Subgroup Generated by One Element is Cyclic

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Theorem

Let $G$ be a group.

Let $a \in G$.


Then $\gen a$, the subgroup generated by $a$, is cyclic:


Proof

By Subgroup Generated by One Element is Set of Powers:

$\gen a = \set {a^n : n \in \Z}$

The result follows by definition of cyclic group.

$\blacksquare$


Also see


Sources