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#############################################################################
##
## injector.gi CRISP Burkhard Höfling
##
## Copyright © 2000, 2005 Burkhard Höfling
##
#############################################################################
##
#M InjectorOp(<grp>, <class>)
##
InstallMethod(InjectorOp, "for pcgs computable groups: use radical", true,
[IsGroup and CanEasilyComputePcgs and IsFinite, IsFittingClass], 0,
function(G, C)
return InjectorFromRadicalFunction(G, U -> Radical(U, C), true);
end);
#############################################################################
##
#M InjectorOp(<grp>, <class>
##
InstallMethod(InjectorOp, "injector function is known", true,
[IsGroup and IsFinite and IsSolvableGroup,
IsFittingClass and HasInjectorFunction],
SUM_FLAGS, # prefer injector function if known
function(G, C)
return InjectorFunction(C)(G);
end);
#############################################################################
##
#M InjectorOp(<grp>, <class>)
##
InstallMethod(InjectorOp, "for FittingSetRep w/o injector function",
function(G, C) return IsIdenticalObj(CollectionsFamily(G), C); end,
[IsGroup and IsFinite and IsSolvableGroup, IsFittingSetRep], 0,
function(G, C)
if not IsFittingSet(G, C) then
Error("<C> must be a Fitting set for <G>");
fi;
return InjectorFromRadicalFunction(G, U -> Radical(U, C), false);
end);
#############################################################################
##
#M InjectorOp(<grp>, <class>)
##
InstallMethod(InjectorOp, "for FittingSetRep if injector function is known",
function(G, C)
return IsIdenticalObj(CollectionsFamily(G), C);
end,
[IsGroup and IsFinite and IsSolvableGroup,
IsFittingSetRep and HasInjectorFunction],
SUM_FLAGS, # prefer injector function if known
function(G, C)
if not IsFittingSet(G, C) then
Error("<C> must be a Fitting set for <G>");
fi;
return InjectorFunction(C)(G);
end);
#############################################################################
##
#M InjectorOp(<grp>, <class>)
##
InstallMethodByNiceMonomorphismForGroupAndClass(InjectorOp,
IsFinite and IsSolvableGroup, IsFittingClass);
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##
#M InjectorOp(<grp>, <class>)
##
InstallMethodByIsomorphismPcGroupForGroupAndClass(InjectorOp,
IsFinite and IsSolvableGroup, IsFittingClass);
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##
#M InjectorOp(<grp>, <class>)
##
CRISP_RedispatchOnCondition(InjectorOp,
"redispatch if group is finite or soluble",
true,
[IsGroup, IsGroupClass],
[IsFinite and IsSolvableGroup],
RankFilter(IsGroup) + RankFilter(IsGroupClass));
#############################################################################
##
#F InjectorFromRadicalFunction(<G>, <radfunc>, <hom>)
##
InstallGlobalFunction(InjectorFromRadicalFunction,
function(G, radfunc, hom)
local natQ, I, J, nat, H, N, C, W, i, gens, nilpser;
I := radfunc(G);
if hom then
natQ := NaturalHomomorphismByNormalSubgroup(G, I);
H := Image(natQ);
else
H := G;
fi;
# compute a normal series from I to G with nilpotent factors
nilpser := [];
while not IsTrivial(H) do
Add(nilpser, H);
H := Residual(H, NilpotentGroups);
od;
if hom then
nilpser := Reversed(nilpser);
else
nilpser := List(Reversed(nilpser), H ->
ClosureGroup(I, H));
fi;
# treat the nilpotent factors
for i in [2..Length(nilpser)] do
Info(InfoInjector, 1, "starting step ",i-1);
H := nilpser[i];
# I is an F-injector of H
Info(InfoInjector, 2, "computing normalizer");
if i > 2 then
if hom then
J := ImagesSet(natQ, I);
nat := NaturalHomomorphismByNormalSubgroup(
NormalizerOfPronormalSubgroup(H, J), J);
N := ImagesSource(nat);
else
N := NormalizerOfPronormalSubgroup(H, I);
fi;
else # otherwise I is trivial
N := H;
fi;
Info(InfoInjector, 3, " normalizer has order ",
Size(N));
Info(InfoInjector, 2, "computing Carter subgroup");
C := NilpotentProjector(N);
Info(InfoInjector, 3, " carter subgroup has order ",
Size(C));
if hom then
if i > 2 then
W := PreImagesSet(nat, C);
else
W := C;
fi;
W := PreImagesSet(natQ, W);
else
W := ClosureGroup(I, C);
fi;
Info(InfoInjector, 3, " preimage of carter subgroup has order ",
Size(W));
Info(InfoInjector, 2, " computing radical");
# the radical has to be computed in the full group
I := radfunc(W);
Info(InfoInjector, 3, " injector has order ", Size(I));
od;
return I;
end);
############################################################################
##
#E
##
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