TITLE: Can any group be embedded in a simple group?
QUESTION [25 upvotes]: 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?

REPLY [45 votes]: 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 $Q_G$.