How can I understand $\Bbb Z\times \Bbb Z/2\Bbb Z$

Question

This may be stupid request, but I would like to have a intuition for the group $\Bbb Z\times \Bbb Z/2\Bbb Z$ in terms of 'real' objects. 'Real' could mean geometric but not necessarily. I perhaps what I'd like is to see it as a set of $\{0,1\}$-indexed integers and for the indices to somehow make 'real' sense.

If this intuition generalized to $\Bbb Z\times\Bbb Z/p\Bbb Z$ or $\Bbb Z\times\Bbb Z/n\Bbb Z$ it would be even better...

Answer 1

I don't know how 'real' this is, but here's one way to visualise $\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.

There are two copies of $\mathbb{Z}$, call them $\mathbb{Z}\times\{0\}$ and $\mathbb{Z}\times\{1\}$.

$$\mathbb{Z}\times\{0\}\qquad :\qquad \dots\qquad -3\qquad -2\qquad -1\qquad 0\qquad 1\qquad 2\qquad 3\qquad \dots$$

$$\mathbb{Z}\times\{1\}\qquad :\qquad \dots\qquad -3\qquad -2\qquad -1\qquad 0\qquad 1\qquad 2\qquad 3\qquad \dots$$

Pick two numbers from the lists. If both numbers belong to the same list, add them and take the corresponding number in the first list; if they belong to different lists, add them and take the corresponding number in the second list.

Answer 2

Here's a simple geometric interpretation. Consider the figure below, with marked fenceposts (or copies of the letter C) at every integer point on a horizontal line.

Its group of symmetries is $\Bbb Z\times \Bbb Z/2\Bbb Z$, generated by the horizontal translation and reflection across the horizontal.