Order of Cyclic Group equals Order of Generator
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Theorem
Let $G$ be a finite cyclic group which is generated by $a \in G$.
Then:
- $\order a = \order G$
where:
Proof
Let $\order a = n$.
From List of Elements in Finite Cyclic Group:
- $G = \set {a_0, a_1, \ldots, a_{n - 1} }$
Hence the result.
$\blacksquare$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 9$: Cyclic Groups
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 43$
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39.2$ Cyclic Groups