A group and the completion of its coset semigroup

Let K1(G) denote the inverse subsemigroup of K(G) consisting of all right cosets of all non-trivial subgroups of G. This paper concentrates on the study of the group Σ(K1(G)) of all units of the completion of K1(G). Characterizations and Representations of Σ(K1(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 K1(G) are uniquely determined by each other, up to isomorphism. This extends the related results obtained by Schein in 1973.