elements of finite order in G

Question

Let G be the quotient group (ℤ×ℤ)/⟨2,4⟩

What are the elements of finite order and how do i find how many are there?

Does this make G cyclic?

I am stuck on how to start this

Answer 1

Consider the the ismorphism $\mathbb Z^2 \to \mathbb Z^2$, given by $\begin{pmatrix}1&1\\2&1\end{pmatrix} \in SL_2(\mathbb Z)$.

This isomorphism maps $\begin{pmatrix}2\\0\end{pmatrix}$ to $\begin{pmatrix}2\\4\end{pmatrix}$, hence

$$\mathbb Z^2 /\langle (2,4) \rangle \cong \mathbb Z^2/\langle (2,0) \rangle = \mathbb Z/2\mathbb Z \times \mathbb Z.$$

Should be a gimme to find the elements of finite order.

Remark. Speaking in terms of the classifacation of fintely generated abelian groups, one would just say that the Smith normal form of the matrix $\begin{pmatrix}2\\4\end{pmatrix}$ is $\begin{pmatrix}2\\0\end{pmatrix}$.