what is conjugacy class?

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- Jan 27th 2011, 05:53 AMluckylawranceconjugacy class
what is conjugacy class?

- Jan 27th 2011, 05:54 AMroninpro
- Jan 27th 2011, 06:38 AMHallsofIvy
Two members of a group, x and y, are said to be "conjugate" if there exists a member of the group, z, such that $\displaystyle zxz^{-1}= y$. That, of course, is the same as saying that $\displaystyle zx= yz$. It is easy to prove that that is an equivalence relation (refexive: x is conjugate to itself by taking z equal to the group identity. symmetric: if x is conjugate to y, so that $\displaystyle zxz^{-1}= y$, then $\displaystyle x= z^{-1}yz= uyu^{-1}$ with $\displaystyle u= z^{-1}$. transitive: if x is conjugate to y and y is conjugate to z then $\displaystyle uxu^{-1}= y$ and $\displaystyle vyv^{-1}= z$. Then $\displaystyle v(uxu^{-1})v^{-1}= (vu)x(vu)^{-1}= z$ so x is conjugate to x).

A "conjugacy class" is the equivalence class using "conjugate" as the equivalence relation.

Notice that if the group is Abelian (commutative) then zx= xz so that if x and y are conjuate, zx= xz= yz and then x= y. That, is, in Abelian groups, a member of the group is conjugate only to itself and the conjugacy classes consist only of the single elements.