%
\Chapter{Nearfields, planar nearrings and weakly divisible nearrings}
%
A *nearfield* is a nearring with $1$ where each nonzero element has a
multiplicative inverse. The (additive) group reduct of a finite
nearfield is necessarily elementary abelian.
For an exposition of nearfields we refer to
\cite{Waehling:Fastkoerper}.
Let $(N,+,
\cdot)$ be a left nearring. For $a,b
\in N$ we define $a
\equiv b$
iff $a
\cdot n = b
\cdot n$ for all $n
\in N$. If $a
\equiv b$, then $a$ and $b$
are called *equivalent multipliers*.
A nearring $N$ is called *planar* if $| N/_{
\equiv} |
\ge 3$ and if
for any two non-equivalent multipliers $a$ and $b$ in $N$, for any $c
\in N$,
the equation $a
\cdot x = b
\cdot x + c$ has a unique solution.
See
\cite{Clay:Nearrings} for basic results on planar nearrings.
All finite nearfields are planar nearrings.
A left nearring $(N,+,
\cdot)$ is called *weakly divisible* if
$
\forall a,b
\in N
\exists x
\in N : a
\cdot x = b$ or $b
\cdot x = a$.
All finite integral planar nearrings are weakly divisible.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Dickson numbers}
\>IsPairOfDicksonNumbers( <q>, <n> )
A pair of Dickson numbers $(q,n)$ consists of a prime power integer $q$
and a natural
number $n$ such that for $p = 4$ or $p$ prime, $p|n$ implies
$p|q-1$.
\beginexample
gap> IsPairOfDicksonNumbers( 5, 4 );
true
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Dickson nearfields}
\>DicksonNearFields( <q>, <n> )
All finite nearfields with 7 exceptions can be obtained via socalled
coupling maps from finite fields. These nearfields are called Dickson
nearfields.
The multiplication map of such a Dickson nearfield is given by a pair of
Dickson numbers $(q,n)$ in the following way:
Let $F = GF(q^n)$ and $w$ be a primitive element of $F$. Let
$H$ be the subgroup of $(F
\setminus\{0
\},
\cdot)$ generated by $w^n$.
Then $
\{w^{(q^i-1)/(q-1)}
\ |
\ 0
\leq i
\leq n-1
\}$ is a set of coset
representatives of $H$ in $F
\setminus\{0
\}$.
For $f
\in Hw^{(q^i-1)/(q-1)}$ and $x
\in F$ define $f*x = f
\cdot x^{q^i}$
and $0*x = 0$. Then $*$ is a nearfield multiplication on the additive group
$(F,+)$.
Note that a Dickson nearfield is not uniquely determined by $(q,n)$, since
$w$ can be chosen arbitrarily. Different choices of $w$ may yield isomorphic
nearfields.
`DicksonNearFields
' returns a list of the non-isomorphic Dickson nearfields
determined by the pair of Dickson numbers $(q,n)$
\beginexample
gap> DicksonNearFields( 5, 4 );
[ ExplicitMultiplicationNearRing ( <pc group of size 625 with
4 generators> , multiplication ),
ExplicitMultiplicationNearRing ( <pc group of size 625 with
4 generators> , multiplication ) ]
\endexample
\>NumberOfDicksonNearFields( <q>, <n> )
`NumberOfDicksonNearFields
' returns the number of non-isomorphic Dickson
nearfields which can be obtained from a pair of Dickson numbers $(q,n)$.
This
number is given by $
\Phi(n)/k$. Here $
\Phi(n)$ denotes the
number
of relatively prime residues modulo $n$ and $k$ is the multiplicative order
of $p$ modulo $n$ where $p$ is the prime divisor of $q$.
\beginexample
gap> NumberOfDicksonNearFields( 5, 4 );
2
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Exceptional nearfields}
\>ExceptionalNearFields( <q> )
There are 7 finite nearfields which cannot be obtained from finite fields
via a Dickson process. They are of size $p^2$ for
$p = 5, 7, 11, 11, 23, 29, 59$. (There exist 2 exceptional nearfields of size
121.)
`ExceptionalNearFields
' returns the list of exceptional nearfields for a given
size <q>.
\beginexample
gap> ExceptionalNearFields( 25 );
[ ExplicitMultiplicationNearRing ( <pc group of size 25 with
2 generators> , multiplication ) ]
\endexample
\>AllExceptionalNearFields()
There are 7 finite nearfields which cannot be obtained from finite fields
via a Dickson process. They are of size $p^2$ for
$p = 5, 7, 11, 11, 23, 29, 59$. (There exist 2 exceptional nearfields of size
121.)
`AllExceptionalNearFields
' without argument returns the list of exceptional
nearfields.
\beginexample
gap> AllExceptionalNearFields();
[ ExplicitMultiplicationNearRing ( <pc group of size 25 with
2 generators> , multiplication ),
ExplicitMultiplicationNearRing ( <pc group of size 49 with
2 generators> , multiplication ),
ExplicitMultiplicationNearRing ( <pc group of size 121 with
2 generators> , multiplication ),
ExplicitMultiplicationNearRing ( <pc group of size 121 with
2 generators> , multiplication ),
ExplicitMultiplicationNearRing ( <pc group of size 529 with
2 generators> , multiplication ),
ExplicitMultiplicationNearRing ( <pc group of size 841 with
2 generators> , multiplication ),
ExplicitMultiplicationNearRing ( <pc group of size 3481 with
2 generators> , multiplication ) ]
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Planar nearrings}
\>PlanarNearRing( <G>, <phi>, <reps> )
A finite *Ferrero pair* is a pair of groups $(N,
\Phi)$ where $
\Phi$ is a
fixed-point-free automorphism group of $(N,+)$.
Starting with a Ferrero pair $(N,
\Phi)$ we can construct a planar nearring
in the following way,
\cite{Clay:Nearrings}:
Select representatives, say $e_{1},
\ldots,e_{t}$, for some or all of the
non-trivial orbits of $N$ under $
\Phi$.
Let $C =
\Phi(e_1)
\cup\ldots\cup\Phi(e_t)$.
For each $x
\in N$ we define $a * x = 0$ for $a
\in N
\setminus C$, and
$a * x=
\phi_{a}(x)$ for $a
\in\Phi(e_{i})
\subset C$ and $
\phi_{a}(e_{i})=a$.
Then $(N,+,*)$ is a (left) planar nearring.
Every finite planar nearring can be constructed from some Ferrero pair
together with a set of orbit representatives in this way.
`PlanarNearRing
' returns the planar nearring on the group determined by
the fixed-point-free automorphism group <phi> and the list of chosen orbit
representatives <reps>.
\beginexample
gap> C7 := CyclicGroup( 7 );;
gap> i := GroupHomomorphismByFunction( C7, C7, x -> x^-1 );;
gap> phi := Group( i );;
gap> orbs := Orbits( phi, C7 );
[ [ <identity> of ... ], [ f1, f1^6 ], [ f1^2, f1^5 ],
[ f1^3, f1^4 ] ]
gap> # choose reps from the orbits
gap> reps := [orbs[2][1], orbs[3][2]];
[ f1, f1^5 ]
gap> n := PlanarNearRing( C7, phi, reps );
ExplicitMultiplicationNearRing ( <pc group of size 7 with
1 generator> , multiplication )
\endexample
\>OrbitRepresentativesForPlanarNearRing( <G>, <phi>, <i> )
%For a fixed Ferrero pair distinct choices of representatives may yield
%isomorphic nearrings.
Let $(N,
\Phi)$ be a Ferrero pair, and let $E =
\{ e_{1},
\ldots,e_{s}
\}$ and
$F =
\{ f_{1},
\ldots,f_{t}
\}$ be two sets of non-zero orbit representatives.
The nearring obtained from $N,
\Phi, E$ by the Ferrero construction
(see `PlanarNearRing
') is isomorphic to the nearring obtained from $N,\Phi, F$
iff there exists an automorphism $
\alpha$ of $(N,+)$ that normalizes $
\Phi$
such that
$
\{ \alpha(e_{1}),
\ldots,
\alpha(e_{s})
\} =
\{ f_{1},
\ldots,f_{t}
\}$.
The function `OrbitRepresentativesForPlanarNearRing
'
returns precisely one set of representatives of cardinality <i> for each
isomorphism class of planar nearrings which can be generated from the
Ferrero pair ( <G>, <phi> ).
\beginexample
gap> C7 := CyclicGroup( 7 );;
gap> i := GroupHomomorphismByFunction( C7, C7, x -> x^-1 );;
gap> phi := Group( i );;
gap> reps := OrbitRepresentativesForPlanarNearRing( C7, phi, 2 );
[ [ f1, f1^2 ], [ f1, f1^5 ] ]
gap> n1 := PlanarNearRing( C7, phi, reps[1] );;
gap> n2 := PlanarNearRing( C7, phi, reps[2] );;
gap> IsIsomorphicNearRing( n1, n2 );
false
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Weakly divisible nearrings}
\>WdNearRing( <G>, <psi>, <phi>, <reps> )
Every finite (left) weakly divisible nearring $(N,+,
\cdot)$ can be constructed
in the following way:
(1) Let $
\psi$ be an endomorphism of the group $(N,+)$ such that Ker
$
\psi =$ Image $
\psi^{r-1}$ for some integer $r, r>0$. (Let $
\psi^0 :=$ id.)
(2) Let $
\Phi$ be an automorphism group of $(N,+)$ such that
$
\psi\Phi\subseteq\Phi\psi$ and $
\Phi$ acts fixed-point-free on
$N
\setminus$ Image $
\psi$.
(That is, for each
$
\varphi\in\Phi$ there exists $
\varphi'\in\Phi$ such that
$
\psi\varphi =
\varphi'\psi$ and for all $n\in N\setminus$ Image $\psi$ the
equality $n^
\varphi = n$ implies $
\varphi =$ id. Note that our functions
operate from the right just like GAP-mappings do.)
(3) Let $E
\subseteq N$ be a complete set of orbit representatives for
$
\Phi$ on $N
\setminus$ Image $
\psi$, such that for all $e_1, e_2
\in E$, for all
$
\varphi\in\Phi$ and for all $1
\leq i
\leq r-1$ the equality
$e_1^{
\varphi\psi^i} = e_2^{
\psi^i}$ implies $
\varphi\psi^i =
\psi^i$.
Then for all $n
\in N, n
\neq 0$, there are $i
\geq 0 ,
\varphi\in\Phi$ and
$e
\in E$ such that $n = e^{
\varphi\psi^i}$; furthermore, for fixed $n$, the
endomorphism $
\varphi\psi^i$ is independent of the choice of $e$ and
$
\varphi$ in the representation of $n$.
For all $x
\in N, e
\in E,
\varphi\in\Phi$ and $i
\geq 0$ define $0
\cdot x := 0$
and
$$ e^{
\varphi\psi^i}
\cdot x := x^{
\varphi\psi^i} $$
Then $(N,+,
\cdot)$ is a zerosymmetric (left) wd nearring.
`WdNearRing
' returns the wd nearring on the group as defined above
by the nilpotent endomorphism <psi>, the automorphism group <phi> and
a list of orbit representatives <reps> where the arguments fulfill the
conditions (1) to (3).
\beginexample
gap> C9 := CyclicGroup( 9 );;
gap> psi := GroupHomomorphismByFunction( C9, C9, x -> x^3 );;
gap> Image( psi );
Group([ f2, <identity> of ... ])
gap> Image( psi ) = Kernel( psi );
true
gap> a := GroupHomomorphismByFunction( C9, C9, x -> x^4 );;
gap> phi := Group( a );;
gap> Size( phi );
3
gap> orbs := Orbits( phi, C9 );
[ [ <identity> of ... ], [ f2 ], [ f2^2 ], [ f1, f1*f2, f1*f2^2 ],
[ f1^2, f1^2*f2^2, f1^2*f2 ] ]
gap> # choose reps from the orbits outside of Image( psi )
gap> reps := [orbs[4][1], orbs[5][1]];
[ f1, f1^2 ]
gap> n := WdNearRing( C9, psi, phi, reps );
ExplicitMultiplicationNearRing ( <pc group of size 9 with
2 generators> , multiplication )
\endexample
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