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