# Definition:Infinite Group

## Definition

A group which is not finite is an infinite group.

That is, an infinite group is a group of infinite order.

That is, a group $\struct {G, \circ}$ is an infinite group if and only if its underlying set $G$ is infinite.

That is, an infinite group is a group with an infinite number of elements.

### Countable

An infinite group whose underlying set $G$ is countable is a countably infinite group.

### Uncountable

An infinite group whose underlying set is uncountable is an uncountable group.

## Also see

• Results about the order of a group can be found here.
• Results about infinite groups can be found here.