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 Template:Harv, with an English translation in Template:Harv).

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 $G$ 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

$\det X_G=\prod _{j=1}^r{P}_j(x_{g_1},x_{g_2},\dots ,x_{g_n}{)}^{\mathrm{deg}{P}_j}$

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

References

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