nLab
commutator

Contents

Definition

In rings and algebras

For AA a ring or associative algebra, the commutator of two elements x,yAx,y \in A is the element

[x,y]xyyx. [x,y] \coloneqq x y - y x \,.

In groups

For GG a group, the group commutator of two elements a,bGa,b \in G is the element

[a,b]a 1b 1ab. [a,b] \coloneqq a^{-1} b^{-1} a b \,.

Properties

Revised on November 19, 2017 15:20:31 by Urs Schreiber (94.220.49.59)