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Octonions, Albert vectors and the group $\mathrm{E}_6(F)$

John N. Bray, Yegor Stepanov, Robert A. Wilson

Abstract

We present a uniform approach to the construction of the groups of type $\mathrm{E}_6$ over arbitrary fields without using Lie theory. This gives a simple description of the group generators and some of the subgroup structure. In the finite case our approach also permits relatively straightforward computation of the group order.

Octonions, Albert vectors and the group $\mathrm{E}_6(F)$

Abstract

We present a uniform approach to the construction of the groups of type over arbitrary fields without using Lie theory. This gives a simple description of the group generators and some of the subgroup structure. In the finite case our approach also permits relatively straightforward computation of the group order.

Paper Structure

This paper contains 12 sections, 44 theorems, 120 equations, 1 table.

Table of Contents

  1. Introduction
  2. Some properties of $\Omega_{2m}(F, Q)$
  3. Octonions
  4. A basis for the split octonion algebra
  5. Albert vectors
  6. Some elements of $\mathrm{SE}_6(F)$
  7. Action of $\mathrm{SE}_6(F)$ on white points
  8. White points in the case of a finite field
  9. The stabiliser of a white point
  10. Simplicity of $\mathord{\mathrm{E}}_6(F)$
  11. Some related geometry
  12. Conclusions

Key Result

Lemma \oldthetheorem

The stabiliser in $\Omega_{2m+2}(F,Q)$ of the vector $v_1$ is a subgroup of shape $W\mathord{{\;\!{:}\;\!}}\Omega_{2m}(F,Q_W)$, and the stabiliser of the pair $(v_1,v_2)$ is a subgroup $\Omega_{2m}(F,Q_W)$.

Theorems & Definitions (69)

  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 59 more