10 Nearfields, planar nearrings and weakly divisible nearrings

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.

    gap> IsPairOfDicksonNumbers( 5, 4 );
    true

10.2 Dickson nearfields

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(qⁿ) and w be a primitive element of F. Let H be the subgroup of (F\{0},·) generated by wⁿ. 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 non-isomorphic 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 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 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

10.3 Exceptional nearfields

There are 7 finite nearfields which cannot be obtained from finite fields via a Dickson process. They are of size p² 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 ) ]

There are 7 finite nearfields which cannot be obtained from finite fields via a Dickson process. They are of size p² 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 ) ]

10.4 Planar nearrings

A finite Ferrero pair is a pair of groups (N,Φ) where Φ is a fixed-point-free 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 e₁,…,e_t, for some or all of the non-trivial orbits of N under Φ. Let C = Φ(e₁)∪…∪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 fixed-point-free 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 )

Let (N,Φ) be a Ferrero pair, and let E = { e₁,…,e_s } and F = { f₁,…,f_t } be two sets of non-zero 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(e₁),…,α(e_s) } = { f₁,…,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

10.5 Weakly divisible nearrings

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 ψ⁰ := id.)

(2) Let Φ be an automorphism group of (N,+) such that ψΦ ⊆ Φψ and Φ acts fixed-point-free 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 GAP-mappings do.)

(3) Let E ⊆ N be a complete set of orbit representatives for Φ on N\ Image ψ, such that for all e₁, e₂ ∈ E, for all φ ∈ Φ and for all 1 ≤ i ≤ r−1 the equality e₁^φψi = e₂^ψi implies φψⁱ = ψⁱ.

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 φψⁱ 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

eφψi·x : = xφψi

Then (N,+,·) 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).

    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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