Cyclic Group/Group Presentation
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Generator of the Cyclic Group of Order $n$
The presentation of a finite cyclic group of order $n$ is:
- $C_n = \gen {a: a^n = e}$
Proof
The elements of $\gen {a: a^n = e}$ are:
- $a, a^2, a^3, \ldots, a^{n - 1}, a^n, a^{n + 1}, a^{n + 2}, \ldots$
However, we have that:
- $a^n = e$
and so the elements of $\gen {a: a^n = e}$ are:
- $a, a^2, a^3, \ldots, a^{n - 1}, e, e a, e a^2, \ldots$
That is:
- $C_n = \set {a, a^2, \ldots, a^{n - 1}, e}$
and the result follows by definition of cyclic group.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Example $4.10$