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On the Brunnian conjecture

Jean-Yves Degos

Abstract

Let $p$ be a primer number, $n \geq 3$ and integer. Let $f(X) = X^n + a_{n-1}X^{n-1} + \cdots +a_1 X + a_0 \in \mathbb{F}_p[X]$ be a primitive polynomial of degree $n$. Let $C_f$ be the companion matrix of $f(X)$, and $G$ the companion matrix of the polynomial $X^n-1$. Define $G_1 := C_f$ and $G_{k+1} = G G_k G^{-1}$ for $0 \leq k \leq n-1$. The so called ``Brunnian Conjecture'' states that: the general linear group $GL(n,p)$ is generated by $G_1, G_2, \ldots, G_n$. In this paper, we prove it for $p \geq 5$ and $n$ not divisible by $p-1$.

On the Brunnian conjecture

Abstract

Let be a primer number, and integer. Let be a primitive polynomial of degree . Let be the companion matrix of , and the companion matrix of the polynomial . Define and for . The so called ``Brunnian Conjecture'' states that: the general linear group is generated by . In this paper, we prove it for and not divisible by .

Paper Structure

This paper contains 3 sections, 6 theorems, 32 equations.

Table of Contents

  1. Background and strategy
  2. Statements and proofs
  3. Further remarks

Key Result

Theorem 1

Let $p$ a prime number, and $n \geq 3$ an integer, and a primitive polynomial of degree $n$. Let $C=C_f$ denote the companion matrix of $f$, namely: Let $G$ denote the companion matrix of the polynomial $X^n-1$, and let: Then, if $p \geq 5$ and $n$ is not divisible by $p-1$, then

Theorems & Definitions (12)

  • Theorem
  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • ...and 2 more