Exercise 7, § 2

Given an associative law on a set E and an element $x \in E$, let A be the set of $x^n$ for $n \in \mathbf{N}^*$; if there is an identity element, let B be the set of $x^n$ for $n \in \mathbf{N}$; if further $x$ is invertible, let C be the set of $x^a$ for $a \in \mathbf{Z}$. Show that, if A (resp.

B, C) is infinite, it is isomorphic (with the law induced by the given law on E) to $\mathbf{N}^*$ (resp. $\mathbf{N}, \mathbf{Z}$) with addition.