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 Waehling:Fastkoerper.
Let (N,+,·) be a left nearring. For a,b ∈ N we define a ≡ b iff a·n = b·n for all n ∈ N. If a ≡ b, then a and b are called equivalent multipliers. A nearring N is called planar if  N/_{ ≡ }  ≥ 3 and if for any two nonequivalent multipliers a and b in N, for any c ∈ N, the equation a·x = b·x + c has a unique solution. See Clay:Nearrings for basic results on planar nearrings.
All finite nearfields are planar nearrings.
A left nearring (N,+,·) is called weakly divisible if ∀a,b ∈ N ∃x ∈ N : a·x = b or b·x = a.
All finite integral planar nearrings are weakly divisible.
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, pn implies pq−1.
gap> IsPairOfDicksonNumbers( 5, 4 ); true
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\{0},·) generated by w^{n}. Then {w^{(qi−1)/(q−1)}  0 ≤ i ≤ n−1 } is a set of coset representatives of H in F\{0}. For f ∈ Hw^{(qi−1)/(q−1)} and x ∈ F define f*x = f·x^{qi} 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 nonisomorphic Dickson nearfields
determined by the pair of Dickson numbers (q,n)
gap> DicksonNearFields( 5, 4 ); [ ExplicitMultiplicationNearRing ( <pc group of size 625 with 4 generators> , multiplication ), ExplicitMultiplicationNearRing ( <pc group of size 625 with 4 generators> , multiplication ) ]
NumberOfDicksonNearFields(
q,
n )
NumberOfDicksonNearFields
returns the number of nonisomorphic Dickson
nearfields which can be obtained from a pair of Dickson numbers (q,n).
This number is given by Φ(n)/k. Here Φ(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.
gap> NumberOfDicksonNearFields( 5, 4 ); 2
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.
gap> ExceptionalNearFields( 25 ); [ ExplicitMultiplicationNearRing ( <pc group of size 25 with 2 generators> , multiplication ) ]
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.
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 ) ]
PlanarNearRing(
G,
phi,
reps )
A finite Ferrero pair is a pair of groups (N,Φ) where Φ is a fixedpointfree automorphism group of (N,+).
Starting with a Ferrero pair (N,Φ) we can construct a planar nearring in the following way, Clay:Nearrings: Select representatives, say e_{1},…,e_{t}, for some or all of the nontrivial orbits of N under Φ. Let C = Φ(e_{1})∪…∪Φ(e_{t}). For each x ∈ N we define a * x = 0 for a ∈ N\C, and a * x=ϕ_{a}(x) for a ∈ Φ(e_{i}) ⊂ C and ϕ_{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 fixedpointfree automorphism group phi and the list of chosen orbit
representatives reps.
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 )
OrbitRepresentativesForPlanarNearRing(
G,
phi,
i )
Let (N,Φ) be a Ferrero pair, and let E = { e_{1},…,e_{s} } and
F = { f_{1},…,f_{t} } be two sets of nonzero orbit representatives.
The nearring obtained from N,Φ, E by the Ferrero construction
(see PlanarNearRing
) is isomorphic to the nearring obtained from N,Φ, F
iff there exists an automorphism α of (N,+) that normalizes Φ
such that
{ α(e_{1}),…,α(e_{s}) } = { f_{1},…,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 ).
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
WdNearRing(
G,
psi,
phi,
reps )
Every finite (left) weakly divisible nearring (N,+,·) can be constructed in the following way:
(1) Let ψ be an endomorphism of the group (N,+) such that Ker ψ = Image ψ^{r−1} for some integer r, r > 0. (Let ψ^{0} := id.)
(2) Let Φ be an automorphism group of (N,+) such that ψΦ ⊆ Φψ and Φ acts fixedpointfree on N\ Image ψ. (That is, for each φ ∈ Φ there exists φ′ ∈ Φ such that ψφ = φ′ψ and for all n ∈ N\ Image ψ the equality n^{φ} = n implies φ = id. Note that our functions operate from the right just like GAPmappings do.)
(3) Let E ⊆ N be a complete set of orbit representatives for Φ on N\ Image ψ, such that for all e_{1}, e_{2} ∈ E, for all φ ∈ Φ and for all 1 ≤ i ≤ r−1 the equality e_{1}^{φψi} = e_{2}^{ψi} implies φψ^{i} = ψ^{i}.
Then for all n ∈ N, n ≠ 0, there are i ≥ 0 ,φ ∈ Φ and e ∈ E such that n = e^{φψi}; furthermore, for fixed n, the endomorphism φψ^{i} is independent of the choice of e and φ in the representation of n.
For all x ∈ N, e ∈ E,φ ∈ Φ and i ≥ 0 define 0·x : = 0
and

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).
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 )
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