6.4). If \(k\) is a cardinal number such that \(\omega\leq k\leq\text{cf}|G|\), where \(\text{cf}|G|\) is the cofinality of \(|G|\), then \(G\) can be partitioned into \(\omega\) subsets which are \(k\)-small (Theorem 5.1). Some assertions about relations between \(S_k\) and closed ideals of the compact right topological semigroup \(\beta G\) associated to \(G\) are proved. Theorem 4.5 of the present paper was published also in the joint paper of Lutsenko and the author (Theorem 2.1), cited above.