Find all subgroups of order four in $\mathbb{Z}_4 \bigoplus \mathbb{Z}_4$.
I found a solution online from this link. The elements of order four are
(0,1) (0,3)
(1,0) (3,0)
(1,1) (3,3)
(1,2) (3,2)
(1,3) (3,1)
(2,1) (2,3)
Since there are 12 of them, there must be six cyclic groups of order 4.
The rest of the groups are of the form $\mathbb{Z}_2 \bigoplus \mathbb{Z}_2$
{0,0} {0,2},{2,0},{2,2}
I see that there are 12 order four elements, but I'm not sure how we can use that to conclude that there must be six cyclic groups of order 4? Furthermore, I'm not quite sure I understand the part about the non-cyclic groups or the notation being used?
I was also wondering in general if there's a systematic method that can be used to find the number of subgroups of a certain order $n$ in any finite abelian group given its decomposition into cyclic groups of prime powers?
Any help would be much appreciated!
Best Answer
It's right that there are six cyclic subgroups of order $4$, since there are $12$ elements of order $4$ and $\varphi (4)=2$ of them in each.
There are three elements of order $2$. Thus only one more subgroup of order $4$, isomorphic the Klein four group.