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Commutator

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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://www.encyclopediaofmath.org/index.php?title=Commutator&oldid=31490
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article