Difference between revisions of "Commutator"
From Encyclopedia of Mathematics
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| − | ''of two elements | + | {{TEX|done}} |
| + | ''of two elements $a$ and $b$ in a group with multiple operators'' | ||
The element | The element | ||
| − | + | $$-a-b+a+b.$$ | |
| − | For groups without multiple operators (here the operation is usually called multiplication), the commutator of the elements | + | For groups without multiple operators (here the operation is usually called multiplication), the commutator of the elements $a$ and $b$ is the element $a^{-1}b^{-1}ab$. The set of all commutators in a group $G$ generates a subgroup, called the [[Commutator subgroup|commutator subgroup]] of $G$. |
| − | In an associative algebra the element | + | In an associative algebra the element $[x,y]=xy-yx$ is called the Lie product, or commutator, of $x$ and $y$. |
Latest revision as of 14:01, 10 April 2014
of two elements $a$ and $b$ in a group with multiple operators
The element
$$-a-b+a+b.$$
For groups without multiple operators (here the operation is usually called multiplication), the commutator of the elements $a$ and $b$ is the element $a^{-1}b^{-1}ab$. The set of all commutators in a group $G$ generates a subgroup, called the commutator subgroup of $G$.
In an associative algebra the element $[x,y]=xy-yx$ is called the Lie product, or commutator, of $x$ and $y$.
How to Cite This Entry:
Commutator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commutator&oldid=16944
Commutator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Commutator&oldid=16944
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article