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#############################################################################
##
## radical.gi CRISP Burkhard Höfling
##
## Copyright © 2000, 2002, 2005 Burkhard Höfling
##
#############################################################################
##
#F InvariantSubgroupsCA(act, ser, avoid, cover, pretest, test, data)
##
## computes the G-invariant normal subgroups N of ser[1] such that
## ser[cover] equals the intersection of N and ser[avoid], N contains
## ser[cover] properly, and N belongs to the class described by the functions
## pretest and test. pretest and test are the functions described in the
## manual(see "OneInvariantSubgroupMaxWrtNProperty").
##
InstallGlobalFunction(InvariantSubgroupsCA,
function(act, ser, avoid, cover, pretest, test, data)
local j, CC, L, newser, norms, bool, newnorms;
if avoid = 1 then
return [];
fi;
norms := InvariantSubgroupsCA(act, ser, avoid-1, cover,
pretest, test, data);
bool := pretest(ser[avoid-1], ser[avoid], ser[cover], data);
if bool <> false then
if avoid = cover then
CC := [ser[avoid-1]];
else
CC := ComplementsMaximalUnderAction(act, ser, avoid-1,
avoid, cover, true);
Info(InfoLattice, 1, Length(CC), " complements found");
fi;
if Length(CC) > 0 then
newser := ShallowCopy(ser);
for L in CC do
if bool = true or test(L, ser[cover], data) then
# L belongs to class
newser[cover-1] := L;
for j in [avoid+1..cover-1] do
# make ser a chief series through L
newser[j-1] := ClosureGroup(ser[j], L);
od;
Add(norms, L);
if avoid > 2 then
Append(norms,
InvariantSubgroupsCA(act,
newser, avoid-1, cover-1,
pretest, test, data));
fi;
fi;
od;
fi;
fi;
return norms;
end);
#############################################################################
##
#O AllInvariantSubgroupsWithNProperty
#O (<act>, <G>, <pretest>, <test>, <data>)
##
InstallMethod(AllInvariantSubgroupsWithNProperty,
"for soluble group", true,
[IsListOrCollection, IsGroup and IsSolvableGroup and IsFinite,
IsFunction, IsFunction, IsObject],
0,
function( act, G, pretest, test, data)
local ser, norms;
if IsTrivial(G) then
return [G];
fi;
ser := CompositionSeriesUnderAction(act, G);
norms := InvariantSubgroupsCA(act, ser,
Length(ser), Length(ser), pretest, test, data);
Add(norms, TrivialSubgroup(G));
return norms;
end);
#############################################################################
##
#M AllInvariantSubgroupsWithNProperty
##
CRISP_RedispatchOnCondition(AllInvariantSubgroupsWithNProperty,
"redispatch if group is finite or soluble",
true,
[IsListOrCollection, IsGroup, IsFunction, IsFunction, IsObject],
[, IsFinite and IsSolvableGroup], # no conditions on other arguments
0);
#############################################################################
##
#M OneInvariantSubgroupMaxWrtNProperty(
#M <act>, <G>, <pretest>, <test>, <data>)
##
InstallMethod(OneInvariantSubgroupMaxWrtNProperty,
"for soluble group", true,
[IsListOrCollection, IsGroup and IsSolvableGroup and IsFinite,
IsFunction, IsFunction, IsObject],
0,
function(act, G, pretest, test, data)
local n, ser, i, j, CC, R, rpos, bool;
if IsTrivial(G) then
return G;
fi;
ser := ShallowCopy(CompositionSeriesUnderAction(act, G));
n := Length(ser);
for rpos in [n-1, n-2..1] do
Info(InfoComplement, 1, "starting step ",n-rpos, "(testing ser)");
bool := pretest(ser[rpos], ser[rpos+1], ser[rpos+1], data);
if bool = fail then
bool := test(ser[rpos], ser[rpos+1], data);
fi;
if not bool then
break;
fi;
od;
if bool then # G has passed test
return G;
fi;
rpos := rpos + 1;
for i in [rpos-2,rpos-3..1] do
# ser[rpos] is the property-radical of ser[i+1]
Info(InfoComplement, 1, "starting step ",n-i+1);
bool := pretest(ser[i], ser[i+1], ser[rpos], data);
if bool <> false then
Info(InfoComplement, 3, "Complementing");
CC := ComplementsMaximalUnderAction(act, ser, i, i+1, rpos, bool <> true);
Info(InfoComplement, 3, Length(CC), "complements found, ",
"(bool = ",bool, ")");
if bool = true then
if CC = fail then
CC := [];
else
CC := [CC];
fi;
fi;
for R in CC do
if bool = true or test(R, ser[rpos], data) then # R is the property-radical of i
Info(InfoComplement, 3, "modifying series...\n");
for j in [i+2..rpos-1] do
ser[j-1] := ClosureGroup(ser[j], R); #make ser a chief series through L
od;
rpos := rpos - 1;
ser[rpos] := R;
break; # no need to check other groups
fi;
od;
fi;
od;
return ser[rpos];
end);
#############################################################################
##
#M OneInvariantSubgroupMaxWrtNProperty
##
CRISP_RedispatchOnCondition(OneInvariantSubgroupMaxWrtNProperty,
"redispatch if group is finite or soluble",
true,
[IsListOrCollection, IsGroup, IsFunction, IsFunction, IsObject],
[, IsFinite and IsSolvableGroup], # no conditions on other arguments
0);
#############################################################################
##
#M AllNormalSubgroupsWithNProperty
##
InstallMethod(AllNormalSubgroupsWithNProperty,
"via AllInvariantSubgroupsWithNProperty", true,
[IsGroup, IsFunction, IsFunction, IsObject], 0,
function(G, pretest, test, data)
return AllInvariantSubgroupsWithNProperty(G, G, pretest, test, data);
end);
#############################################################################
##
#M OneNormalSubgroupMaxWrtNProperty
##
InstallMethod(OneNormalSubgroupMaxWrtNProperty,
"via OneInvariantSubgroupMaxWrtNProperty", true,
[IsGroup, IsFunction, IsFunction, IsObject], 0,
function(G, pretest, test, data)
return OneInvariantSubgroupMaxWrtNProperty(G, G, pretest, test, data);
end);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "if only in is known", true,
[IsGroup and IsFinite and CanEasilyComputePcgs, IsFittingClass], 0,
function(G, C)
return OneInvariantSubgroupMaxWrtNProperty(G, G,
function(U, V, R, class)
if SmallestRootInt(Index(U, V)) in Characteristic(class) then
return fail; # cannot decide
else
return false; # never in C
fi;
end,
function(S, R, class)
return S in class;
end,
C);
end);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "if injector is known", true,
[IsGroup and IsFinite and IsSolvableGroup, IsFittingClass and HasInjectorFunction], 2,
function(G, C)
return Core(G, Injector(G, C));
end);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "if radical function is known", true,
[IsGroup and IsFinite and IsSolvableGroup, IsFittingClass and HasRadicalFunction],
SUM_FLAGS, # high preference
function(G, C)
return RadicalFunction(C)(G);
end);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "for Fitting product", true,
[IsGroup and IsFinite and IsSolvableGroup, IsFittingProductRep], 0,
function(G, C)
local nat;
nat := NaturalHomomorphismByNormalSubgroup(G, Radical(G, C!.bot));
return PreImagesSet(nat, Radical(ImagesSource(nat), C!.top));
end);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "for intersection of classes", true,
[IsGroup and IsFinite and IsSolvableGroup, IsFittingClass and IsClassByIntersectionRep],
function(G, C)
local D, R, l;
R := G;
l := [];
for D in C!.intersected do
if HasRadicalFunction(D) then
R := RadicalFunction(D)(R);
else
Add(l, D);
fi;
od;
if Length(l) > 0 then
# compute a normal subgroup of R which is maximal with respect
# to belonging to all classes in l. Since every normal subgroup
# of the C-Residual of G belongs to l, the C-residual of G
# contains the group returned by OneNormalSubgroupMaxWrtNProperty,
# even though the intersection of the classes in l need not itself
# be a Fitting class.
return OneInvariantSubgroupMaxWrtNProperty(G, R,
function(U, V, T, data)
local p;
p := Factors(Index(U, V))[1];
if ForAll(data, D -> p in Characteristic(D)) then
return fail; # cannot decide
else
return false; # never in
fi;
end,
function(S, T, data)
return ForAll(data, D -> S in D);
end,
l);
else
return R;
fi;
end);
#############################################################################
##
#M RadicalOp
##
InstallMethodByNiceMonomorphismForGroupAndClass(RadicalOp,
IsFinite and IsSolvableGroup, IsFittingClass);
#############################################################################
##
#M RadicalOp
##
InstallMethodByIsomorphismPcGroupForGroupAndClass(RadicalOp,
IsFinite and IsSolvableGroup, IsFittingClass);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "generic method for FittingSetRep",
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 OneInvariantSubgroupMaxWrtNProperty(G, G,
ReturnFail,
function(S, R, data)
return S in data;
end,
C);
end);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "for FittingSetRep with injector function",
function(G, C)
return IsIdenticalObj(CollectionsFamily(G), C);
end,
[IsGroup and IsFinite and IsSolvableGroup,
IsFittingSetRep and HasInjectorFunction],
0,
function(G, C)
if not IsFittingSet(G, C) then
Error("<C> must be a Fitting set for <G>");
fi;
return Core(G, Injector(G, C));
end);
#############################################################################
##
#M RadicalOp
##
InstallMethod(RadicalOp, "for FittingSetRep with radical function",
function(G, C)
return IsIdenticalObj(CollectionsFamily(G), C);
end,
[IsGroup and IsFinite and IsSolvableGroup,
IsFittingSetRep and HasRadicalFunction],
SUM_FLAGS, # highly preferable
function(G, C)
if not IsFittingSet(G, C) then
Error("<C> must be a Fitting set for <G>");
fi;
return RadicalFunction(C)(G);
end);
#############################################################################
##
#M RadicalOp
##
CRISP_RedispatchOnCondition(RadicalOp,
"redispatch if group is finite or soluble",
true,
[IsGroup, IsGroupClass], [IsFinite and IsSolvableGroup],
RankFilter(IsGroup) + RankFilter(IsGroupClass));
############################################################################
##
#E
##
[ Dauer der Verarbeitung: 0.34 Sekunden
(vorverarbeitet)
]
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