TITLE: Is every finite group a normal subgroup of a symmetric group?
QUESTION [11 upvotes]: By Cayley's theorem, we know that for any finite group $G$, there exists $N \in \mathbb{N}$ such that $G$ is isomorphic to a subgroup of $S_N$, the symmetric group on $N$ letters. Can we prove that for every finite group $G$ there is some symmetric group $S_N$ such that $G$ is isomorphic to a $normal$ subgroup of $S_N$?

REPLY [3 votes]: In general (i.e. for $N \neq 4$), the only normal subgroups of $S_N$ are $S_N$ itself, $A_N$, and $1.$  Therefore no, because most $G$ will not map to one of these.  ($S_4$ has an additional normal subgroup, the Klein $4$-group hiding in it.)