Frobenius determinant theorem
Encyclopedia
In mathematics, the Frobenius determinant theorem is a discovery 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 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 fact, and this theorem became known as the Frobenius determinant theorem.
have elements 
, and let 
be associated with each element of 
. Define the matrix 
with entries 
. Then
where r is the number of conjugacy classes of G.
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, who wrote a letter to F. G. Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
about it (reproduced in , with an English translation in ).
If one takes the multiplication table of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G and replaces each entry g with the variable xg, and subsequently takes the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
, 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 fact, and this theorem 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 r is the number of conjugacy classes of G.