The Group of Units#

If \(N\) is a nearfield and \(F = {\cal K}(N)\) is its kernel, \(N\) is a vector space over \(F\) and for all \(u\in N^\times\), the map \(x \mapsto x\circ u\) is an \(F\)-linear transformation. This action of \(N^\times\) 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 \(N^\times\) 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 \(\zeta\) be a primitive element of \(K = {\bf F}_{q^v}\) and put \(A = \langle \zeta^v\rangle\). Then \(A\) is a group of order \(m = (q^v-1)/v\) and the elements \(s_i = \zeta^{(q^i-1)/(q-1)}\) (\(1\le i \le v\)) are coset representatives for \(A\) in \(K^\times\). Let \(\Phi\) denote the Frobenius automorphism \(x \mapsto x^q\) of \(K\) and define \(\rho : K^\times \to {\operatorname{Gal}}(K/{\bf F}_{p})\) by \(\rho(u) = \Phi^i\) if \(u\in s_iA\); that is, letting automorphisms of \(K\) act on the right, we have \(x^{\rho(u)} = x^{q^i}\). The map \(\rho\) is not a homomorphism. However, its image is the cyclic group of order \(v\) generated by \(\Phi = \rho(\zeta)\) and the fixed field of \(\hbox{im}\rho\) is \({\bf F}_{q}\); thus \(\hbox{im}\rho\) may be identified with \({\operatorname{Gal}}(K/{\bf F}_{q})\).

The underlying set of \(N\) is identified with \(K\) and multiplication in \(N\) is defined to be \(w\circ u = w^{\rho(u)}u\).

The group \(U\) of units of the Dickson nearfield \(D = D(p,h,v,\zeta)\) has generators \(a\) and \(b\) and relations \(a^m = 1\), \(b^v = a^t\) and \(b^{-1}ab = a^q\), where \(q = p^h\), \(m = (q^v-1)/v\) and \(t = m/(q-1)\). Furthermore, Ellers and Karzel [Ellers and Karzel, 1964] show that \(\gcd(v,t) = \gcd(q-1,t) \le 2\). Equality holds if and only if \(v \equiv 2 \pmod 4\) and \(q\equiv 3\pmod 4\) and this in turn is equivalent to the Sylow \(2\)-subgroup of \(U\) being a generalised quaternion group.

The centre of \(D\) is \({\bf F}_{q}\) and its group of units is generated by \(\zeta^{vt}\).

UnitGroup(N): Nfd -> GrpMat, Map#
UnitGroup(GrpPerm, N): Nfd -> GrpPerm#
UnitGroup(GrpPC, N): NfdDck -> GrpPC#
UnitGroup(GrpPC, N): NfdZss -> GrpPC#: The unit group of the nearfield \(N\).

Example: unitgrp (ex-110490)#

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

Run in calculator

Order(x): NfdElt -> RngIntElt#: 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 \(E\cdot U\) is sharply two-transitive. All sharply two-transitive groups occur in this way.

AffineGroup(N): Nfd -> GrpMat#
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 \(\Gamma = {\operatorname{Gal}}(K/{\bf F}_{p})\) and \(S = \Gamma\ltimes K^\times\) is the semidirect product of \(\Gamma\) and \(K^\times\), then \(D^\times \to S : w \mapsto \rho(w)w\) is an embedding of the multiplicative group \(D^\times\) of \(D = D(p,h,v,\zeta)\) in \(S\), where multiplication in \(S\) is defined by

\[(\gamma_1 a_1)(\gamma_2 a_2) = \gamma_1\gamma_2 a_1^{\gamma_2} a_2.\]

If \(U\) is the image of \(D^\times\) in \(S\), then \(\Gamma\cap U = 1\), \(\Gamma U = S\) and \(K^\times \cap U = A = \langle\zeta^v\rangle\). In fact, from the definition of \(\rho\), we have \(UK^\times = \Gamma_0\ltimes K^\times\), where \(\Gamma_0 = {\operatorname{Gal}}(K/{\bf F}_{q})\). This is the extended unit group of the Dickson nearfield \(D\).

ExtendedUnitGroup(D): NfdDck -> GrpMat#: The extended unit group of a Dickson nearfield.