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In sagemath, the general orthogonal group over a binary field is defined by

> G = GO(n, GF(2), e)

in which G indicates the n*n binary orthogonal matrices in GF(2).(When n is odd, that is for sure). What I don't get is when n is even. It says that parameter 'e' is relevant only when n is even. The sagemath description for e is:

e – +1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms.

What exactly is the mathematical meaning of e? Simple example gives

> G = GO(4, GF(2), 1)
> G.order() = 72
> H = GO(4, GF(2), -1)
> H.order() = 120
> inter = G.intersection(H)
> inter.order() = 12

Would this mean that the set of all binary orthogonal matrices of size 4 is constructed by the union of G and H(so that they have order 72+120-12 = 180)? More generally, how can i find the whole group of binary orthogonal matrices of size n(n even)?

In sagemath, the general orthogonal group over a binary field is defined by

> G = GO(n, GF(2), e)

in which G indicates the n*n binary orthogonal matrices in GF(2).(When n is odd, that is for sure). What I don't get is when n is even. It says that parameter 'e' is relevant only when n is even. The sagemath description for e is:

e – +1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms.

What exactly is the mathematical meaning of e? Simple example gives

> G = GO(4, GF(2), 1)
> G.order() = 72
> H = GO(4, GF(2), -1)
> H.order() = 120
> inter = G.intersection(H)
> inter.order() = 12

Would this mean that the set of all binary orthogonal matrices of size 4 is constructed by the union of G and H(so that they have order 72+120-12 = 180)? More generally, how can i find the whole group of binary orthogonal matrices of size n(n even)?even)?

Edit) Now I'm confused about the whole general orthogonal group. When I did

> G = GO(3, GF(2))
> set = [matrix(element) * matrix(element).transpose() for element in G]

set does not check out to be all identites. I thought that the general orthogonal group had the definition of $$GO(n, GF(q)) = {Q \in GL(n, GF(q)) | QQ^T = Q^TQ = I_n}$$ Is this not what the GO command does?

In sagemath, the general orthogonal group over a binary field is defined by

> G = GO(n, GF(2), e)

in which G indicates the n*n binary orthogonal matrices in GF(2).(When n is odd, that is for sure). What I don't get is when n is even. It says that parameter 'e' is relevant only when n is even. The sagemath description for e is:

e – +1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms.

What exactly is the mathematical meaning of e? Simple example gives

> G = GO(4, GF(2), 1)
> G.order() = 72
> H = GO(4, GF(2), -1)
> H.order() = 120
> inter = G.intersection(H)
> inter.order() = 12

Would this mean that the set of all binary orthogonal matrices of size 4 is constructed by the union of G and H(so that they have order 72+120-12 = 180)? More generally, how can i find the whole group of binary orthogonal matrices of size n(n even)?

Edit) Now I'm confused about the whole general orthogonal group. When I did

> G = GO(3, GF(2))
> set = [matrix(element) * matrix(element).transpose() for element in G]

set does not check out to be all identites. I thought that the general orthogonal group had the definition of $$GO(n, GF(q)) = {Q \in GL(n, GF(q)) | QQ^T = Q^TQ = I_n}$$ Is this not what the GO command does?

In sagemath, the general orthogonal group over a binary field is defined by

> G = GO(n, GF(2), e)

in which G indicates the n*n binary orthogonal matrices in GF(2).(When n is odd, that is for sure). What I don't get is when n is even. It says that parameter 'e' is relevant only when n is even. The sagemath description for e is:

e – +1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms.

What exactly is the mathematical meaning of e? Simple example gives

> G = GO(4, GF(2), 1)
> G.order() = 72
> H = GO(4, GF(2), -1)
> H.order() = 120
> inter = G.intersection(H)
> inter.order() = 12

Would this mean that the set of all binary orthogonal matrices of size 4 is constructed by the union of G and H(so that they have order 72+120-12 = 180)? More generally, how can i find the whole group of binary orthogonal matrices of size n(n even)?

Edit) Now I'm confused about the whole general orthogonal group. When I did

> G = GO(3, GF(2))
> set = [matrix(element) * matrix(element).transpose() for element in G]

set does not check out to be all identites. I thought that the general orthogonal group had the definition of $$GO(n, GF(q)) = {Q \in GL(n, GF(q)) | QQ^T = Q^TQ = I_n}$$ Is this not what the GO command does?

In sagemath, the general orthogonal group over a binary field is defined by

> G = GO(n, GF(2), e)

in which G indicates the n*n binary orthogonal matrices in GF(2).(When n is odd, that is for sure). What I don't get is when n is even. It says that parameter 'e' is relevant only when n is even. The sagemath description for e is:

e – +1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms.

What exactly is the mathematical meaning of e? Simple example gives

> G = GO(4, GF(2), 1)
> G.order() = 72
> H = GO(4, GF(2), -1)
> H.order() = 120
> inter = G.intersection(H)
> inter.order() = 12

Would this mean that the set of all binary orthogonal matrices of size 4 is constructed by the union of G and H(so that they have order 72+120-12 = 180)? More generally, how can i find the whole group of binary orthogonal matrices of size n(n even)?

Edit) Now I'm confused about the whole general orthogonal group. When I did

> G = GO(3, GF(2))
> set = [matrix(element) * matrix(element).transpose() for element in G]

set does not check out to be all identites. I thought that the general orthogonal group had the definition of $$GO(n, GF(q)) = {Q \{Q \in GL(n, GF(q)) | QQ^T = Q^TQ = I_n}$$ I_n\}$$ Is this not what the GO command does?

In sagemath, the general orthogonal group over a binary field is defined by

> G = GO(n, GF(2), e)

in which G indicates the n*n binary orthogonal matrices in GF(2).(When n is odd, that is for sure). What I don't get is when n is even. It says that parameter 'e' is relevant only when n is even. The sagemath description for e is:

e – +1 or -1, and ignored by default; only relevant for finite fields and if the degree is even: a parameter that distinguishes inequivalent invariant forms.

What exactly is the mathematical meaning of e? Simple example gives

> G = GO(4, GF(2), 1)
> G.order() = 72
> H = GO(4, GF(2), -1)
> H.order() = 120
> inter = G.intersection(H)
> inter.order() = 12

Would this mean that the set of all binary orthogonal matrices of size 4 is constructed by the union of G and H(so that they have order 72+120-12 = 180)? More generally, how can i find the whole group of binary orthogonal matrices of size n(n even)?

Edit) Now I'm confused about the whole general orthogonal group. When I did

> G = GO(3, GF(2))
> set = [matrix(element) * matrix(element).transpose() for element in G]

set does not check out to be all identites. I thought that the general orthogonal group had the definition of $$GO(n, GF(q)) = \{Q \in GL(n, GF(q)) | QQ^T = Q^TQ = I_n\}$$ Is this not what the GO command does?