I want to visualize infinite cyclic group. I know it looks like $(\mathbb{Z},+)$.But actually I want to visualize it as a limiting case of finite cyclic group when the order of group gradually gets larger.
I want to draw a similarity with the following fact:
Suppose I am at a point on a circle and I can traverse on the circle once to reach the same point by completing a cycle.Similarly in case of finite cyclic groups,we keep increasing the power of the generating element and at a finite stage return to the same element again by completing a cycle.Now I come back to my circle example,if now standing at the same point on the circle,the radius of the circle is gradually increased then in the limiting case it becomes a straight line,then it resembles an infinite cyclic group because now you start traversing from that point on the line or infinite circle whatever,you move away and away from the starting point but never cycle back to the same element again,this is similar to infinite cyclic group in the sense,keep multiplying an element by generator $a$ and the power keeps increasing but we never reach back to the same element.Is this a way in which I can visualize?
This is definitely a valid way to think about $\mathbb{Z}$. In fact, this is how I’ve heard people talk about the infinite dihedral group
$$\langle r,s~|~s^2 = 1, srs = r^{-1}\rangle$$
where $r$ is thought of as shifting the integer line down (rotating an infinite cyclic group) and $s$ flips the line over. Compare this to the usual dihedral group
$$\langle r,s~|~r^n=s^2=1, srs=r^{-1}\rangle$$
which gives the symmetries of a regular $n$-gon sitting on a plane, where $r$ is rotation and $s$ is a flip.