Let a finite abelian group $G$ have invariant factors $n_1, n_2, . . . , n_k$ . What are the invariant factors of $G \times G$?
What I have done: I used as an example the group $G=\mathbb{Z}_3\times\mathbb{Z}_3\times\mathbb{Z}_4 $ whose invariant factors are $3$ and $12$. Then $G\times G$ has the invariant factors $3, 3, 12$ and $12$. Can I say that in general the invariant factors of $G\times G$ are just a combination of them? (That is, If $G$ have invariant factors $n_1, n_2, . . . , n_k$, then the invariant factors of $G\times G$ are $n_1, n_1, n_2, n_2, . . . , n_k,n_k$)