Show that, for every $n$, $A_{n+2}$ has a subgroup isomorphic to $S_n$

Question

Show that, for every $n$, $A_{n+2}$ has a subgroup isomorphic to $S_n$

Also, in general, when I construct an isomorphism, what's necessary to show that it's well-defined? Is showing it is a bijection and homomorphism enough? And are there any rules to follow when I try to construct a homomophism or isomorphism? I mean when I'm asked to show a certain group is isomorphic to another, it's always difficult for me to find the mapping.

Answer 1

Hint: The even permutations in $S_n$ are already there as the ones that fix two given elements. Now try to add to these the same ones, but which instead of fixing those two elements, transposes them. Show that you now have the desired subgroup.

Edit: As pointed out by Anon, I messed up the hint. For the second part, one should take an odd permutation and compose it with the transposition of those two given elements.