Let $S\_\omega$ be the group of bijections $f:\omega\to\omega$, and let $F\_\omega = \{\pi\in S\_\omega: \exists N\in \omega(\pi(k) = k \text{ for all } k\geq N)\}$. It is easy to see that $F\_\omega$ is a normal subgroup of $S\_\omega$.


Is $S\_\omega/F\_\omega$ isomorphic to a subgroup of $S\_\omega$?



Is $S_\omega/F_\omega$ embeddable to $S_\omega$?
No, it is not. 


McKenzie (1971) observed that the "direct sum" of $\ge\aleph\_1$ non-abelian groups cannot be embedded into $S\_\omega$ (indeed, it yields an ascending chain of centralizers in $S\_\omega$ of type $\omega\_1$, and this is not possible).


On the other hand, the direct sum of $\aleph\_1$ (or $2^{\aleph\_0}$) countable groups can always be embedded in $S\_\omega/F\_\omega$: this easily follows from the fact that in a countable set there exist $2^{\aleph\_0}$ infinite subsets with pairwise finite intersection.


