Conjugation (group theory)

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

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

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.

Inner automorphism

For a given element y in G let $T_y$ denote the operation of conjugation by y. It is easy to see that the function composition $T_y \circ T_z$ is just $T_{yz}$ .

Conjugation $T_y$ preserves the group operations:

$T_y(1) = 1^y = y^{-1} 1 y = 1 ; \,$

$T_y(uv) = y^{-1}uvy = y^{-1}uyy^{-1}vy = u^y v^y = T_y(u) T_y(v) ; \,$

$T_y(u)^{-1} = (y^{-1} u y)^{-1} = y^{-1}u^{-1}y = T_y(u)^{-1} . \,$

Since $T_y$ is thus a bijective function, with inverse function $T_{y^{-1}}$ , it is an automorphism of G, termed an inner automorphism. The inner automorphisms of G form a group $Inn(G)$ and the map $y \mapsto T_y$ is a homomorphism from G onto $Inn(G)$ . The kernel of this map is the centre of G.