Existence of Unique Subgroup Generated by Subset/Singleton Generator
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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