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Discovery 2019-01-01

Let $u$ be order 3 octonion of norm $1$ and trace $-1$ (in case of characteristic 2 we have $-1=1$; norm is $x\bar x$, trace is $x+\bar x$). Then element $L_uR_{u^{-1}}$ is automorphism. This way we obtain order 3 generators of $G_2(q)$. I tested it in GAP for $q=2,3,4,5,7,8,9$. I don't know how to prove it yet but it seems to be rather easy. I define finite octonions as $\mathbb H_q+\mathbb H_q\iota$ using Cayley-Dickson formula. We define quaternions $\mathbb H_q=M_2\mathbb F_q$ as matrices $2\times 2$ over finite field. Conjugation is defined as adjugate matrix - exchange elements on the diagonal and reverse sign off-diagonal. The complex numbers algebra is finite field $\mathbb F_{q^2}$ represented by subalgebra generated by matrix $\pmatrix{n&1\\ 1&0}$ of order $q+1$ when $n$ is order $q-1$ generator of the field multiplicative group. This matrix satisfy equation $x^2=nx+1$.

Having $\mathbb C_q=\mathbb F_{q^2}$ we can also define quaternions using Cayley-Dickson formula $\mathbb H_q=\mathbb C_q+\mathbb C_qj$. Conjugation is then defined as $\overline{x+yj}=\bar{x}-yj$. It coincides with Frobenius automorphism $x\to x^q$ of the field $\mathbb F_{q^2}$.

Similar law is valid for compact octonions. Take $u=e^{2πi/3}$ for some imaginary octonion $i$. Then $L_uR_{u^{-1}}$ is automorphism fixing complex plane $\langle1,i\rangle$ and rotating by angle $\frac{2π}{3}$ in perpendicular space.

End of discovery

Addition 2019-04-09

Let $T_u$ denote $L_uR_{\bar u}$. Then we have following formula $T_u^{T_v}=T_{u^v}$. In this formula $g^h=h^{-1}gh$ valid in the group or in octonion algebra (elements of norm 1 are invertible).

Recipe for $J_2$

Let $u,v,w$ be three octonions of norm 1 and trace 1 over field $F_4$ such that order of product of each pair is 5. It is equivalent to the fact that pair generate group $A_5$ or 4-dimensional subalgebra $M_2F_4$. In such case elements $T_u, T_v, T_w$ generate group $J_2$ of size 604800. From the formula above we have that 3A conjugacy class in the group corresponds to constellation of 560 order 3 octonions closed by conjugation $x\to \bar yxy $. What is the meaning of such constellation ? I am looking for geometrical object preserved by group $J_2\subset G_24$.

Discovery 2019-01-01

Let $u$ be order 3 octonion of norm $1$ and trace $-1$ (in case of characteristic 2 we have $-1=1$; norm is $x\bar x$, trace is $x+\bar x$). Then element $L_uR_{u^{-1}}$ is automorphism. This way we obtain order 3 generators of $G_2(q)$. I tested it in GAP for $q=2,3,4,5,7,8,9$. I don't know how to prove it yet but it seems to be rather easy. I define finite octonions as $\mathbb H_q+\mathbb H_q\iota$ using Cayley-Dickson formula. We define quaternions $\mathbb H_q=M_2\mathbb F_q$ as matrices $2\times 2$ over finite field. Conjugation is defined as adjugate matrix - exchange elements on the diagonal and reverse sign off-diagonal. The complex numbers algebra is finite field $\mathbb F_{q^2}$ represented by subalgebra generated by matrix $\pmatrix{n&1\\ 1&0}$ of order $q+1$ when $n$ is order $q-1$ generator of the field multiplicative group. This matrix satisfy equation $x^2=nx+1$.

Having $\mathbb C_q=\mathbb F_{q^2}$ we can also define quaternions using Cayley-Dickson formula $\mathbb H_q=\mathbb C_q+\mathbb C_qj$. Conjugation is then defined as $\overline{x+yj}=\bar{x}-yj$. It coincides with Frobenius automorphism $x\to x^q$ of the field $\mathbb F_{q^2}$.

Similar law is valid for compact octonions. Take $u=e^{2πi/3}$ for some imaginary octonion $i$. Then $L_uR_{u^{-1}}$ is automorphism fixing complex plane $\langle1,i\rangle$ and rotating by angle $\frac{2π}{3}$ in perpendicular space.

End of discovery

Addition 2019-04-09

Let $T_u$ denote $L_uR_{\bar u}$. Then we have following formula $T_u^{T_v}=T_{u^v}$. In this formula $g^h=h^{-1}gh$ valid in the group or in octonion algebra (elements of norm 1 are invertible).

Recipe for $J_2$

Let $u,v,w$ be three octonions of norm 1 and trace 1 over field $F_4$ such that order of product of each pair is 5 (for fixed $u$ there are 2016 such $v$; for fixed $u,v$ there are 960 such $w$). It is equivalent to the fact that pair generate group $A_5$ or 4-dimensional subalgebra $M_2F_4$. In such case elements $T_u, T_v, T_w$ generate group $J_2$ of size 604800. From the formula above we have that 3A conjugacy class in the group corresponds to constellation of 560 order 3 octonions closed by conjugation $x\to \bar yxy $. What is the meaning of such constellation ? I am looking for geometrical object preserved by group $J_2\subset G_24$.

Discovery 2019-01-01

Let $u$ be order 3 octonion of norm $1$ and trace $-1$ (in case of characteristic 2 we have $-1=1$; norm is $x\bar x$, trace is $x+\bar x$). Then element $L_uR_{u^{-1}}$ is automorphism. This way we obtain order 3 generators of $G_2(q)$. I tested it in GAP for $q=2,3,4,5,7,8,9$. I don't know how to prove it yet but it seems to be rather easy. I define finite octonions as $\mathbb H_q+\mathbb H_q\iota$ using Cayley-Dickson formula. We define quaternions $\mathbb H_q=M_2\mathbb F_q$ as matrices $2\times 2$ over finite field. Conjugation is defined as adjugate matrix - exchange elements on the diagonal and reverse sign off-diagonal. The complex numbers algebra is finite field $\mathbb F_{q^2}$ represented by subalgebra generated by matrix $\pmatrix{n&1\\ 1&0}$ of order $q+1$ when $n$ is order $q-1$ generator of the field multiplicative group. This matrix satisfy equation $x^2=nx+1$.

Having $\mathbb C_q=\mathbb F_{q^2}$ we can also define quaternions using Cayley-Dickson formula $\mathbb H_q=\mathbb C_q+\mathbb C_qj$. Conjugation is then defined as $\overline{x+yj}=\bar{x}-yj$. It coincides with Frobenius automorphism $x\to x^q$ of the field $\mathbb F_{q^2}$.

Similar law is valid for compact octonions. Take $u=e^{2πi/3}$ for some imaginary octonion $i$. Then $L_uR_{u^{-1}}$ is automorphism fixing complex plane $\langle1,i\rangle$ and rotating by angle $\frac{2π}{3}$ in perpendicular space.

End of discovery

Discovery 2019-01-01

Let $u$ be order 3 octonion of norm $1$ and trace $-1$ (in case of characteristic 2 we have $-1=1$; norm is $x\bar x$, trace is $x+\bar x$). Then element $L_uR_{u^{-1}}$ is automorphism. This way we obtain order 3 generators of $G_2(q)$. I tested it in GAP for $q=2,3,4,5,7,8,9$. I don't know how to prove it yet but it seems to be rather easy. I define finite octonions as $\mathbb H_q+\mathbb H_q\iota$ using Cayley-Dickson formula. We define quaternions $\mathbb H_q=M_2\mathbb F_q$ as matrices $2\times 2$ over finite field. Conjugation is defined as adjugate matrix - exchange elements on the diagonal and reverse sign off-diagonal. The complex numbers algebra is finite field $\mathbb F_{q^2}$ represented by subalgebra generated by matrix $\pmatrix{n&1\\ 1&0}$ of order $q+1$ when $n$ is order $q-1$ generator of the field multiplicative group. This matrix satisfy equation $x^2=nx+1$.

Having $\mathbb C_q=\mathbb F_{q^2}$ we can also define quaternions using Cayley-Dickson formula $\mathbb H_q=\mathbb C_q+\mathbb C_qj$. Conjugation is then defined as $\overline{x+yj}=\bar{x}-yj$. It coincides with Frobenius automorphism $x\to x^q$ of the field $\mathbb F_{q^2}$.

Similar law is valid for compact octonions. Take $u=e^{2πi/3}$ for some imaginary octonion $i$. Then $L_uR_{u^{-1}}$ is automorphism fixing complex plane $\langle1,i\rangle$ and rotating by angle $\frac{2π}{3}$ in perpendicular space.

End of discovery

Addition 2019-04-09

Let $T_u$ denote $L_uR_{\bar u}$. Then we have following formula $T_u^{T_v}=T_{u^v}$. In this formula $g^h=h^{-1}gh$ valid in the group or in octonion algebra (elements of norm 1 are invertible).

Recipe for $J_2$

Let $u,v,w$ be three octonions of norm 1 and trace 1 over field $F_4$ such that order of product of each pair is 5. It is equivalent to the fact that pair generate group $A_5$ or 4-dimensional subalgebra $M_2F_4$. In such case elements $T_u, T_v, T_w$ generate group $J_2$ of size 604800. From the formula above we have that 3A conjugacy class in the group corresponds to constellation of 560 order 3 octonions closed by conjugation $x\to \bar yxy $. What is the meaning of such constellation ? I am looking for geometrical object preserved by group $J_2\subset G_24$.

Discovery 2019-01-01

Let $u$ be order 3 octonion of norm $1$ and trace $-1$ (in case of characteristic 2 we have $-1=1$; norm is $x\bar x$, trace is $x+\bar x$). Then element $L_uR_{u^{-1}}$ is automorphism. This way we obtain order 3 generators of $G_2(q)$. I tested it in GAP for $q=2,3,4,5,7,8,9$. I don't know how to prove it yet but it seems to be rather easy. I define finite octonions as $\mathbb H_q+\mathbb H_q\iota$ using Cayley-Dickson formula. We define quaternions $\mathbb H_q=M_2\mathbb F_q$ as matrices $2\times 2$ over finite field. Conjugation is defined as adjugate matrix - exchange elements on the diagonal and reverse sign off-diagonal. The complex numbers algebra is finite field $\mathbb F_{q^2}$ represented by subalgebra generated by matrix $\pmatrix{n&1\\ 1&0}$ of order $q+1$ when $n$ is order $q-1$ generator of the field multiplicative group. This matrix satisfy equation $x^2=nx+1$.

Having $\mathbb C_q=\mathbb F_{q^2}$ we can also define quaternions using Cayley-Dickson formula $\mathbb H_q=\mathbb C_q+\mathbb C_qj$. Conjugation is then defined as $\overline{x+yj}=\bar{x}+yj$. It coincides with Frobenius automorphism $x\to x^q$ of the field $\mathbb F_{q^2}$.

Similar law is valid for compact octonions. Take $u=e^{2πi/3}$ for some imaginary octonion $i$. Then $L_uR_{u^{-1}}$ is automorphism fixing complex plane $\langle1,i\rangle$ and rotating by angle $\frac{2π}{3}$ in perpendicular space.

End of discovery

Discovery 2019-01-01

Let $u$ be order 3 octonion of norm $1$ and trace $-1$ (in case of characteristic 2 we have $-1=1$; norm is $x\bar x$, trace is $x+\bar x$). Then element $L_uR_{u^{-1}}$ is automorphism. This way we obtain order 3 generators of $G_2(q)$. I tested it in GAP for $q=2,3,4,5,7,8,9$. I don't know how to prove it yet but it seems to be rather easy. I define finite octonions as $\mathbb H_q+\mathbb H_q\iota$ using Cayley-Dickson formula. We define quaternions $\mathbb H_q=M_2\mathbb F_q$ as matrices $2\times 2$ over finite field. Conjugation is defined as adjugate matrix - exchange elements on the diagonal and reverse sign off-diagonal. The complex numbers algebra is finite field $\mathbb F_{q^2}$ represented by subalgebra generated by matrix $\pmatrix{n&1\\ 1&0}$ of order $q+1$ when $n$ is order $q-1$ generator of the field multiplicative group. This matrix satisfy equation $x^2=nx+1$.

Having $\mathbb C_q=\mathbb F_{q^2}$ we can also define quaternions using Cayley-Dickson formula $\mathbb H_q=\mathbb C_q+\mathbb C_qj$. Conjugation is then defined as $\overline{x+yj}=\bar{x}-yj$. It coincides with Frobenius automorphism $x\to x^q$ of the field $\mathbb F_{q^2}$.

Similar law is valid for compact octonions. Take $u=e^{2πi/3}$ for some imaginary octonion $i$. Then $L_uR_{u^{-1}}$ is automorphism fixing complex plane $\langle1,i\rangle$ and rotating by angle $\frac{2π}{3}$ in perpendicular space.

End of discovery