Any finite group $G$ can be embedded into $A\_{|G|+2}$ via Cayley's theorem ($G\hookrightarrow S\_{|G|}\hookrightarrow A\_{|G|+2}$). If $G$ is not assumed to be finite, is it still always possible to embedd it into a simple group?



Can any group be embedded in a simple group?
Yes. Assume $G$ is infinite. Cayley still embeds $G$ into the group $S\_G$
of permutations of $G$. This group is no longer simple: there is a normal 
subgroup, call it $N\_G$, consisting of all permutations that fix
the complement of a subset of $G$ of cardinality smaller than that of $G$.
But by a theorem of Baer, Schreier and Ulam, *every* normal subgroup 
of $S\_G$, other than $S\_G$ itself, is contained in $N\_G$.
Hence $Q\_G := S\_G / N\_G$ is simple. Moreover the composite map 
$G \to S\_G \to Q\_G$ is still an embedding because no non-identity
element of $G$ has any fixed points. We have thus embedded $G$
into the simple group $Q\_G$.


