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 "x""g", 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 formula_1 have elements formula_2, and let formula_3 be associated with each element of formula_1. Define the matrix formula_5 with entries formula_6. Then
 formula_7
where the formula_8's are pairwise non-proportional irreducible polynomials and formula_9 is the number of conjugacy classes of "G".
References.
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