< back to index

Cyclic group

Definition

A cyclic group is a group generated by a single element.

$C_n = \langle g \rangle = \{1, g, g^2, ..., g^{n-1}\}$ is the cyclic group of order $n$.

Cyclic groups are sometimes defined to be finite.

Properties

A cyclic group is abelian.

Types

Up to isomorphism, there is one cyclic group $C_n$ of order $n$ for $n \ge 1$, and there is one infinite cyclic group that is isomorphic to the group of integers $\mathbb{Z}$ under addition.

In a prime cyclic group $C_p$, all the elements apart from the identity are generators. The prime cyclic groups are simple groups.

Examples

Rotational symmetries of regular polygons form finite cyclic groups.

Related theorems

• Cyclic decomposition theorem (module theory)

all these pages adapted with probably insufficient credit from my university's lecture notes