Proposition 18

A finite monogenous group of order a is isomorphic to $\mathbf{Z}/a\mathbf{Z}$. An infinite monogenous group is isomorphic to $\mathbf{Z}$.

Let G be a monogenous group (written multiplicatively) and x a generator of G. The identity $x^m x^n = x^{m+n}$ (§ 1, no. 3, formula (1)) shows that the mapping $n \mapsto x^n$ is a homomorphism of $\mathbf{Z}$ into G. Its image is a subgroup of G containing x and hence it is G. By no. 5, Theorem 3, the group G is isomorphic to the quotient of $\mathbf{Z}$ by a subgroup, which is necessarily of the form $a\mathbf{Z}$ (Proposition 17). If $a > 0$, the group G is finite of order a and if $a = 0$, the group G is isomorphic to $\mathbf{Z}$.