If N is a nearfield and F = (K)(N) is its kernel, N is a vector space over F and for all u∈N x , the map x |-> x u is an F-linear transformation. This action of N x on the non-zero elements of the vector space is transitive and fixed-point-free.
Similarly, we may regard N as a vector space over its prime field and again the elements of N x act as linear transformations. In the following code the vector space E could be either a vector space over the kernel or a vector space of the prime field. The default setting is to use the kernel. But if the parameter LargeMatrices is set to true when a regular nearfield is first defined, the prime field will be used. For irregular nearfields the kernel coincides with the prime field.
Let (p, h, v) be the Dickson triple for N, let ζ be a primitive element of K = GF(qv) and put A = < ζv >. Then A is a group of order m = (qv - 1)/v and the elements si = ζ(qi - 1)/(q - 1) (1≤i ≤v) are coset representatives for A in K x . Let Φ denote the Frobenius automorphism x |-> xq of K and define ρ : K x to Gal(K/GF(p)) by ρ(u) = Φi if u∈siA; that is, letting automorphisms of K act on the right, we have xρ(u) = xqi. The map ρ is not a homomorphism. However, its image is the cyclic group of order v generated by Φ = ρ(ζ) and the fixed field of imρ is GF(q); thus imρ may be identified with Gal(K/GF(q)).
The underlying set of N is identified with K and multiplication in N is defined to be w u = wρ(u)u.
The group U of units of the Dickson nearfield D = D(p, h, v, ζ) has generators a and b and relations am = 1, bv = at and b - 1ab = aq, where q = ph, m = (qv - 1)/v and t = m/(q - 1). Furthermore, Ellers and Karzel [EK64] show that gcd(v, t) = gcd(q - 1, t) ≤2. Equality holds if and only if v ≡ 2 mod 4 and q ≡ 3 mod 4 and this in turn is equivalent to the Sylow 2-subgroup of U being a generalised quaternion group.
The centre of D is GF(q) and its group of units is generated by ζvt.
The unit group of the nearfield N.
> N := DicksonNearfield(3^3,13); > zeta := N`prim; > x := N!(zeta^((3^39-1) div (3^13-1))); > S := sub< N | x >; > U := UnitGroup(S); > IsAbelian(U); true > Factorisation(#N); [ <3, 39> ] > Factorisation(#S); [ <3, 13> ] > Factorisation(#Kernel(N)); [ <3, 3> ] > S; Nearfield S of Dickson type defined by the pair (1594323, 1) Order = 1594323
The order of the unit x of a nearfield.
As a matrix group, the unit group U of a nearfield acts regularly on the non-zero vectors of the underlying vector space E and consequently the affine group E.U is sharply two-transitive. All sharply two-transitive groups occur in this way.
The sharply two-transitive affine group associated with a nearfield, returned as a matrix group.
If Γ = Gal(K/GF(p)) and S = Γltimes K x is the semidirect product of Γ and K x , then D x to S : w |-> ρ(w)w is an embedding of the multiplicative group D x of D = D(p, h, v, ζ) in S, where multiplication in S is defined by
(γ1 a1)(γ2 a2) = γ1γ2 a1γ2 a2.
If U is the image of D x in S, then Γ∩U = 1, Γ U = S and K x ∩U = A = < ζv >. In fact, from the definition of ρ, we have UK x = Γ0ltimes K x , where Γ0 = Gal(K/GF(q)). This is the extended unit group of the Dickson nearfield D.
The extended unit group of a Dickson nearfield.