Order of Cyclic Group equals Order of Generator

Theorem

Let $G$ be a finite cyclic group which is generated by $a \in G$.

Then:

$\order a = \order G$

where:

$\order a$ denotes the order of $a$ in $G$

$\order G$ denotes the order of $G$.

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