What Do the Elements of $\mathbb{Z}_2 \times \mathbb{Z}$ Look Like?

[mathmari]

Hey!

To show that in the additive group $\mathbb{Z}_2\times\mathbb{Z}$ there are non-zero elements $A,B$ of infinite order such that $A+B$ has finite order, we have to find such $A$ and $B$, right? (Wondering)

How do the elements of $\mathbb{Z}_2\times\mathbb{Z}$ look like? (Wondering)

 

[mathmari]

The elements of $\mathbb{Z}_2\times\mathbb{Z}$ are of the form $(x,y)$ where $x\in \mathbb{Z}_2$ and $y\in \mathbb{Z}$, right? (Wondering)

And the order of such an element is $n$ for which $(nx,ny)=(0,0)$, right?

For $A=(1,k)$ and $B=(0,-k)$ for $k\in \mathbb{Z}_{>0}$, we have that $A+B=(1,0)$, right?

$A$ and $B$ have infinite order, since the second coordinate is never equal to $0$ for $k>0$, and $A+B$ has finite order, and specifically the order is $2$, since $2(1,0)=(2,0)=(0,0)$.

Is this correct? (Wondering)

 

[I like Serena]

Yep. All correct. (Happy)

 

[mathmari]

Great! Thank you! (Sun)