#############################################################################
##
#W maxgroup.gi NilMat Alla Detinko
#W Bettina Eick
#W Dane Flannery
##
##
## This file contains methods to construct a nilpotent maximal absolutely
## irreducible subgroup of GL(n,q).
##
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##
#F MonomialSylow(p,a[,po,t]) . . . . . . . . . . . .construct monomial group
##
## This function constructs a list of generators for a monomial p-subgroup
## of GL(p^a, F) isomorphic to a Sylow p-subgroup of S_(p^a). If 2 arguments
## are given, then F = Q is used, otherwise F = GF(po^t).
##
BindGlobal( "MonomialSylow", function(arg)
local s,g,p,a;
p := arg[1];
a := arg[2];
s := SylowSubgroup(SymmetricGroup(p^a),p);
g := GeneratorsOfGroup(s);
if Length(arg) = 2 then
return List(g, x -> PermutationMat(x,p^a,Rationals));
else
return List(g, x -> PermutationMat(x,p^a,GF(arg[3],arg[4])));
fi;
end );
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##
#F CyclicSylow(m,p) . . . . . . . . . . . . . . construct cyclic Sylow group
##
## For a given element m of a field and a prime p construct a generator of
## the Sylow p-subgroup of <m>; if p does not divide |<m>| returns 1.
##
BindGlobal( "CyclicSylow", function(m,p)
local o, t, o1;
o := Order(m);
t := PLength(o,p); #equals 0 if p does not divide o
o1 := o/(p^t);
return m^o1;
end );
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##
#F AbelianSylow(p,a) . . . . . . . . . . . . . . .
#F TwoSylow(p,a) . . . . . . . . . . . . . . .
##
## Constructs generators for a Sylow 2-subgroup G of GL(2,q) for 4|q-3.
## This group has the form G = <A,g> and A is an abelian subgroup of G of
## index 2.
##
## First construct A, which is isomorphic to the Sylow 2-subgroup of F(i)
## via a regular representation, i.e. an element i of order 4 should be
## represented by the matrix [[0,1],[-1,0]].
##
BindGlobal( "AbelianSylow", function(po,t)
local a,c,r,i,F,F1,B,B1,B2,vec;
F := GF(po,t);
F1 := GF(F,2);
a := PrimitiveRoot(F1);
c := CyclicSylow(a,2);
r := Order(c);
i := c^(r/4); #element of order 4 in GF(F,2);exists as 4|q^2-1
B := Basis(F1); #canonical basis of F1/F
B1 := [a^0,i];
B2 := RelativeBasis(B,B1);#change of canonical basis to B1
vec := BasisVectors(B2);
return List(vec, x -> Coefficients(B2,x*c));# this is A=<C> in GL(2,q)
end );
BindGlobal( "TwoSylow", function(po,t)
local A,g;
A := AbelianSylow(po,t);
g := DiagonalMat([Z(po,t)^0,(-1)*Z(po,t)^0]);
return [A,g];
end );
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##
#F KroneckerProductLists( L, M )
##
## This function constructs the list of pairwise Kronecker products of the input
## lists L and M.
##
BindGlobal( "KroneckerProductLists", function(L,M)
local i,LM;
LM := [];
for i in [1..Length(M)] do
Append( LM, List(L, x -> KroneckerProduct(x, M[i])));
od;
return LM;
end );
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##
#F SylowSubgroupGL(p,a,po,t)
##
## Construct generators of the Sylow p-subgroup of GL(p^a,q), p|q-1. Such
## subgroups are absolutely irreducible.
##
BindGlobal( "SylowSubgroupGL", function(p,a,po,t)
local q,c,C,w,S,h,H,K,l,L,n;
# set up
q := po^t;
if not IsInt((q-1)/p) then return fail; fi;
w := Z(q);
n := p^a;
#First construct the Sylow subgroup in monomial case
if p>2 or q mod 4 = 1 then
c := CyclicSylow(w,p);
C := IdentityMat(n,w);
C[n][n] := c;
S := MonomialSylow(p,a,po,t);
Add(S,C);
return S;
fi;
# now we have p=2 and q mod 4 = 3, i.e. if we have the non-monomial case
h := TwoSylow(po,t); #primitive 2-Sylow subgroup of GL(2,q);2 generators
if a=1 then return h;fi;
H := IdentityMat(n,w);
H[1][1] := h[1][1][1];
H[1][2] := h[1][1][2];
H[2][1] := h[1][2][1];
H[2][2] := h[1][2][2];
K := IdentityMat(n,w);
K[1][1] := h[2][1][1];
K[1][2] := h[2][1][2];
K[2][1] := h[2][2][1];
K[2][2] := h[2][2][2];
l := MonomialSylow(p,a-1,po,t);
L := List(l, x -> KroneckerProduct(x,IdentityMat(2,w)));
Add(L,H);
Add(L,K);
return L;
end );
#########################################################################
##
#F MaximalAbsolutelyIrreducibleNilpotentMatGroup . . .
##
## This function constructs a nilpotent maximal absolutely irreducible
## subgroup G of GL(n, q) or returns 'fail' if such subgroups do not exist.
## The group GL(n,q) contains absolutely irreducible nilpotent subgroups
## if p|q-1 for each prime divisor p of n.
##
## Let n = p1^a1...pk^ak be the prime facorization of n. Then G
## is a Kronecker product of groups G1,...,Gk, where Gi = Hi \cdot F*,
## Hi is a Sylow pi-subgroup of GL(pi^ai,F). Sylow p-subgroups of GL(p^a,F)
## are absolutely irreducible monomial (recall p|q-1), besides the case
## p=2, 4|q-3 when Sylow 2-subgroups of GL(2,q) are primitive and
## a Sylow 2-subgroups of GL(2^a,q) is imprimitive with 2-dimension
## systems of imprimitivity (a>1).
##
InstallGlobalFunction( MaximalAbsolutelyIrreducibleNilpotentMatGroup,
function(n,po,t)
local q,w,l1,l2,k,r,H,W,i;
q := po^t;
w := Z(q);
#First construct prime factorization of n.
l1 := Filtered([2..n],x -> IsPrimeInt(x));
l2 := Filtered(l1, x -> IsInt(n/x));
r := List(l2, x -> PLength(n,x));
k := Length(l2);
# an easy case
if not ForAll(l2,x -> IsInt((q-1)/x)) then return fail; fi;
# construct a list of lists of generators
H := List([1..k], x -> SylowSubgroupGL(l2[x], r[x],po,t));
# build up kronecker-products
W := H[1];
for i in [2..k] do
W := KroneckerProductLists(W,H[i]);
od;
Add( W, IdentityMat(n,w)*w );
return Group(W);
end );
