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Example: Examples of Cyclic Groups

(related to Part: Group Theory)

The following are examples of cyclic groups:

• The integers under addition $(\mathbb Z, + )$ form a cyclic group with the generator $1$ and the generator $-1$. There are no other generators.
• Given a positive integer $n$, the $n$-th roots of unity in the complex numbers form a cyclic group of order $n$, i.e. the complex exponential function $\exp(2\pi i/n)$ is a generator of this group. Also, if $r$ is relatively prime to $n$, then $\exp(2\pi ir/n)$ is a generator of this group.

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References

Bibliography

1. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition