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.

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:

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

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

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

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.