Commutator: Difference between revisions
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In a [[ring (mathematics)|ring]], the commutator of elements ''x'' and ''y'' may be defined as | In a [[ring (mathematics)|ring]], the commutator of elements ''x'' and ''y'' may be defined as | ||
:<math> [x,y] = x y - y x . \, </math> | :<math> [x,y] = x y - y x . \, </math>[[Category:Suggestion Bot Tag]] |
Latest revision as of 12:00, 31 July 2024
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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