Order of each element in the given group

Question

I am a little bit confused with this problem: let $G:=\mathbb{Z}_{2} \times \mathbb{Z}_{4}$. I have to find the order of each element in the group

Since $\mathbb{Z}_{2} \times \mathbb{Z}_{4}=(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)$

Does it mean that I have to multiply by a number s.t. each of the sets becomes (0,0)?

Answer 1

I am not sure that I perfectly understand what you mean in the last sentence but I can suggest the following; The order of an element $x$ is the least positive integer $n$ s.t. $x^n=1$ (in multiplicative notation) respectively $n\cdot x=0$ (in additive notation). Note that $n\cdot x$ means just $x+x+\dots+x$ $n$-times.

Take a random element and try to calculate such an $n$, as the Wanderer suggested.

Answer 2

Consider the element $(0,1)$. The group $\mathbb{Z}_2 \times \mathbb{Z}_4$ is additive, so the order is $4$. In fact, if you sum $(0,1)+(0,1)+(0,1)+(0,1)$, you will obtain the pair $(0,0)$, identity element for the above group.