# 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](../../references.md#cite-ellers-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.

```magma
> 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

```

## `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.