A Note on Locally Compact Subsemigroups of Compact Groups

TL;DR

This paper provides an elementary proof that locally compact subsemigroups of compact groups are closed subgroups and explores implications for certain pseudocompact semigroups.

Contribution

It offers a new elementary proof for the structure of locally compact subsemigroups within compact groups and derives consequences for pseudocompact semigroups.

Findings

Locally compact subsemigroups of compact groups are closed subgroups.

Commutative cancellative pseudocompact semigroups with open shifts are compact groups.

Abstract

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.