20.3.1 Computation of Automorphisms of Group G₉

We can deduce relations by using generator relations from the structure description of G₉ and elementary calculations:

Now, we shall investigate the automorphism group

Let be defined by:

As ψ is automorphism, ψ should preserve the relation x₂x₁x₃ = x₃x₁, that is

As n ≢ 0(mod p), because if n ≡ 0(mod p) then order of (ψ(x₂)) ≠ p.

Also, x₄x₃ = x₃x₄ then ψ(x₄)ψ(x₃) = ψ(x₃)ψ(x₄)

Further, x₁x₄ = x₄x₁ implies ψ(x₁)ψ(x₄) = ψ(x₄)ψ(x₁)

Now, ys − uw ≡ 0(mod p) and yi − kw ≡ 0(mod p) are two homogeneous equations with determinant iu − sk ≠ 0. Hence, these equations have only trivial solutions. ⇒ y = w ≡ 0(mod p).

Next, we check the kernel of ψ. Let be any element of so that it is mapped to identity.

(20.1)

(20.2)

(20.3)

(20.4)

Equations (20.1) and (20.3) are the system of two homogeneous equations with iu − sk ≠ 0 determinant, which gives only a trivial solution. Hence, h₁ = h₃ = 0. From (20.4), we get

As

we get

Further, from (20.2) we get nh₂ ≡ 0(mod p), but n ≢ 0(mod p) ⇒ h₂ = 0. So, if and ψ(h) = 1, it must be that h = 1.

Hence, the kernel is trivial and ψ is an automorphism with the constraints on the parameters already deduced. Now it is very easy to calculate the order of the automorphism group. We have p choices for all j, l, t, v, x, and p − 1 choices for z and i, u, s, and k will in some sense be equivalent to a matrix in GL(F_p), which gives (p² −p)(p² − 1) choices for the elements. Therefore, we have that is really defined as:

20.3.2 Computation of Automorphisms of Group G₁₀

By using our relations from the structure of G₁₀ and elementary calculations, we can deduce some useful relations:

We begin our study of automorphism group by using the above relations.

Let be defined by:

Since ψ is an automorphism, ψ should preserve the relation x₄x₃ = x₂x₃x₄, hence

(20.5)

(20.6)

(20.7)

(20.8)

using (20.7) and (20.8) in (20.5) and (20.6), we get

(20.9)

(20.10)

Also,

(20.11)

(20.12)

(20.13)

(20.14)

Using Equations (20.13) and (20.14) in (20.11) and (20.12), we get

(20.15)

(20.16)

Since j, k, l ≡ 0(mod p), therefore i ≢ 0(mod p) because if i ≡ 0(mod p), then Ord(x₁) ≠ p. Further, from (20.15), i = zn ≢ 0(mod p), we have z ≢ 0(mod p) and n ≢ 0(mod p).

Next, we have x₂x₃ = x₃x₂

Now, n ≢ 0(mod p) gives v ≡ 0(mod p)

From (20.10), we can see zu ≡ n + vy(mod p), therefore zu ≡ n(mod p) or n ≡ zu(mod p).

Also, from (20.15) i ≡ zn(mod p), we get i ≡ zzu(mod p). Thus, we have

Further, as n ≡ zu(mod p) and n ≢ 0(mod p), therefore u ≢ 0 (mod p). Substituting the value of v in (20.9), we get Now, we shall find the constraints on the parameters for which ψ has a trivial kernel. Let

(20.17)

(20.18)

(20.19)

(20.20)

From (20.20), we have h₄ ≡ 0(mod p) for z ≢ 0(mod p) and from (20.19), we get uh₃ ≡ 0(mod p). Since u ≢ 0(mod p), therefore h₃ = 0. Further, from (20.18), we have nh₂ ≡ 0(mod p). As n ≢ 0(mod p), we have h₂ = 0. From (20.17), ih₁ ≡ 0(mod p), but i ≢ 0(mod p), thus h₁ = 0.

So, if and ψ(h) = 1, it must be that h = 1, hence the kernel is trivial and ψ is an automorphism with the constraints we have already deduced. Thus, for ψ to be an automorphism there are p choices for all s, t, w, x, y, and p − 1 choices for both u and z. Therefore, we have that is really defined as:

20.3.3 Computation of Automorphisms of G₁₁

We can deduce some useful relations by using the generator relations of G₁₁ and elementary calculations:

With these relations, we begin our study of the automorphisms of

Let be defined by:

Now, ord(x₁) = ord(ψ(x₁)) = p².

Therefore,

Also, the order of is p, so the order of ψ(x₂) is

Order of x₃ is also p, so Order of ψ(x₃) is p

As ψ is an automorphism, ψ should preserve the relation thus we get

Next, we find the constraints for parameters using the kernel. Let be any element of that it is mapped to identity.

(20.21)

(20.22)

(20.23)

From (20.21), we get

Next, from (20.22) we have

Also, from (20.23) we obtain sz ≡ 0(mod p), but p f s, therefore z ≡ 0(mod p).

So, if and ψ(h) = 1, it must be that h = 1, hence the kernel is trivial and ψ is an automorphism with the restrictions we have already deduced on the parameters. Now, we have here p² − p choices for i and p choices for all l, q, j, k, n, and p − 1 choices for s. Therefore, we have that is really defined as: