# Conjugation (group theory)

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*.