Frobenius determinant theorem
In mathematics, the Frobenius determinant theorem was a conjecture made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in , with an English translation in ). If one takes the multiplication table of a finite group G and replaces each entry g with the variable xg, and subsequently takes the determinant, then the determinant factors as a product of n irreducible polynomials, where n is the number of conjugacy classes. Moreover, each polynomial is raised to a power equal to its degree. Frobenius proved this surprising conjecture, and it became known as the Frobenius determinant theorem. Formal statement
Let a finite group  have elements , and let  be associated with each element of . Define the matrix  with entries . Then

 

where the 's are pairwise non-proportional irreducible polynomials and  is the number of conjugacy classes of G.

References

  Review

 

Theorems in algebra
Determinants
Theorems in group theory
Matrix theory