Surjective symmetry classes



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Surjective symmetry classes

  In order to derive the number of surjective fixed points of we use the preceeding corollaries together with an application of the Principle of Inclusion and Exclusion in order to get rid of the nonsurjective fixed points. We denote by the set of points contained in the -th cyclic factor of and put, for each index set :

Then, by the Principle of Inclusion and Exclusion, we obtain for the desired number of surjective fixed points of the following expression:

Now we recall that

This set can be identified with , where denotes the product of the cyclic factors of the numbers of which lie in , and where is the set of points contained in these cyclic factors. Thus

We can make this more explicit by an application of gif which yields:

 

Putting these things together we conclude

. Corollary   The number of surjective fixed points of is

where the middle sum is taken over all the sequences of natural numbers such that (they correspond to all possible choices of out of , where of the chosen cyclic factors of are -cycles). Hence the numbers of surjective fixed points of and of amount to:

and

where the sum is taken over all the sequences , and .

An application of the Cauchy-Frobenius Lemma finally yields the desired numbers of surjective symmetry classes:

. Theorem   The number of surjective -classes is

where the inner sum is taken over the sequences described in the corollary above. This implies, by restriction, the equations

and

where the last sum is to be taken over all the sequences such that and .

Try to compute the number of surjective symmetry classes for various group actions.

Exercises

E .   Prove that .



next up previous contents
Next: Various combinatorial numbers Up: Actions Previous: Injective symmetry classes



Herr Fripertinger
Sun Feb 05 18:28:26 MET 1995