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.

Inner automorphism

For a given element y in G let denote the operation of conjugation by y. It is easy to see that the function composition is just .

Conjugation preserves the group operations:

Since is thus a bijective function, with inverse function , it is an automorphism of G, termed an inner automorphism. The inner automorphisms of G form a group and the map is a homomorphism from G onto . The kernel of this map is the centre of G.