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#############################################################################
##
#W primitive.gi NilMat Alla Detinko
#W Bettina Eick
#W Dane Flannery
##
##
## This file contains a method to determine all nilpotent primitive matrix
## groups in GL(n,q) up to conjugacy.
##
#############################################################################
##
#F SingerCycle( n, po, l ) . . . . . . . . . . . . a Singer cycle in GL(n,po)
##
BindGlobal( "SingerCycle", function(n,po,l)
local F, F1, a, B;
F := GF(po^l);
F1 := GF(F, n);
a := PrimitiveRoot(F1);
B := Basis(F1);
return List(BasisVectors(B), x -> Coefficients(B, x*a));
end );
#############################################################################
##
#F OrdersPAG( n, po, l ) . . . . . . . . . . orders of primitive abelian grps
##
BindGlobal( "OrdersPAG", function(n, po, l)
local q, qn, l1, l2, K1, K2, y, d;
q := po^l;
qn := q^n - 1;
l1 := Filtered([1..n], IsPrime);
l1 := Filtered(l1, x -> IsInt(n/x));
l2 := List(l1, r -> r*(q^(n/r) - 1));
K1 := DivisorsInt(qn);
K2 := [];
for y in K1 do
if ForAll(l2, x -> not IsInt(x/y))
then Add(K2,y);
fi;
od;
return K2;
end );
#############################################################################
##
#F PrimitiveAbelianGens( n, po, l ) . . . . . gens for primitive abelian grps
##
BindGlobal( "PrimitiveAbelianGens", function (n, po, l)
local qn, K2, a, Anq;
qn := po^(l*n) - 1;
K2 := OrdersPAG(n, po, l);
a := SingerCycle(n, po, l);
return List(K2, x -> a^(qn/x));
end );
#############################################################################
##
#F OddPrimitiveAbelianGens( n, po, l ) . . . . gens for odd prim. abel. grps
##
BindGlobal( "OddPrimitiveAbelianGens", function (n, po, l)
local qn, a, K2, K3;
qn := po^(l*n) - 1;
K2 := OrdersPAG(n, po, l);
K3 := Filtered(K2, x -> x mod 2 = 1);
a := SingerCycle(n, po, l);
return List(K3, x -> a^(qn/x));
end );
#############################################################################
##
#F SpecialMatrix( po, t )
##
## Construct a matrix g = [[x,y],[y,-x]] with x^2+y^2 = -1.
##
BindGlobal( "SpecialMatrix", function(po,t)
local z,i,o,j,x,y,g,n;
z := Z(po,t);
o := z^0;
n := po^t-1;
i := 1;
x := z;
y := z;
while x^2 + y^2 <> -o and i <= n do
i := i+1;
x := z^i;
j := 1;
y := z;
while x^2 + y^2 <> -o and j<= n do
j := j+1;
y := z^j;
od;
od;
g := [[x,y],[y,-x]];
return g;
end );
############################################################################
##
#F NilpotentPrimitiveMatGroups( n, po, l )
##
## Returns a complete and irredundant list of conjugacy class representatives
## of the nilpotent primitive subgroups of GL(n,qo^l).
##
BindGlobal( "NilPrimMatGroups", function(n, po, l)
local q, m, Pnq, Cnq, t, e, d, s, a, b, i;
# set up
q := po^l;
m := n/2;
# start with abelian primitive groups
Pnq := List( PrimitiveAbelianGens(n, po,l), x -> GroupByGenerators([x]));
if po = 2 or n mod 2 = 1 or IsInt(n/4) or q mod 4 = 1 then return Pnq; fi;
# consider abelian primitive groups of odd order in deg n/2
Cnq := OddPrimitiveAbelianGens(m, po, l);
if Cnq = [] then return Pnq; fi;
Cnq := List(Cnq, x -> KroneckerProduct(x, IdentityMat(2, GF(q))));
# set up for additions
t := PLength((q^n - 1), 2);
e := IdentityMat(m, GF(q));
d := KroneckerProduct(e, DiagonalMat([Z(q)^0,-1*Z(q)^0]));
s := KroneckerProduct(e, SpecialMatrix(po, l));
a := KroneckerProduct(e, AbelianSylow(po, l*m));
# extend list
b := a^(2^(t-2));
Append( Pnq, List(Cnq, x -> GroupByGenerators([x, a, d])));
Append( Pnq, List(Cnq, x -> GroupByGenerators([x, b, s])));
# this is all in certain cases
if po mod 8 = 3 then return Pnq; fi;
# extend list further
for i in [1..(t-3)] do
b := a^(2^i);
Append( Pnq, List(Cnq, x -> GroupByGenerators([x, b, d])));
Append( Pnq, List(Cnq, x -> GroupByGenerators([x, b, s])));
od;
return Pnq;
end );
InstallGlobalFunction( SizesOfNilpotentPrimitiveMatGroups, function(n, po, l)
local q, Anq, Pnq, m, i, Cnq, t, N1nq, N2nq, L1, N3nq;
q := po^l;
m := n/2;
t := PLength((q^n - 1), 2);
# start with abelian primitive groups
Pnq := OrdersPAG(n, po, l);
if po = 2 or n mod 2 = 1 or IsInt(n/4) or q mod 4 = 1 then return Pnq; fi;
# consider abelian primitive groups of odd order in deg n/2
Cnq := Filtered(OrdersPAG(m, po, l), x -> x mod 2 = 1);
Append( Pnq, List( Cnq, x -> 2^(t+1)*x ));
Append( Pnq, List( Cnq, x -> 8*x));
# this is all in certain cases
if po mod 8 = 3 then return Pnq; fi;
# extend list further
for i in [1..(t-3)] do
Append( Pnq, List(Cnq, x -> 2^(t-i+1) * x) );
Append( Pnq, List(Cnq, x -> 2^(t-i+1) * x) );
od;
return Pnq;
end );
InstallGlobalFunction( NilpotentPrimitiveMatGroups, function(n,po,l)
local Pnq, i, Onq;
Pnq := NilPrimMatGroups(n,po,l);
Onq := SizesOfNilpotentPrimitiveMatGroups(n, po, l);
for i in [1..Length(Pnq)] do
SetIsNilpotentGroup( Pnq[i], true );
SetSize( Pnq[i], Onq[i] );
od;
return Pnq;
end );
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