which of the following group is isomorphic with : $\frac{ \Bbb{Z ×Z }} { \langle (2,2) \rangle } $?

Question

which of the following group is isomorphic with : $ \frac{ \Bbb{Z ×Z }} { \langle (2,2) \rangle} $ ?

1- $\Bbb{Z} $

2- $\Bbb{Z×Z} $

3-$\Bbb{Z_2 ×Z_2} $

4- $\Bbb{Z_2 × Z} $

The group $ \frac{ \Bbb{Z ×Z }} { \langle (2,2) \rangle} $ is infinite and is not cyclic then the "1" ,"3" is false .

Answer 1

Define a homomorphism $\phi:\mathbb Z\times\mathbb Z\rightarrow\mathbb Z_2\times\mathbb Z$ as $\phi(m,n)=(m\pmod2,m-n).$ Then $\phi$ is surjective, and $\ker\phi=\{(m,n)\in\mathbb Z\times\mathbb Z\mid m=n\text{ and }2\mid m\}=\left<(2,2)\right>.$ By the isomorphism theorem, we have...

Hope this helps.