Finite groups in $$G^{LUC}$$ G L U C

Abstract

Let \(G\) be a locally compact \(\sigma \)-compact Abelian group and let \(G^{LUC}\) denote the largest semigroup compactification of \(G\). We show that for every finite group \(Q\) in \(G^*=G^{LUC}{\setminus } G\) with identity \(u\), there is a finite group \(F\) in \(G\) such that \(Q=Fu\). In particular, if \(G\) contains no nontrivial finite group, neither does \(G^{LUC}\).