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Function s which are constant for members of the same conjugacy class are called Class Function s.


DEFINITION


Suppose ''G'' is a group. Two elements ''a'' and ''b'' of ''G'' are called conjugate if there exists an element ''g'' in ''G'' with
gag


(In Linear Algebra , for Matrices this is called Similarity .)

It can be readily shown that conjugacy is an Equivalence Relation and therefore partitions ''G'' into Equivalence Class es. (This means that every element of the group belongs to precisely one conjugacy class, and the classes Cl(''a'') and Cl(''b'') are equal if and only if ''a'' and ''b'' are conjugate, and Disjoint otherwise.) The equivalence class that contains the element ''a'' in ''G'' is
:Cl(''a'') = {''x'' ∈ ''G'' : there exists ''g'' in ''G'' such that ''x'' = ''gag''−1}
and is called the conjugacy class of ''a''. The '''class number''' of ''G'' is the number of conjugacy classes.


EXAMPLES

The Symmetric Group ''S3'', consisting of all 6 Permutation s of three elements, has three conjugacy classes:
  • no change (abc -> abc)

  • interchanging two (abc -> acb, abc -> bac, abc -> cba)

  • a cyclic permutation of all three (abc -> bca, abc -> cab)


The symmetric group ''S4'', consisting of all 24 permutations of four elements, has five conjugacy classes:
  • no change

  • interchanging two

  • a cyclic permutation of three

  • a cyclic permutation of all four

  • interchanging two, and also the other two


See also the Proper Rotations Of The Cube , which can be characterized by permutations of the body diagonals.


PROPERTIES


  • The identiy element is always in its own class, that is Cl(''e'') = {''e''}


  • If ''G'' is Abelian , then ''gag''−1 = ''a'' for all ''a'' and ''g'' in ''G''; so Cl(''a'') = {''a''} for all ''a'' in ''G''; the concept is therefore not very useful in the abelian case.


  • If two elements ''a'' and ''b'' of ''G'' belong to the same conjugacy class (i.e., if they are conjugate), then they have the same Order . More generally, every statement about ''a'' can be translated into a statement about ''b''=''gag''−1, because the map φ(''x'') = ''gxg''−1 is an Automorphism of ''G''.


  • An element ''a'' of ''G'' lies in the Center Z(''G'') of ''G'' if and only if its conjugacy class has only one element, ''a'' itself. More generally, if C''G''(''a'') denotes the '' Centralizer '' of ''a'' in ''G'', i.e., the Subgroup consisting of all elements ''g'' such that ''ga'' = ''ag'', then the Index : C''G''(''a'') is equal to the number of elements in the conjugacy class of ''a''.



CONJUGACY CLASS EQUATION


If ''G'' is a finite group, then the previous paragraphs, together with the Lagrange's Theorem , imply that the number of elements in every conjugacy class Divides the Order Of ''G'' .

Furthermore, for any group ''G'', we can define a representative set ''S'' = {''x''''i''} by picking one element from each conjugacy class of ''G'' that has more than one element. Then ''G'' is the Disjoint Union of Z(''G'') and the conjugacy classes Cl(''x''''i'') of the elements of ''S''. One can then formulate the following important class equation:
  Where The Sum Extends Over ''H''<sub>''i''</sub> C<sub>''G''</sub>(''x''<sub>''i''</sub>) for each ''x''<sub>''i''</sub> in ''S'' Note that : ''H''<sub>''i''</sub> is the number of elements in conjugacy class ''i'', a proper Divisor of ''G'' bigger than one If the divisors of ''G'' are known, then this equation can often be used to gain information about the size of the center or of the conjugacy classes
  Since The Order Of Any Subgroup Of ''G'' Must Divide The Order Of ''G'', It Follows That Each ''H''<sub>''i''</sub> Also Has Order Some Power Of ''p''<sup>( ''k''<sub>''i''</sub> )</sup> But Then The Class Equation Requires That ''G'' ''p''<sup>''n''</sup> = Z(''G'') + &sum<sub>''i''</sub> (''p''<sup>( ''k''<sub>''i''</sub> )</sup>) From this we see that ''p'' must divide Z(''G''), so Z(''G'') > 1
  :Cl(''S'') : N(''S'')