Let $S_\omega$ be the group of bijections of the countable ordinal $\omega:=\{0,1,2,\dots\}$ and $Alt_\omega$ be the subgroup of $S_\omega$ consisting of even permutations of $\omega$ (i.e., the compositions of even number of transpositions). By Onofri-Schreier-Ulam Theorem, $Alt_\omega$ is the smallest non-trivial normal subgroup in the permutation group $S_\omega$.

It is known that

$\bullet$ the group $Alt_\omega$ is not isomorphic to a subgroup of a compact topological group;

These two results motivate the following

Problem 1.Is the group $Alt_\omega$ isomorphic to a dense subgroup of a non-discrete locally compact topological group?

Since the group $Alt_\omega$ is simple, this problem can be reformulated in the following form.

Problem 2.Let $h:Alt_\omega\to G$ be a homomorphism of $Alt_\omega$ to a locally compact topological group $G$. Is the image $h(Alt_\omega)$ discerete?

**Remark.** By the answer of @YCor to this question,
for every homomorphism $h:S_\omega\to G$ to a locally compact topological group $G$ the image $h(S_\omega)$ is discrete.