- Dickson numbers
- Dickson nearfields
- Exceptional nearfields
- Planar nearrings
- 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 <[>Waehling:Fastkoerper].

Let `(N,+,cdot)` be a left nearring. For `a,b inN` we define `a equivb`
iff `acdotn = bcdotn` for all `ninN`. If `a equivb`, then `a` and `b`
are called **equivalent multipliers**.
A nearring `N` is called **planar** if `| N/ _{equiv} | ge3` and if
for any two non-equivalent multipliers

All finite nearfields are planar nearrings.

A left nearring `(N,+,cdot)` is called **weakly divisible** if
`foralla,binN existsxinN : acdotx = b` or `bcdotx = a`.

All finite integral planar nearrings are weakly divisible.

`IsPairOfDicksonNumbers( `

`, `

` )`

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

`DicksonNearFields( `

`, `

` )`

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

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( `

`, `

` )`

`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`.

gap> NumberOfDicksonNearFields( 5, 4 ); 2

`ExceptionalNearFields( `

` )`

There are 7 finite nearfields which cannot be obtained from finite fields
via a Dickson process. They are of size `p ^{2}` for

`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

`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( `

`, `

`, `

` )`

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, <[>Clay:Nearrings:]
Select representatives, say `e _{1},...,e_{t}`, for some or all of the
non-trivial orbits of

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 generators> , multiplication )

`OrbitRepresentativesForPlanarNearRing( `

`, `

`, `

` )`

Let `(N,Phi)` be a Ferrero pair, and let `E = { e _{1},...,e_{s} }` and

`PlanarNearRing`

) is isomorphic to the nearring obtained from
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( `

`, `

`, `

`, `

` )`

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

(2) Let `Phi` be an automorphism group of `(N,+)` such that
`psiPhisubseteqPhipsi` and `Phi` acts fixed-point-free on
`Nsetminus` Image `psi`.
(That is, for each
`varphiinPhi` there exists `varphi'inPhi` such that
`psivarphi= varphi'psi` and for all `ninNsetminus` 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 `EsubseteqN` be a complete set of orbit representatives for
`Phi` on `Nsetminus` Image `psi`, such that for all `e _{1}, e_{2}inE`, for all

Then for all `ninN, nneq0`, there are `igeq0 ,varphiinPhi` and
`einE` such that `n = e ^{varphipsi^i}`; furthermore, for fixed

For all `xinN, einE,varphiinPhi` and `igeq0` define `0cdotx := 0`
and

` e ^{varphipsi^i}cdotx := x^{varphipsi^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).

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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SONATA manual

November 2012