A group and the completion of its coset semigroup

TL;DR

This paper investigates the structure of the group of units in the completion of a semigroup formed by right cosets of subgroups of a periodic group, revealing conditions under which the group and semigroup uniquely determine each other.

Contribution

It characterizes and represents the units of the completion of the coset semigroup for certain periodic groups, extending Schein's 1973 results.

Findings

The group of units is characterized for periodic groups with permuting minimal subgroups.

G and its coset semigroup are mutually determined by each other for most such groups.

Extends classical results on coset semigroups and their completions.

Abstract

Let ${\cal K}_1(G)$ denote the inverse subsemigroup of ${\cal K}(G)$ consisting of all right cosets of all non-trivial subgroups of $G$. This paper concentrates on the study of the group $\Sigma({\cal K}_1(G))$ of all units of the completion of ${\cal K}_1(G)$. The characterizations and the representations of $\Sigma({\cal K}_1(G))$ 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 ${\cal K}_1(G)$ are uniquely determined by each other, up to isomorphism. This extends the related results obtained by Schein in 1973.