Is there always a non-cyclic abelian subgroup of S2n with order n^2?

Question

Let $ n \in N$. Does there always exist a non-cyclic abelian subgroup of $S_{2n}$, the symmetric group on 2n letters, with $n^2$ elements? How could a computer give an example of such subgroup in? For instance in $S_6$ how could a computer find a non-cyclic abelian subgroup with order 9?

Answer 1

Yes, you can take $\langle (12\cdots n), ((n+1)\cdots 2n)\rangle$ : this is a group isomorphic to $(\mathbb{Z}/n\mathbb{Z})^2$.