7.68 ConjugacyClasses

ConjugacyClasses( G )

ConjugacyClasses returns a list of the conjugacy classes of elements of the group G. The elements in the list returned are conjugacy class domains as created by ConjugacyClass (see ConjugacyClass). Because conjugacy classes are domains, all set theoretic functions can be applied to them (see Domains).

    gap> a5 := Group( (1,2,3), (3,4,5) );;  a5.name := "a5";;
    gap> ConjugacyClasses( a5 );
    [ ConjugacyClass( a5, () ), ConjugacyClass( a5, (3,4,5) ), 
      ConjugacyClass( a5, (2,3)(4,5) ), ConjugacyClass( a5, (1,2,3,4,5) ), 
      ConjugacyClass( a5, (1,2,3,5,4) ) ] 

ConjugacyClasses first checks if G.conjugacyClasses is bound. If the component is bound, it returns that value. Otherwise it calls G.operations.ConjugacyClasses( G ), remembers the returned value in G.conjugacyClasses, and returns it.

The default function called this way is GroupOps.ConjugacyClasses. This function takes random elements in G and tests whether such a random element g lies in one of the already known classes. If it does not it adds the new class ConjugacyClass( G, g ) (see ConjugacyClass). Also after adding a new class it tests whether any power of the representative gives rise to a new class. It returns the list of classes when the sum of the sizes is equal to the size of G.

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