# conjugacy class

• Jan 27th 2011, 05:53 AM
luckylawrance
conjugacy class
what is conjugacy class?
• Jan 27th 2011, 05:54 AM
roninpro
• Jan 27th 2011, 06:38 AM
HallsofIvy
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.