# Group property

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article is about a general term. A list of important particular cases (instances) is available at Category:Group properties

## Definition

### Symbol-free definition

A **group property** is a map from the collection of all groups to the two-element set (true, false) with the property that any two isomorphic groups get mapped to the same thing.

### Caution

A group property must be decidable purely based on the abstract group structure, and should *not* be dependent on additional structure (like topology, or analytic or algebraic structure). Even if such additional structure occurs in the definition, it should have a universal or existential quantification associated to it.

## Examples

### Important examples of group properties

Being Abelian is a group property: a group is Abelian if any two elements in it commute. Every group either *is Abelian* or *is not Abelian*.

Other examples are being nilpotent, simple, finite. Being *connected* is not a group property because it requires additional structure on the group, and one could have isomorphic groups one of which is connected and the other is not.