Definition 15

A group is called monogenous if it admits a system of generators consisting of a single element. A finite monogenous group is called cyclic.

Every monogenous group is commutative (no. 3, Corollary 2 to Proposition 2). Every quotient group of a monogenous group is monogenous (no. 3, Corollary 3 to Proposition 2).

The additive group $\mathbf{Z}$ is monogenous: it is generated by $\{1\}$. For every positive integer $a$, the group $\mathbf{Z}/a\mathbf{Z}$ is monogenous, for it is a quotient of $\mathbf{Z}$.