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#############################################################################
##
#W sylow.gi NilMat Alla Detinko
#W Bettina Eick
#W Dane Flannery
##
##
## This file contains a method to compute a list of Sylow subgroups of a
## given nilpotent matrix group over GF(q).
##
#############################################################################
##
#F SylowSubgroupsOfNilpotentFFMatGroup(G) . . . . . determine Sylow subgroups
##
## This function takes a nilpotent matrix group G over GF(q) and returns
## its Sylow system.
##
## Note that this function does not check if the given group G is nilpotent
## and it may return wrong results if it is not.
##
## Note that the generic method to compute the Sylow system of nilpotent
## groups is not so bad. Hence the function below is only a small improvement
## that uses the results of the nilpotency test in a sensible form.
##
InstallGlobalFunction( SylowSubgroupsOfNilpotentFFMatGroup, function(G)
local F, p, n, l, J, S, U, P, B, C, W, bC, oC, bB, oB, sB, o, syl, q, t;
# check a trivial case
if ForAll(GeneratorsOfGroup(G), x -> x = One(G)) then return []; fi;
# catch arguments
F := FieldOfMatrixGroup(G);
p := Characteristic(F);
n := DimensionOfMatrixGroup(G);
l := ClassLimit(n, F);
# compute Jordan and Pi-primary splitting
J := JordanSplitting(G);
S := J[1];
U := J[2];
P := PiPrimarySplitting(S);
B := P[1];
C := P[2];
# now G = U x C x B
# U is a p-subgroup, C is abelian and
# Sylow subgroups of B are available unless B is abelian
# get generators and relevant primes for C
bC := GeneratorsOfGroup(C);
oC := Set(PrimeFactors(List(GeneratorsOfGroup(C), Order)));
# get generators and relevant primes for B
if HasSylowSystem(B) then
sB := SylowSystem(B);
oB := List(sB, PrimePGroup);
else
oB := Set(PrimeFactors(List(GeneratorsOfGroup(B), Order)));
bB := GeneratorsOfGroup(B);
fi;
# do a check
if p in oB or p in oC then
Error("something wrong with Sylow subgroups");
fi;
# get relevant primes for G
if ForAny( GeneratorsOfGroup(U), x -> x <> x^0 ) then
o := Union( [p], oC, oB );
else
o := Union( oC, oB );
fi;
# determine Sylow subgroups corresponding to o
syl := [];
for q in o do
# the p-Sylow subgroup is U (unless it is trivial)
if q = p then
SetIsPGroup(U, true);
SetPrimePGroup(U, q);
Add( syl, U );
else
# otherwise join p-Sylow subgroups of C and B
t := [];
if q in oB then
if HasSylowSystem(B) then
W := sB[Position(oB,q)];
else
W := SylowSubgroupOfNilpotentMatGroupFF( B, bB, q );
fi;
Append( t, GeneratorsOfGroup(W) );
fi;
if q in oC then
W := SylowSubgroupOfNilpotentMatGroupFF( C, bC, q );
Append( t, GeneratorsOfGroup(W) );
fi;
# create group and store info
W := GroupByGenerators(t);
SetIsPGroup(W, true);
SetPrimePGroup(W, q);
Add( syl, W );
fi;
od;
return syl;
end );
##
## need high value, otherwise GAP chooses the method for solvable groups
## using a special pc sequence to compute the Sylow system.
##
InstallMethod( SylowSystem, [IsFFEMatrixGroup and IsNilpotentGroup], NICE_FLAGS,
SylowSubgroupsOfNilpotentFFMatGroup );
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