Subgroup Generated by One Element is Set of Powers

Theorem
Let $G$ be a group.

Let $a \in G$.

Then the subgroup generated by $a$ is the set of powers:
 * $\gen a = \set {a^n : n \in \Z}$

Proof
By definition, the subgroup generated by $a$ is the intersection of all subgroups containing $a$.

By Powers of Element form Subgroup, the set $H = \set {a^n : n \in \Z}$ is a subgroup.

Thus $\gen a \subseteq H$.

By Power of Element in Subgroup, $H \subseteq \gen a$.

By definition of set equality, $\gen a = H$.

Also see

 * Definition:Cyclic Group