Generators of a direct product of $\Bbb Z_2$ with $\Bbb Z_4$.

Question

Given two groups :

$$\mathbb{Z} _{2} \;( =\mathbb{Z}/2\mathbb{Z})$$

$$\mathbb{Z}_{4} \;( =\mathbb{Z}/4\mathbb{Z}) $$

We define the direct product :

$$G =\mathbb{Z} _{2} \times \mathbb{Z}_{4} $$

I know $G$ is not cyclic as there is no element of order eight.

But can we find two elements that generate $G$?

Thanks in advance for your help

Answer 1

Two generators that go together are $([1]_2, [0]_4)$ and $([0]_2, [1]_4)$, where $[a]_n:=\{a+bn\mid b\in\Bbb Z\}.$

Answer 2

$\Bbb Z_2\times\Bbb Z_4=\langle (1,0),(0,1)\rangle $.