why does an infinite cyclic group need to be infinite on both sides? i'm watching these group theory lectures and i'm confused about minute 2:30
https://www.youtube.com/watch?v=WbzwNYN2dH4
Groups are being explained as a buch of simbols that produce permutations on another set (the symbol "1" moves every element on another set by 1, -1 does the opposite), and their composition is explained as doing one permutation after another.
at this point infinite cyclic groups are being explained, and the lecturer is trying to make a point into why the integers under addition are an infinite cyclic group, he tries to do this by counter example.
The idea is that the integers are a cyclic group that is infinite on both sides, if they weren't then the group would not be a permutation so it would not even be a group.
I hope this picture makes it a bit clearer, here the group is infinite to the right, I'm trying to abstract the symbols but this is supposed to be a cyclic group, the only way to get $C$, for example, is to compose $A *A*A$ or $A^3$ .

This means that there is no $A^n = A^1$ (as it would happen with a finite cyclic group) so the only operation that can get $A^1$ is $A^0 * A^1$.
So my question is, why is this arraingment not even a group, what group axioms are being broken?
Please let me know if any clarification is needed.

The existence of inverses fails.
It's exactly the situation between $\mathbb Z$, which is a groups, and $\mathbb N$, which is not.
