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#############################################################################
##
## residual.gi CRISP Burkhard Höfling
##
## Copyright © 2000-2002, 2005, 2006, 2008 by Burkhard Höfling
##
#############################################################################
##
#M NormalSubgroups(<grp>)
##
InstallMethod(NormalSubgroups,
"normal complement method for pcgs computable groups",
true,
[IsGroup and CanEasilyComputePcgs],
1,
function(G)
return AllInvariantSubgroupsWithQProperty(G, G,
ReturnTrue, ReturnTrue, rec());
end);
#############################################################################
##
#M NormalSubgroups(<grp>)
##
InstallMethod(NormalSubgroups,
"normal complement method for pc groups",
true,
[IsPcGroup],
1,
function(G)
return AllInvariantSubgroupsWithQProperty(G, G,
ReturnTrue, ReturnTrue, rec());
end);
#############################################################################
##
#M NormalSubgroups(<grp>)
##
InstallMethod(NormalSubgroups,
"normal complement method for soluble perm groups",
true,
[IsPermGroup and IsSolvableGroup],
0,
function(G)
return AllInvariantSubgroupsWithQProperty(G, G,
ReturnTrue, ReturnTrue, rec());
end);
#############################################################################
##
#M NormalSubgroups(<grp>)
##
InstallMethod(NormalSubgroups,
"via IsomorphismPcGroup",
true,
[IsGroup and IsFinite and IsSolvableGroup],
0,
function(G)
local iso, im;
if CanEasilyComputePcgs(G) then
TryNextMethod();
fi;
iso := IsomorphismPcGroup(G);
im := ImagesSource(iso);
return List(NormalSubgroups(im), N -> PreImagesSet(iso, N));
end);
#############################################################################
##
#M NormalSubgroups(<grp>)
##
InstallMethod(NormalSubgroups,
"via nice monomorphism",
true,
[IsGroup and IsFinite and IsHandledByNiceMonomorphism],
0,
function(G)
local iso, im;
iso := NiceMonomorphism(G);
im := NiceObject(G);
return List(NormalSubgroups(im), N -> PreImagesSet(iso, N));
end);
#############################################################################
##
#M NormalSubgroups(<grp>)
##
CRISP_RedispatchOnCondition(NormalSubgroups,
"redispatch if group is finite or soluble",
true,
[IsGroup],
[IsFinite and IsSolvableGroup],
# rank this method fairly high - presumably all fast methods for computing
# the normal subgroups need to know if the group is finite and soluble
RankFilter(IsPcGroup and IsPermGroup and IsSolvableGroup));
#############################################################################
##
#M CharacteristicSubgroups(<grp>)
##
InstallMethod(CharacteristicSubgroups,
"normal complement method for pcgs computable groups",
true,
[IsGroup and CanEasilyComputePcgs],
0,
function(G)
return AllInvariantSubgroupsWithQProperty(
AutomorphismGroup(G), G, ReturnTrue, ReturnTrue, rec());
end);
#############################################################################
##
#M CharacteristicSubgroups(<grp>)
##
InstallMethod(CharacteristicSubgroups,
"via IsomorphismPcGroup",
true,
[IsGroup and IsFinite and IsSolvableGroup],
0,
function(G)
local iso, im;
if CanEasilyComputePcgs(G) then
TryNextMethod();
fi;
iso := IsomorphismPcGroup(G);
im := ImagesSource(iso);
return List(CharacteristicSubgroups(im),
N -> PreImagesSet(iso, N));
end);
#############################################################################
##
#M CharacteristicSubgroups(<grp>)
##
InstallMethod(CharacteristicSubgroups,
"via NiceMonomorphism",
true,
[IsGroup and IsFinite and IsHandledByNiceMonomorphism],
0,
function(G)
local iso, im;
iso := NiceMonomorphism(G);
im := NiceObject(G);
return List(CharacteristicSubgroups(im),
N -> PreImagesSet(iso, N));
end);
#############################################################################
##
#M CharacteristicSubgroups(<grp>)
##
CRISP_RedispatchOnCondition(CharacteristicSubgroups,
"redispatch if group is finite or soluble",
true,
[IsGroup],
[IsFinite and IsSolvableGroup],
RankFilter(IsGroup and IsFinite and IsSolvableGroup)-1);
#############################################################################
##
#M AllInvariantSubgroupsWithQProperty(<act>, <grp>, <pretest>, <test>, <data>)
##
## note that act must induce the full inner automorphism group
##
InstallMethod(AllInvariantSubgroupsWithQProperty,
"for soluble group",
true,
[IsListOrCollection, IsGroup and IsSolvableGroup and IsFinite, IsFunction, IsFunction, IsObject],
RankFilter(IsPcGroup),
function(act, G, pretest, test, data)
local i, norms, ser, new, N, cmpl, bool, testser;
norms := [G];
if IsTrivial(G) then
return norms;
fi;
testser := true;
ser := CompositionSeriesUnderAction(act, G);
for i in [3..Length(ser)] do
new := [];
for N in norms do
bool := pretest(ser[i-1], ser[i], N, data);
if bool <> false then
cmpl := InvariantComplementsOfElAbSection(
act, N, ser[i-1], ser[i], true);
if bool = true then
Append(new, cmpl);
else
Append(new, Filtered(cmpl, C -> test(C, N, data)));
fi;
fi;
od;
Append(norms, new);
if testser then # ser[i-2] has passed the test
bool := pretest(ser[i-2], ser[i-1], ser[i-2], data);
if bool = fail then
bool := test(ser[i-1], ser[i-2], data);
fi;
if bool then
Add(norms, ser[i-1]);
else
testser := false;
fi;
fi;
Info(InfoLattice, 1, "Step ",i,", ",Length(norms)," invariant subgroups");
od;
if testser then # ser[i-2] has passed the test
bool := pretest(ser[Length(ser)-1],
ser[Length(ser)], ser[Length(ser)-1], data);
if bool = fail then
bool := test(ser[Length(ser)], ser[Length(ser)-1], data);
fi;
if bool then
Add(norms, ser[Length(ser)]);
fi;
fi;
return norms;
end);
#############################################################################
##
#M AllInvariantSubgroupsWithQProperty(<act>, <grp>, <pretest>, <test>, <data>)
##
CRISP_RedispatchOnCondition(AllInvariantSubgroupsWithQProperty,
"redispatch if group is finite or soluble",
true,
[IsListOrCollection, IsGroup, IsFunction, IsFunction, IsObject],
[, IsFinite and IsSolvableGroup], # no conditions on other arguments
0);
#############################################################################
##
#M OneInvariantSubgroupMinWrtQProperty(<act>, <grp>, <pretest>, <test>, <data>)
##
InstallMethod(OneInvariantSubgroupMinWrtQProperty,
"for soluble group",
true,
[IsListOrCollection, IsGroup and IsSolvableGroup and IsFinite,
IsFunction, IsFunction, IsObject],
0,
function(act, G, pretest, test, data)
local i, k, M, ser, gens, new, N, cmpl, bool, R;
if IsTrivial(G) then
return G;
fi;
ser := CompositionSeriesUnderAction(act, G);
for k in [2..Length(ser)] do
bool := pretest(ser[k-1], ser[k], ser[k-1], data);
if bool = fail then
bool := test(ser[k], ser[k-1], data);
fi;
if not bool then
break;
fi;
od;
if bool then
return TrivialSubgroup(G);
fi;
M := ser[k-1];
gens := GeneratorsOfGroup(G);
for i in [k..Length(ser)-1] do
bool := pretest(ser[i], ser[i+1], M, data);
if bool <> false then
cmpl := InvariantComplementsOfElAbSection(
act, M, ser[i], ser[i+1], bool <> true);
if bool = true then
if cmpl <> fail then
M := cmpl;
fi;
else
for R in cmpl do
if test(R, M, data) then
M := R;
break;
fi;
od;
fi;
fi;
Info(InfoLattice, 1, "Step ",i,"done");
od;
return M;
end);
############################################################################
##
#M OneInvariantSubgroupMinWrtQProperty(<act>, <grp>, <pretest>, <test>, <data>)
##
CRISP_RedispatchOnCondition(OneInvariantSubgroupMinWrtQProperty,
"redispatch if group is finite or soluble",
true,
[IsListOrCollection, IsGroup, IsFunction, IsFunction, IsObject],
[,IsFinite and IsSolvableGroup], # no conditions on other arguments
0);
#############################################################################
##
#M AllNormalSubgroupsWithQProperty
##
InstallMethod(AllNormalSubgroupsWithQProperty,
"try AllInvariantSubgroupsWithQProperty", true,
[IsGroup, IsFunction, IsFunction, IsObject], 0,
function(G, pretest, test, data)
return AllInvariantSubgroupsWithQProperty(G, G, pretest, test, data);
end);
#############################################################################
##
#M OneNormalSubgroupMinWrtQProperty
##
InstallMethod(OneNormalSubgroupMinWrtQProperty,
"try OneInvariantSubgroupMinWrtQProperty", true,
[IsGroup, IsFunction, IsFunction, IsObject], 0,
function(G, pretest, test, data)
return OneInvariantSubgroupMinWrtQProperty(G, G, pretest, test, data);
end);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
InstallMethod(ResidualOp, "for group and formation with ResidualFunction", true,
[IsGroup and IsFinite and IsSolvableGroup,
IsOrdinaryFormation and HasResidualFunction],
SUM_FLAGS, # highly preferable
function(G, C)
return ResidualFunction(C)(G);
end);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
InstallMethod(ResidualOp, "for group and formation product", true,
[IsGroup and IsFinite and IsSolvableGroup, IsFormationProductRep], 0,
function(G, C)
return Residual(Residual(G, C!.top), C!.bot);
end);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
InstallMethod(ResidualOp, "for group and intersection of formations", true,
[IsGroup and IsFinite and CanEasilyComputePcgs, IsOrdinaryFormation and IsClassByIntersectionRep],
function(G, C)
local D, R, l;
R := TrivialSubgroup(G);
l := [];
# first form the join of all residuals that can be computed cheaply
for D in C!.intersected do
if HasResidualFunction(D) then
R := ClosureGroup(R, ResidualFunction(D)(G));
elif HasLocalDefinitionFunction(D) then
R := ClosureGroup(R, Residual(G, D));
else
Add(l, D);
fi;
od;
if Length(l) > 0 then # treat the remaining classes
# Note that, although the intersection of the classes in l
# need not be a formation, any subgroup returned by
# OneInvariantSubgroupMinWrtQProperty lies below the C-residual of G.
R := ClosureGroup(R, OneInvariantSubgroupMinWrtQProperty(G, G,
ReturnFail,
function(R, S, data)
local fac;
fac := data.grp/R;
return ForAll(data.list, D -> fac in D);
end,
rec(grp := G, list := l)));
fi;
return R;
end);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
InstallMethod(ResidualOp, "generic method for pcgs computable group", true,
[IsGroup and IsFinite and CanEasilyComputePcgs, IsOrdinaryFormation], 0,
function(G, C)
return OneInvariantSubgroupMinWrtQProperty(G, G,
ReturnFail,
function(R, S, data)
return data.grp/R in data.class;
end,
rec(grp := G, class := C));
end);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
InstallMethod(ResidualOp, "for locally defined formation", true,
[IsGroup and IsFinite and CanEasilyComputePcgs,
IsSaturatedFormation and HasLocalDefinitionFunction],
0,
function(G, C)
return OneInvariantSubgroupMinWrtQProperty(G, G,
function(U, V, R, data)
local mpcgs, p, pos;
mpcgs := ModuloPcgs(U, V);
p := RelativeOrders(mpcgs)[1]; # exponent of U/V
if p in Characteristic(data.class) then
pos := PositionSorted(data.primes, p);
if pos > Length(data.primes) or data.primes[pos] <> p then
# add the information from LocalDefinitionFunction
Add(data.primes, p, pos);
Add(data.locgens, LocalDefinitionFunction(data.class)(data.grp, p), pos);
fi;
# test if generators of local residual acts centrally
return CentralizesLayer(data.locgens[pos], mpcgs);
else
return false;
fi;
end,
ReturnFail, # will produce an error message if accidentally called
rec(grp := G,
class := C,
primes := [], # primes for which the local function has been called
locgens := [] # corresponding results of calls to local function
));
end);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
InstallMethodByNiceMonomorphismForGroupAndClass(ResidualOp,
IsFinite and IsSolvableGroup, IsOrdinaryFormation);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
InstallMethodByIsomorphismPcGroupForGroupAndClass(ResidualOp,
IsFinite and IsSolvableGroup, IsOrdinaryFormation);
#############################################################################
##
#M ResidualOp(<grp>, <class>)
##
CRISP_RedispatchOnCondition(ResidualOp,
"redispatch if group is finite or soluble",
true,
[IsGroup, IsGroupClass],
[IsFinite and IsSolvableGroup],
RankFilter(IsGroup) + RankFilter(IsGroupClass));
############################################################################
##
#E
##
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(vorverarbeitet)
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