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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.
UnitGroup(GrpPerm, N) : Nfd -> GrpPerm
UnitGroup(GrpPC, N) : NfdDck -> GrpPC
UnitGroup(GrpPC, N) : NfdZss -> GrpPC
The unit group of the nearfield N.
In this example we construct the group of units of a subnearfield.
> 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.
AffineGroup(GrpPerm, N) : Nfd -> GrpPerm
AffineGroup(GrpPC, N) : NfdDck -> GrpPC
AffineGroup(GrpPC, N) : NfdZss -> GrpPC
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.
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