Conjugation (group theory)

In group theory, conjugation is an operation between group elements. The conjugate of x by y is:

$x^{y}=y^{-1}xy.\,$

If x and y commute then the conjugate of x by y is just x again. The commutator of x and y can be written as

$[x,y]=x^{-1}x^{y},\,$

and so measures the failure of x and y to commute.

Two elements are said to be conjugate if one is obtained as a conjugate of the other: the resulting relation of conjugacy is an equivalence relation, whose equivalence classes are the conjugacy classes.