Permutation group and their subgroups

Question

I need to verify these two statements.

1. Every finite group is a subgroup of $A_n$ for some $n\geq 1$.
2. Every finite group is a quotient of $A_n$ for some $n\geq 1$.

I think first one is false since an odd permutation in $S_n$ should remain odd in every other $S_m$ for $m\geq n$. I am not sure if I proceed right.

Answer 1

You should know something about the normal subgroups (i.e. the possible kernels of homomorphisms) of $A_{n}$, for $n \ge 5$.

As to the first question, do you know that every finite group is (isomorphic to) a subgroup of $S_{n}$? Because then there is a (rather straightforward) way of showing that $S_{n}$ is isomorphic to a subgroup of $A_{n+2}$.