Group in which every element is a commutator

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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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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.


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