Existence of Unique Subgroup Generated by Subset/Singleton Generator

Theorem

Let $\struct {G, \circ}$ be a group.

Let $a \in G$.

Then $H = \gen a = \set {a^n: n \in \Z}$ is the unique smallest subgroup of $G$ such that $a \in H$.

That is:

$K \le G: a \in K \implies H \subseteq K$

Proof

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

Let $K \le G: a \in K$.

Then $\forall n \in \Z: a^n \in K$.

Thus, $H \subseteq K$.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 37.5$ Some important general examples of subgroups