% \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 <G> 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 generators> , 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 <G> 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 %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: