Conjugation (group theory)

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

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

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

${\displaystyle [x,y]=x^{-1}x^{y},\,}$

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 ${\displaystyle T_{y}}$ denote the operation of conjugation by y. It is easy to see that the function composition ${\displaystyle T_{y}\circ T_{z}}$ is just ${\displaystyle T_{yz}}$.

Conjugation ${\displaystyle T_{y}}$ preserves the group operations:

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

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