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 (Dedekind 1968), with an English translation in (Curtis 2003, p. 51)).
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.
Let a finite group
Define the matrix
's are pairwise non-proportional irreducible polynomials and
is the number of conjugacy classes of G.[1]