Find an example of a non-cyclic abelian group of permutations with n^2 elements for each n in the natural numbers, n >3.

Question

I'm really stuck on this problem below from my textbook:

Find an example of a non-cyclic abelian group of permutations with $n^2$ elements for each n in the natural numbers, $n > 3$.

It seems quite vague, and I'm not really sure where to begin. If anyone could help, I would be very appreciative.

Thanks,

Jack

Answer 1

Take $$G=\langle (12\dots n)\rangle\times \langle(n+1 \dots 2n)\rangle=H\times K$$ Note that $|H|=|K|=n$, hence $G$ is not cyclic.
Since $H,K$ are abelian, so is $G$.
Also $|G|=|H||K|=n^2$.