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##
## finite near-fields:
##
############################################################################
##
#M IsPairOfDicksonNumbers( <q>, <n> )
##
InstallMethod(
IsPairOfDicksonNumbers,
"default",
true,
[IsInt, IsInt],
0,
function( q, n )
local primes, x;
primes := Set( Factors( n ) );
if n mod 4 = 0 then
primes[1] := 4;
fi;
return (IsPrimePowerInt( q ) and ForAll( primes, x -> (q-1) mod x = 0 ));
end );
############################################################################
##
#M NumberOfDicksonNearFields( <q>, <n> )
##
InstallMethod(
NumberOfDicksonNearFields,
"default",
true,
[IsInt, IsInt],
0,
function( q, n )
return QuoInt( Phi( n ), Size( GroupByPrimeResidues( [SmallestRootInt( q )], n ) ) );
end );
############################################################################
##
#M DicksonNearFields( <q>, <n> )
##
## returns the list of all non-isomorphic near-fields determined by the
## Dickson pair ( <q>, <n> )
##
InstallMethod(
DicksonNearFields,
"default",
true,
[IsInt, IsInt],
0,
function( q, n )
local primes,
F, basis, l, p, z,
a, s, rems, ks, bs, b, powers,
G, gens, g, imgs,
alpha, beta, phi, endos, mul, Ns, N,
x, y, e, f, i;
primes := Set( Factors( n ) );
if n mod 4 = 0 then
primes[1] := 4;
fi;
if not IsPrimePowerInt( q ) or ForAny( primes, x -> (q-1) mod x <> 0 ) then
Error( " (q, n) are not Dickson numbers \n" );
fi;
F := GF( q^n );
basis := Basis( F );
l := Length( basis );
p := RootInt( q^n, l );
z := PrimitiveElement( F );
## determine the non-isomorphic nearfields
a := z^n;
ks := List( RightCosets(Units(Integers mod n),GroupByPrimeResidues([p],n)),
x -> Int( Representative(x) ) );
bs := List( ks, x -> z^x );
Info( InfoNearRing, 1, Length( bs ), " non-isomorphic nearfields \n" );
G := ElementaryAbelianGroup( q^n );
gens := GeneratorsOfGroup( G );
g := gens[1];
imgs := [];
for i in List( basis, x -> Coefficients( basis, a*x ) ) do
Add( imgs, Product( List( [1..l], x -> gens[x]^Int(i[x]) ) ) );
od;
alpha := GroupHomomorphismByImagesNC( G, G, gens, imgs );
Ns := [];
for b in bs do
imgs := [];
for i in List( basis, x -> Coefficients( basis, b*x^q ) ) do
Add( imgs, Product( List( [1..l], x -> gens[x]^Int(i[x]) ) ) );
od;
beta := GroupHomomorphismByImagesNC( G, G, gens, imgs );
Info( InfoNearRing, 2, "autos are built \n" );
Add( Ns, PlanarNearRing( G, Group( alpha, beta ), [g] ) );
Info( InfoNearRing, 1, "new near field \n" );
od;
return Ns;
end );
############################################################################
##
#M ExceptionalNearFields( <q> )
##
## returns the list of all non-isomorphic finite near-fields of size <q>
## which are not Dickson near-fields
##
InstallMethod(
ExceptionalNearFields,
"default",
true,
[IsInt],
0,
function( q )
local p, G, gens, a, b, alpha, phis,
phi, endos, mul, Ns, N,
x, y, e, f, i;
if not q in [25, 49, 121, 529, 841, 3481] then
Error( "there is no exceptional near field of this order" );
fi;
p := RootInt( q );
G := ElementaryAbelianGroup(q);
gens := GeneratorsOfGroup( G );
a := gens[1]; b := gens[2];
alpha := GroupHomomorphismByImagesNC( G, G, gens, [b, a^-1] );
if q = 25 then
phis := [Group( alpha,
GroupHomomorphismByImagesNC( G, G, gens, [a*b^-1, a^-2*b^-2] ) )];
elif q = 49 then
phis := [Group( alpha,
GroupHomomorphismByImagesNC( G, G, gens, [a*b^-1, a^3*b^-2] ) )];
elif q = 121 then
phis := [Group( alpha,
GroupHomomorphismByImagesNC( G, G, gens, [a*b^-5, a^5*b^-2] ),
GroupHomomorphismByImagesNC( G, G, gens, [a^4, b^4] ) ),
Group( alpha,
GroupHomomorphismByImagesNC( G, G, gens, [a^2*b, a^4*b^-3] ) )];
elif q = 529 then
phis := [Group( alpha,
GroupHomomorphismByImagesNC( G, G, gens, [a*b^12, a^-6*b^-2] ),
GroupHomomorphismByImagesNC( G, G, gens, [a^2, b^2] ) )];
elif q = 841 then
phis := [Group( alpha,
GroupHomomorphismByImagesNC( G, G, gens, [a*b^-12, a^-7*b^-2] ),
GroupHomomorphismByImagesNC( G, G, gens, [a^-13, b^-13] ) )];
elif q = 3481 then
phis := [Group( alpha,
GroupHomomorphismByImagesNC( G, G, gens, [a^9*b^-10, a^15*b^-10] ),
GroupHomomorphismByImagesNC( G, G, gens, [a^4, b^4] ) )];
fi;
return List( phis, phi -> PlanarNearRing( G, phi, [a] ) );
end );
############################################################################
##
#M AllExceptionalNearFields( <q> )
##
## returns the list of all non-isomorphic finite near-fields
## which are not Dickson near-fields
##
InstallMethod(
AllExceptionalNearFields,
"default",
true,
[],
0,
function( )
return Concatenation( List( [25, 49, 121, 529, 841, 3481],
ExceptionalNearFields ) );
end);
##
## planar near-rings
##
############################################################################
##
#M PlanarNearRing( <G>, <phi>, <reps> )
##
## returns the planar near-ring determined by the group <G>
## with a group of fixed-point-free automorphisms <phi> and
## a set of orbit representatives <reps>
##
InstallMethod(
PlanarNearRing,
"nearfield",
true,
[IsGroup, IsGroup, IsList],
10,
function( G, phi, reps )
local p, e, autos, mul, a, i, N,
f, j, x, y;
## assuming Size(phi) = Size(G) - 1, that is, nearring is a nearfield
if Size(phi) <> Size(G) - 1 then
TryNextMethod();
fi;
p := SmallestRootInt(Size(G));
e := reps[1];
autos := [];
for f in phi do
i := ExtRepOfObj( e^f );
autos[Sum( List( [1..QuoInt(Length(i),2)], j -> p^(i[2*j-1]-1)*i[2*j]) )] := f;
od;
mul := function( a, x )
if a = Identity( G ) then
return a;
else
## a = e^f, with f in autos
i := ExtRepOfObj(a);
return Image( autos[Sum( List( [1..QuoInt(Length( i ),2)],
j -> p^(i[2*j-1]-1)*i[2*j]) )], x );
fi;
end;
N := ExplicitMultiplicationNearRingNC( G, mul );
SetIsNearField( N, true );
return N;
end );
InstallMethod(
PlanarNearRing,
"default",
true,
[IsGroup, IsGroup, IsList],
0,
function( G, phi, reps )
local t, autos, orbits, mul, a, N,
x, y;
t := Length( reps );
autos := AsList( phi );
orbits := List( reps, x -> List( autos, y -> x^y ) );
mul := function( a, x )
local z,
i, j;
i := 1;
z := Identity( G );
while i in [1..t] do
j := Position( orbits[i], a );
if IsInt( j ) then
z := x^autos[j];
# stop search and return <z>
i := t;
fi;
i := i+1;
od;
return z;
end;
N := ExplicitMultiplicationNearRingNC( G, mul );
SetIsPlanarNearRing( N, true );
return N;
end );
############################################################################
##
#M OrbitRepresentativesForPlanarNearRing( <G>, <phi>, <I> )
##
##
## returns all sets of <I> representatives of the orbits of <phi> on <G>
## so that they generate all non isomorphic planar nearrings out of the
## Ferrero pair ( <G>, <phi> )
InstallMethod(
OrbitRepresentativesForPlanarNearRing,
"default",
true,
[IsGroup, IsGroup, IsInt],
0,
function( G, phi, I )
local A, N, S,
orbs, combs, lanes, reps, tuples,
o, So, t,
i, j, x, y;
t := (Size(G)-1)/Size(phi);
if I > t then
Error( "more representatives than orbits" );
elif t = 1 then
return [[AsList( G )[2]]];
fi;
A := AutomorphismGroup( G );
N := Normalizer( A, phi );
orbs := List( Orbits( phi, Difference( AsList( G ), [Identity(G)] ) ),
Set );
## consider combinations of I orbits to choose one representative
## from each of these orbits
combs := Combinations( orbs, I );
## determine the orbits of N acting on combs
lanes := Orbits( N, combs, OnSetsDisjointSets );
reps := [];
for i in [1..Length(lanes)] do
## choose one combination of I distinct orbits,
## ie, choose one element o from each lane
o := lanes[i][1];
So := Stabilizer( N, o, OnSetsDisjointSets );
## take one element of each orbit in o to obtain I representatives of distinct
## orbits
tuples := List( Cartesian( o ), Set );
## choose representatives of the action of So (the stabilizer of o in N)
## on tuples
reps := Concatenation( reps,
List( Orbits( So, tuples, OnSets ), Representative ) );
od;
return reps;
end );
##
## weakly divisible near-rings
##
############################################################################
##
#M WdNearRing( <G>, <psi>, <phi>, <reps> )
##
## returns the weakly divisible near-ring determined by the group <G>
## with a group of fixed-point-free automorphisms <phi>, a set of
## orbit representatives <reps> and an endomorphism <psi>
##
InstallMethod(
WdNearRing,
"default",
true,
[IsGroup, IsMapping, IsGroup, IsList],
0,
function( G, psi, phi, reps )
local ker_psi, # kernel of <psi>
q, # size of the kernel of <psi>
r, # smallest number such that psi^r = 0
autos, # elements of <phi>
t, # number of right identities, <reps>
psi_autos,
autos_psi,
elements, # list of (sorted) group elements
psi_i, # psi^i
# reps_psi,
multendos, # list of endormophisms determining the
# multiplication, in 1 to 1 correspondence
# with elements
mul, # multiplication function of the near-ring <nr>
nr, # near-ring
x, y, e, a, i, j;
ker_psi := Kernel( psi );
q := Size( ker_psi );
r := LogInt( Size(G), q );
t := Length( reps );
# if q = Size( G ) then
# Error( "ideal of nilpotent elements is trivial, planar near-ring \n" );
# elif
if q^r <> Size(G) or Image(psi^(r-1)) <> ker_psi or
q^(r-1)*(q-1) <> t*Size(phi) then
Error( "psi is not feasible \n" );
fi;
autos := AsList( phi );
psi_autos := List( autos, x -> psi*x );
autos_psi := Set( List( autos, x -> x*psi ) );
if not IsSubset( autos_psi, psi_autos ) then
# necessary condition for associativity is violated
# (mappings operate from the right)
Error( "<phi> is not feasible \n" );
fi;
elements := AsSSortedList( G );
multendos := [];
# determine multiplication by the invertible elements, G - Image(psi):
for e in reps do
for a in autos do
j := Position( elements, e^a );
if IsBound( multendos[j] ) then
# phi is not fixed-point-free on G - Image(psi)
Error( "<phi> is not fixed-point-free on G - Image(psi) or <reps> are not feasible \n" );
fi;
multendos[j] := a;
od;
od;
# determine the multiplication by elements in Image(psi) - Image(psi^2),
# if they exist:
if r > 1 then
for e in reps do
j := Position( elements, e^psi );
if not IsBound( multendos[j] ) then
for a in autos_psi do
j := Position( elements, e^a );
multendos[j] := a;
od;
elif multendos[j] <> psi then
Error( "<reps> are not feasible, Image(psi) - Image(psi^2 \n" );
fi;
od;
fi;
psi_i := psi;
# reps_psi := Set( List( reps, x -> x^psi ) );
for i in [2..r-1] do
# determine the multiplication by the elements Image(psi^i) - Image(psi^(i+1)):
psi_i := psi_i*psi;
autos_psi := Set( List( autos_psi, x -> x*psi ) );
# reps_psi := Set( List( reps_psi, x -> x^psi ) );
for e in reps do
j := Position( elements, e^psi_i );
if not IsBound( multendos[j] ) then
for a in autos_psi do
j := Position( elements, e^a );
multendos[j] := a;
od;
elif multendos[j] <> psi_i then
Print( "elements = ", elements, "\n" );
Print( "multendos =", multendos, "\n" );
Print( " i = ", i, ", e = ", e, " j = ", j, "\n" );
Error( "<reps> are not feasible \n" );
fi;
od;
od;
# multiplication by 0:
j := Position( elements, Identity(G) );
if IsBound(multendos[j]) then
Print( multendos[j], "\n" );
Error( "<reps> are not feasible \n" );
fi;
multendos[j] := GroupHomomorphismByFunction( G, G, x -> Identity(G) );
if ForAny( [1..Size(G)], j -> not IsBound( multendos[j] ) ) then
Error( "mul not defined \n" );
fi;
mul := function( a, x )
return Image( multendos[Position( elements, a )], x );
end;
nr := ExplicitMultiplicationNearRingNC( G, mul );
SetIsWdNearRing( nr, true );
return nr;
end );
###########################################################################
##
#O IsWdNearRing( <nr> )
##
## wd for right nearrings means: for all <a>,<b> in <nr> there is an <x>
## such that <x>*<a> = <b> or <x>*<b> = <a>
InstallMethod(
IsWdNearRing,
"default",
true,
[IsNearRing],
0,
function( nr )
local elms, n, a, B, b, wd,
i, x;
elms := AsList( nr );
n := Size( nr );
# <B[i]> are the elements of <nr> which are obtained by a multiplication
# <elms>[i]*<elms>
B := List( elms, a -> Set( a*elms ) );
wd := true; i := 1;
while wd and i <= n do
b := Filtered( [1..n], x -> x >= i and not elms[x] in B[i] );
# for any element <elms[x]> not in <B[i]> there has to be <elms[i]> in
# <B[x]>
wd := ForAll( b, x -> elms[i] in B[x] );
i := i+1;
od;
SetIsWdNearRing( nr, wd );
return wd;
end );
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2026-03-28
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