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 .

Try to come up with a homomorphism onto one of these groups with the right kernel. — , Jun 04 '16 at 07:16
Related: https://math.stackexchange.com/questions/539224, https://math.stackexchange.com/questions/1047076 — Watson, Jun 04 '16 at 19:15

score 1 · Accepted Answer · answered Jun 04 '16 at 07:17

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.

answered Jun 04 '16 at 07:17

awllower

16,536

Why "2" is false? – amir bahadory Jun 04 '16 at 07:29
An easy way to see why "2" is false is that there is an element of order $2$ in $\frac{ \Bbb{Z \times Z }} { \langle (2,2) \rangle},$ but there is no such element in $\Bbb{Z\times Z}.$ – awllower Jun 04 '16 at 09:01
$ (1,1) + \langle (2,2) \rangle $ is element with order 2. Very thanks. – amir bahadory Jun 04 '16 at 12:18

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

1 Answers1