# Group in which every element is a commutator

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

A **group in which every element is a commutator** is a group in which every element is a commutator.

Note that this property does not depend on whether we use the left or right convention for commutators.

## Examples

All finite simple non-abelian groups have this property (though this is far from obvious, and relies on the classification of finite simple groups). In particular:

## Relation with other properties

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Perfect group | The commutators form a generating set | (obvious) | perfect not implies every element is a commutator | |FULL LIST, MORE INFO |