# Infinite group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

An **infinite group** is a group whose order is infinite, i.e., the underlying set of the group has infinitely many elements. The order of the group may be any infinite cardinal (see there exist groups of every order).

Every group is either an infinite group or a finite group.

In general, group theory results are proved either for finite groups or for all groups. It is generally rare for a result to be proved specifically for infinite groups, but there do exist some results of that sort.