[GAP Forum] G2(3) from OctaveAlgebra(GF(3))

[Marek Mitros]

Hello Wiliam and forum,

There are 702 imaginary unitary elements in octave algebra over GF(3). By
imaginary unitary element I mean such x that conjugated(x) = -x and norm
(x) = 1. For each u imaginary unitary there are 252 perpendicular imaginary
unitary. By "perpendicular" I mean such element v that u*v is also
imaginary. For such pair u,v matrix (L(u)R(v))^2 is automorphism of
octonions.

If you take 8 such random matrices then they will generate G2(3) of
size 4245696.

Regards,
Marek

[Marek Mitros]

I should maybe clarify that L(u) is matrix of left multiplication by
element u and R(v) is right multiplication by element v. Element
(L(u)R(v))^2 is involution fixing quaternion subalgebra generated by u and
v.

Regards,
Marek

Post by Marek Mitros
Hello Wiliam and forum,
There are 702 imaginary unitary elements in octave algebra over GF(3). By
imaginary unitary element I mean such x that conjugated(x) = -x and norm
(x) = 1. For each u imaginary unitary there are 252 perpendicular imaginary
unitary. By "perpendicular" I mean such element v that u*v is also
imaginary. For such pair u,v matrix (L(u)R(v))^2 is automorphism of
octonions.
If you take 8 such random matrices then they will generate G2(3) of
size 4245696.
Regards,
Marek