Statement about equal cardinalities
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A statement about equal cardinalities is a statement that the cardinalities of two sets, obtained in different but related ways, are equal. Statements about equal cardinalities often arise in linear representation theory, but they may also arise in other branches of group theory.
Some typical examples:
- The number of conjugacy classes of a finite group over the complex numbers, equals the number of irreducible representations. This has been proved
- The McKay conjecture, which relates the number of irreducible representations of a certain type in a group, and in the normalizer of a Sylow subgroup
Statements about equal cardinalities may tempt one to think that it is possible to establish an explicit bijection between the sets whose cardinalities have been equated. However, this is not always true. Nonetheless, the equal cardinalities may still be viewed in some sense as arising due to a map, not between the sets themselves, but between structures constructed from the sets. These structures could be:
- Vector spaces, or free modules over a ring, with the sets as generating sets. For instance, there is a natural isomorphism between the vector space having as basis the conjugacy classes of a finite group, and the vector space having as basis the irreducible characters.
- Vector spaces, with some further structure put on.