Corollary 47.4.12.label Let $G$ be an infinite group, then $\beta G$ admits a minimal idempotent in $\beta G \setminus G$.
Proof. Let $L \subset (\beta G \setminus G)$ be a minimal left ideal, then $L$ is a non-empty compact Hausdorff right topological semigroup. By Ellis’ Theorem, there exists $e \in L$ with $e^{2} = e$.$\square$
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