Subgroup Generated by One Element is Set of Powers
Jump to navigation
Jump to search
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, $\gen a = H$.
$\blacksquare$