Commutator: Difference between revisions

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:<math> [x,y] = x^{-1} y^{-1} x y \, </math>
:<math> [x,y] = x^{-1} y^{-1} x y \, </math>


(although variants on this definition are possible).  Elements ''x'' and ''y'' commute if and only if the commutator [''x'',''y''] is equal to the group identity.  The '''commutator subgroup''' or '''derived group''' of ''G'' is the [[subgroup]] generated by all commutators, written <math>G^{(1)}</math> or <math>[G,G]</math>.  It is [[normal subgroup|normal]] and indeed [[characteristic subgroup|characteristic]] and the quotient ''G''/[''G'',''G''] is [[Abelian group|abelian]].  A quotient of ''G'' by a normal subgroup ''N'' is abelian if and only if ''N'' contains the commutator subgroup.
(although variants on this definition are possible).  Elements ''x'' and ''y'' commute if and only if the commutator [''x'',''y''] is equal to the group [[identity element|identity]].  The '''commutator subgroup''' or '''derived group''' of ''G'' is the [[subgroup]] generated by all commutators, written <math>G^{(1)}</math> or <math>[G,G]</math>.  It is [[normal subgroup|normal]] and indeed [[characteristic subgroup|characteristic]] and the quotient ''G''/[''G'',''G''] is [[Abelian group|abelian]].  A quotient of ''G'' by a normal subgroup ''N'' is abelian if and only if ''N'' contains the commutator subgroup.


Commutators of higher order are defined [[iteration|iteratively]] as
Commutators of higher order are defined [[iteration|iteratively]] as

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In algebra, the commutator of two elements of an algebraic structure is a measure of whether the algebraic operation is commutative.

Group theory

In a group, written multiplicatively, the commutator of elements x and y may be defined as

(although variants on this definition are possible). Elements x and y commute if and only if the commutator [x,y] is equal to the group identity. The commutator subgroup or derived group of G is the subgroup generated by all commutators, written or . It is normal and indeed characteristic and the quotient G/[G,G] is abelian. A quotient of G by a normal subgroup N is abelian if and only if N contains the commutator subgroup.

Commutators of higher order are defined iteratively as

The higher derived groups are defined as , and so on.

Ring theory

In a ring, the commutator of elements x and y may be defined as