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Orthogonal irreducible representations of finite solvable groups in odd dimension

Mikko Korhonen

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.

Orthogonal irreducible representations of finite solvable groups in odd dimension

Abstract

We prove that if is a finite irreducible solvable subgroup of an orthogonal group with odd, then preserves an orthogonal decomposition of into -spaces. In particular is monomial. This generalizes a theorem of Rod Gow.
Paper Structure (2 sections, 4 theorems, 10 equations)

This paper contains 2 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem 1.1

Suppose that $n$ is odd. Let $G \leq O(V, Q)$ be finite irreducible solvable. Then there exists an orthogonal decomposition such that $\dim W_i = 1$ for all $1 \leq i \leq n$ and $G$ acts on $\{W_1, \ldots, W_n\}$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:mainthm']}
  • proof : Proof of Corollary \ref{['cor:cor1']}
  • proof : Proof of Corollary \ref{['cor:cor2']}