# Cyclic group

1,016pages on
this wiki

A cyclic group is a group in which it is possible to cycle through all elements of the group starting with a particular element $g$ of the group known as the generator and using only the group operation and the inverse axiom. That is, every element of group can be expressed as an integer power (or multiple if the operation is addition) of $g$.

A group $\left(G,\cdot\right)$ is said to be cyclic if there exists an element $g\in G$, called a generator of $G$, such that $\left\{ g^{n}:n\in\mathbb{Z}\right\} =G$.

The trivial subgroup of (G,·) is cyclic by itself.

## Alternate definition

If one wishes to consider a definition that foregos any notion of the integers, one may wish to view a generator of a group as being an element whose membership in any subgroup implies all other members of the group must also be members of the subgroup. That is, the only subgroup containing a generator is the whole group itself.

A group $\left(G,\cdot\right)$ is said to be cyclic if there exists an element $g\in G$, called a generator of $G$, such that for any subgroup $H \le G$, if $g \in H$, then $H=G$.

## Cyclic subgroups

A cyclic subgroup (H,·) of a group (G,·), denoted as [a], is a cyclic group where $a\in G$ and $H = \left\{ g^{n}:n\in\mathbb{Z}\right\}$.