Conjugation (group theory)

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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

x y = y − 1 x y . {\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

[ x , y ] = x − 1 x y , {\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[edit]

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

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

T y ( 1 ) = 1 y = y − 1 1 y = 1 ; {\displaystyle T_{y}(1)=1^{y}=y^{-1}1y=1;\,}
T y ( u v ) = y − 1 u v y = y − 1 u y y − 1 v y = u y v y = T y ( u ) T y ( v ) ; {\displaystyle 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 . {\displaystyle T_{y}(u)^{-1}=(y^{-1}uy)^{-1}=y^{-1}u^{-1}y=T_{y}(u)^{-1}.\,}

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