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](https://math.stackexchange.com/questions/166358/normal-subgroups-of-infinite-symmetric-group/2645097#2645097), $Alt\_\omega$ 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;](https://mathoverflow.net/questions/3420/countable-subgroups-of-compact-groups/62857#62857)


$\bullet$ [the group $S\_\omega$ is not isomorphic to a dense subgroup of a non-discrete locally compact topological groups.](https://mathoverflow.net/questions/293614/is-each-locally-compact-group-topology-on-the-permutation-group-discrete/293619#293619)


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](https://mathoverflow.net/questions/293614/is-each-locally-compact-group-topology-on-the-permutation-group-discrete/293638#293638), 
for every homomorphism $h:S\_\omega\to G$ to a locally compact topological group $G$ the image $h(S\_\omega)$ is discrete. 



Is $Alt_\omega$ a dense subgroup of a non-discrete locally compact topological group?
Corollary 1.5 of [this Vissarion Belyaev's paper](http://mi.mathnet.ru/smj1670) says that



> 
> A group containing an infinite inert residually finite subgroup
>  embeds into a non-discrete locally compact group.
> 
> 
> 


Here, a subgroup $H$ of a group $G$ is called *inert* if $H\cap g^{-1}Hg$ has a finite index in $H$ for any $g\in G$.


So, in the alternating group $Alt\_\omega$, for instance, the (abelian) subgroup generated by the 3-cycles $(1\;2\;3),(4\;5\;6),(7\;8\;9),\dots$
is inert. infinite, and residually finite . 


Thus, the answer to Question 1 is **yes** (and to Question 2 is **no**).


