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[edit]

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:

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

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

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

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.