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 $$G$$ have elements $$g_1, g_2,\dots,g_n$$, and let $$x_{g_i}$$ be associated with each element of $$G$$. Define the matrix $$X_G$$ with entries $$a_{ij}=x_{g_i g_j}$$. Then


 * $$ \det X_G = \prod_{j=1}^r P_j(x_{g_1},x_{g_2},\dots,x_{g_n})^{\deg P_j}$$

where the $$P_{j}$$'s are pairwise non-proportional irreducible polynomials and $$r$$ is the number of conjugacy classes of G.