Proving that $S_n$ and $A_n \times A_n$ are not isomorphic for $n \geq 5$ sanity check.

Question

My proof was: $|S_n| = n! \neq n!/2 \cdot n!/2 = |A_n \times A_n|$ for $n \geq 5$, so the groups are not isomorphic.

But you can get 15 points for this question, whereas you get only 7 for proving that there is no simple group of order 2015, which is much more difficult. Is my proof wrong?

Answer 1

Your proof is correct. What's the doubt? And it is correct for every $n>1$.