A group and the completion of its coset semigroup

Let ${\mathcal{K}}_1(G)$ denote the inverse subsemigroup of $\mathcal{K}(G)$ consisting of all right cosets of all non-trivial subgroups of $G$. This paper concentrates on the study of the group $\mathrm{\Sigma }({\mathcal{K}}_1(G))$ of all units of the completion of ${\mathcal{K}}_1(G)$. The characterizations and the representations of $\mathrm{\Sigma }({\mathcal{K}}_1(G))$ are given when $G$ is a periodic group whose minimal subgroups permute with each other. Based on these, for such groups $G$ except some special $p$-groups, it is shown that $G$ and its coset semigroup ${\mathcal{K}}_1(G)$ are uniquely determined by each other, up to isomorphism. This extends the related results obtained by Schein in 1973.