Commutator

The commutator of two elements a,b of a group is defined as

[a,b] = a^-1b^-1ab;

The subgroup generated by all commutators of the group is the commutator subgroup. Note the word "generated": the set of all commutators of the group is, generally, not a group (in any interesting case). While clumsily defined, the commutator subgroup is important.

An abelian group has only trivial commutators, as a⋅b⋅a^-1⋅b^-1=1. Hence its commutator subgroup is {1}. The converse is also (trivially) true.

The relationship between this definition and the one given by ariels comes from Lie algebras: suppose we approximate a as 1+δA and b as 1+εB. Then to first order in δ and ε,

aba^-1b^-1 goes to (1+δA)(1+εB)(1-δA)(1+εB) = 1+(AB-BA)δε = 1+[ A,B ]δε

When using matrices, one defines the commutator of two n x n matrices A and B to be AB-BA

Com"mu*ta`tor (?), n. Elec.

A piece of apparatus used for reversing the direction of an electrical current; an attachment to certain electrical machines, by means of which alternating currents are made to be continuous or to have the same direction.

© Webster 1913.