Cyclic Group

Definition

A cyclic group is a group which can be generated by a single element. That is \(G\) is cyclic if and only if there exists a \(g \in G\) such that:

\[ G = \langle g \rangle.\]

The cyclic group of order \(m\) is sometimes represented by \(C_m\), although a cyclic group may be infinite. Note also that we say the cyclic group of order \(m\) since cyclic groups of equal order are isomorphic.

The name cyclic indicates, in the finite case, the fact that powers of the generator eventually recur in a repeating pattern, and every element can be expressed as a power of the generator.

Example

Consider the subgroup of the \((\mathbb{C}, \times)\) generated by \(\{e^{\frac{2i\pi}{3}}\}\). In this cyclic group, the powers of the generator are, starting from \(0\), given by

\[ 1, e^{\frac{2i\pi}{3}}, e^{\frac{4i\pi}{3}}, 1, e^{\frac{2i\pi}{3}}, e^{\frac{4i\pi}{3}}, 1, \dots\]

Example

\((\mathbb{Z}, +)\) is a cyclic group generated by \(1\).