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#############################################################################
##
#W finiteness.gi NilMat Alla Detinko
#W Bettina Eick
#W Dane Flannery
##
##
## This file contains methods for check whether a given f.g. nilpotent
## matrix group over Q is finite. Further, it contains methods to determine
## the order of a f.g. nilpotent matrix group over GF(q) or Q.
##
#############################################################################
##
#F IsFiniteNilpotentMatGroup( G ) . . . . . . . . . . . . decide finiteness
##
## This function takes a f.g. nilpotent matrix group G over Q and tests
## whether it is finite. The test uses results of the nilpotency test
## IsNilpotentMatGroup and hence it is particularly fast, if this test
## has been used for checking the nilpotence of G.
##
## Note that this function does not check whether G is nilpotent and it
## may return wrong results if it is not.
##
InstallGlobalFunction( IsFiniteNilpotentMatGroup, function(G)
local n, g, d, a, p, t, pcgs, kern, F;
# the trivial case
F := FieldOfMatrixGroup(G);
if IsFinite(F) then return true; fi;
if ForAll(GeneratorsOfGroup(G), x -> x = One(G)) then return true; fi;
# set up
g := GeneratorsOfGroup(G);
d := List(g, JordanDecomposition);
n := DimensionOfMatrixGroup(G);
a := NullMat(n,n,Rationals);
# check whether G contains unipotent part
if ForAny(d, x -> not x[2] = a) then return false; fi;
# compute kernel of congruence hom
p := DetermineAdmissiblePrime(g);
t := InducedByField(g, GF(p));
pcgs := CPCS_finite_word( t, n+2 );
kern := POL_NormalSubgroupGeneratorsOfK_p( pcgs, g );
# check whether kernel is trivial
kern := Filtered(kern, x -> not x = One(G));
return (Length(kern) = 0);
end );
InstallMethod( IsFinite, [IsCyclotomicMatrixGroup and IsNilpotentGroup],
function(G)
if IsRationalMatrixGroup(G) then return IsFiniteNilpotentMatGroup(G); fi;
TryNextMethod();
end );
###############################################################################
##
#F SizeByKnownClass(G). . . . . . . .determine the order by the nilpotency class
##
## This function takes a nilpotent matrix group over GF(q) generated by
## semisimple matrices and determines its order.
BindGlobal( "SizeByKnownClass", function(G)
local n, F, l, C, s, c, a;
n := DimensionOfMatrixGroup(G);
F := FieldOfMatrixGroup(G);
l := ClassLimit(n, F);
C := G;
s := 1;
while not IsAbelian(C) do
a := SecondCentralElement(G,C,l);
if IsBool(a) then return fail; fi;
C := OrbitStabGens(C, a);
s := s * C!.index;
od;
return s * Size(C);
end );
#############################################################################
##
#F SizeOfNilpotentMatGroupFF( G ) . . . . . . . . . . . . determine the order
##
## This function takes a nilpotent matrix group G over GF(q) and determines
## its order. The function uses results of the nilpotency test
## IsNilpotentMatGroupFF and hence it is particularly fast, if this test
## has been used for checking the nilpotence of G.
##
## Note that this function does not check whether G is nilpotent and it
## may return wrong results if it is not.
##
## The calls to 'Size' below may need some further refinement and improve-
## ments.
BindGlobal( "SizeOfNilpotentMatGroupFF", function(G)
local J, S, U, P, B, C, syl;
# catch a trivial case
if ForAll(GeneratorsOfGroup(G), x -> x = One(G)) then
return 1;
elif Length(Set(GeneratorsOfGroup(G))) = 1 then
return Order(GeneratorsOfGroup(G)[1]);
fi;
# get available info from nilpotency testing
J := JordanSplitting(G);
S := J[1];
U := J[2];
P := PiPrimarySplitting(S);
B := P[1];
C := P[2];
# now G = B x C x U
if IsAbelian(B) then
return Size(U) * Size(C) * Size(B);
else
# in this case a Sylow system for B is known from nilpotency testing
syl := SylowSubgroupsOfNilpotentFFMatGroup(B);
return Size(U) * Size(C) * Product(List(syl, SizeByKnownClass));
fi;
end );
#############################################################################
##
#F SizeOfNilpotentMatGroupRN( G ) . . . . . . . . . . . . determine the order
##
## This function takes a nilpotent matrix group G over Q and determines its
## order.
##
## Note that this function does not check whether G is nilpotent and it may
## return wrong results if it is not.
##
BindGlobal( "SizeOfNilpotentMatGroupRN", function(G)
local g, p, t, H;
g := GeneratorsOfGroup(G);
p := DetermineAdmissiblePrime(g);
t := InducedByField(g, GF(p));
H := MakeMatGroup(DimensionOfMatrixGroup(G), GF(p), t);
return SizeOfNilpotentMatGroupFF(H);
end );
#############################################################################
##
#F SizeOfNilpotentMatGroup( G ) . . . . . . . . . . . . . determine the order
##
InstallGlobalFunction( SizeOfNilpotentMatGroup, function(G)
if IsFFEMatrixGroup(G) then
return SizeOfNilpotentMatGroupFF(G);
fi;
if IsCyclotomicMatrixGroup(G) and IsRationalMatrixGroup(G) then
return SizeOfNilpotentMatGroupRN(G);
fi;
return fail;
end );
##
## we cannot install a similar method to finite field groups, as the
## algorithm for those groups uses a call to 'Size' for some smaller
## subgroups.
##
InstallMethod( Size, true, [IsCyclotomicMatrixGroup and IsNilpotentGroup], 0,
function(G)
if IsRationalMatrixGroup(G) then
return SizeOfNilpotentMatGroupRN(G);
fi;
TryNextMethod();
end);
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