Orthogonal irreducible representations of finite solvable groups in odd dimension

TL;DR

The paper proves that finite irreducible solvable groups acting orthogonally on odd-dimensional spaces must preserve a decomposition into one-dimensional subspaces, making them monomial, thus generalizing Gow's theorem.

Contribution

It extends Gow's theorem by showing such groups preserve an orthogonal decomposition into 1-spaces in odd dimensions.

Findings

Finite irreducible solvable subgroups in odd dimensions are monomial.

Such groups preserve an orthogonal decomposition into 1-spaces.

Generalization of Gow's theorem to broader class of groups.

Abstract

We prove that if $G$ is a finite irreducible solvable subgroup of an orthogonal group $O(V,Q)$ with $\dim V$ odd, then $G$ preserves an orthogonal decomposition of $V$ into $1$-spaces. In particular $G$ is monomial. This generalizes a theorem of Rod Gow.