# Group property

## Contents

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

## 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.