Brauer's permutation lemma

This article gives the statement, and proof, of a particular subgroup in a group being conjugacy-closed: in other words, any two elements of the subgroup that are conjugate in the whole group, are also conjugate in the subgroup
View a complete list of such instances/statements

Statement

Brauer's permutation lemma has the following equivalent forms:

1. If a row permutation and a column permutation have the same effect on a nonsingular matrix, then they must have the same cycle type i.e., the same number of cycles of each length in their cycle decompositions.
2. The symmetric group is a conjugacy-closed subgroup in the general linear group over any field of characteristic zero
3. If two permutation matrices are conjugate in the general linear group over a field of characteristic zero, then they have the same cycle type (i.e., the same number of cycles of each length in their cycle decompositions), i.e., are conjugate in the symmetric group itself
4. If two permutation representations of a cyclic group are conjugate in the general linear group over a field of characteristic zero, they are also conjugate in the symmetric group

Facts

• Analogue of Brauer's permutation lemma fails over rationals for every non-cyclic finite group: If $G$ is a non-cyclic finite group, we can find two permutation representations $\varphi_1, \varphi_2$ of $G$ that are equivalent as linear representations over the rational numbers but not as permutation representations.
• Symmetric group of degree six or higher is not weak subset-conjugacy-closed in general linear group over rationals

Applications

Brauer's permutation lemma helps us exploit the conjugacy class-representation duality in two interesting ways. Let $C(G)$ denote the set of conjugacy classes of a finite group $G$ and $I(G)$ denote the set of indecomposable linear representations of $G$. Let $\chi(c,\rho)$ denote the trace of $\rho(g)$ where $g \in G$ (i.e. the character value).

Note that since the field has characteristic zero, the irreducible representations are the same as indecomposable representations.

Consider the matrix with rows indexed by indecomposable representations, columns indexed by conjugacy classes, and the entry in row $\rho$ and column $c$ is $\chi(c,\rho)$.

We can now consider two different actions:

• Application of Brauer's permutation lemma to Galois automorphisms on conjugacy class-representation duality
• Application of Brauer's permutation lemma to group automorphisms on conjugacy class-representation duality