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
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39$. Cyclic groups