$\mathbb{Z}\times \mathbb{Z}_{2}$ is a cyclic group?

Question

I think that $\mathbb{Z}\times \mathbb{Z}_{2}$ isn't a cyclic group becuase we don't have any $(a,b)\in \mathbb{Z}\times \mathbb{Z}_{2}$ that can create the group $\mathbb{Z}\times \mathbb{Z}_{2}$.

I'm right?

Thank you!

Answer 1

In very short: suppose

$$\Bbb Z\times\Bbb Z_2=\langle\;(a,b)\;\rangle\;,\;\;a\in\Bbb Z\;,\;b\in\Bbb Z_2$$

Well, then how'd you solve $\;(a+1,b)=n(a,b)\;,\;\;n\in\Bbb Z\;?$

Answer 2

Hint: $\mathbb{Z}/2 \times \mathbb{Z}/2$ is a quotient of $\mathbb{Z} \times \mathbb{Z}/2$.