True /False question based on quotient groups of $S_{n} $ and $A_{n} $.

Question

I am trying assignment questions of Abstract algebra and I need help in following True/ False question.

Which one of following is true?

1. Every finite group is subgroup of $A_{n} $ for some $n\geq 1.$

2. Every finite group is quotient of $A_{n} $ for some $n\geq 1$.

3. No finite group is quotient of $S_{n} $ for $n\geq 3.$

I think 2 can't be true as quotient group of $A_{n} $ will also have even cardinality and Group can be of odd cardinality.

For 3 . I need to know about all quotient groups of $S_{n } $ which are $S_{n} $ and {0,1} and so $Z_{2} $ is an abelian group asked in 3 . Hope I am right!!

Can anyone please tell in detail on how I can prove 1.

Answer 1

1. This is true. Every finite group is isomorphic to some subgroup of some $S_n$ and $S_n$ is isomorphic to a subgroup of $A_{n+2}$.
2. This is false. It follows from the fact that $A_n$ is simple if $n>4$.
3. The group $S_n$ itself is a quotient of $S_n$, for every $n\in\Bbb N$.