Group in which every element is a commutator
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW 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 |