Subgroup Generated by One Element is Cyclic

Theorem
Let $G$ be a group.

Let $a \in G$.

Then $\gen a$, the subgroup generated by $a$, is cyclic:

Proof
By Subgroup Generated by One Element is Set of Powers:
 * $\gen a = \set {a^n : n \in \Z}$

The result follows by definition of cyclic group.

Also see

 * Definition:Cyclic Group