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Let $G \leq \mathrm{Sym} (X)$ for a countable set $X$. Call a colouring of$X$ asymmetric, if the identity is the only element of $G$ which preserves allcolours. The motion (also called minimal degree) of $G$ is the minimal numberof elements moved by an element $g \in G \setminus\{\mathrm{id}\}$. We showthat every locally compact, closed permutation group with infinite motionadmits an asymmetric $2$-colouring. This generalises a recent result by Babaiand confirms a conjecture by Imrich, Smith, Tucker, and Watkins from 2015.

Asymmetric colouring of locally compact permutation groups