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 iteratively as | Commutators of higher order are defined [[iteration|iteratively]] as | ||
:<math> [x_1,x_2,\ldots,x_{n-1},x_n] = [x_1,[x_2,\ldots,[x_{n-1},x_n]\ldots]] . \,</math> | :<math> [x_1,x_2,\ldots,x_{n-1},x_n] = [x_1,[x_2,\ldots,[x_{n-1},x_n]\ldots]] . \,</math> | ||
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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 11:00, 31 July 2024
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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,y] = x^{-1} y^{-1} x y \, }
(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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G^{(1)}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [G,G]} . 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
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_1,x_2,\ldots,x_{n-1},x_n] = [x_1,[x_2,\ldots,[x_{n-1},x_n]\ldots]] . \,}
The higher derived groups are defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G^{(1)} = [G,G]} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G^{(2)} = [G^{(1)},G^{(1)}]} and so on.
Ring theory
In a ring, the commutator of elements x and y may be defined as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x,y] = x y - y x . \, }