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:
 * $\langle a \rangle = \{a^n : n \in \Z\}$

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

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

Thus $\langle a \rangle \subseteq H$.

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

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

Also see

 * Definition:Cyclic Group