Quelle samples.gi Sprache: unbekannt

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##  samples.gi                       CRISP                   Burkhard Höfling
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##  Copyright © 2000, 2005 Burkhard Höfling
##

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#V  AllPrimes
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BindGlobal("AllPrimes", Class(x -> IsInt(x) and IsPrimeInt(x)));
SetName(AllPrimes, "<set of all primes>");

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#V  TrivialGroups
##
BindGlobal("TrivialGroups", SaturatedFittingFormation( rec(
    \in := IsTrivial,
    rad := TrivialSubgroup,
    res := G -> G,
    proj := TrivialSubgroup,
    locdef := ReturnFail,
    char := [],
    bound := ReturnTrue
    )));
SetIsSubgroupClosed(TrivialGroups, true);
SetName(TrivialGroups, "<class of all trivial groups>");

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#V  NilpotentGroups
##
BindGlobal("NilpotentGroups", SaturatedFittingFormation( rec(
    \in := IsNilpotentGroup,
    rad := FittingSubgroup,
    res := G -> NormalClosure(G,
        SubgroupNC(G, NormalGeneratorsOfNilpotentResidual(G))),
    proj := NilpotentProjector,
    locdef := function(G, p) return GeneratorsOfGroup(G); end,
    char := AllPrimes,
    bound := G -> G <> Socle(G)
    )));
SetIsSubgroupClosed(NilpotentGroups, true);
SetName(NilpotentGroups, "<class of all nilpotent groups>");

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#M  NilpotentProjector
##
InstallMethod(NilpotentProjector, "for finite soluble groups", true,
    [IsFinite and IsGroup and IsSolvableGroup], 0,
    function(G)
        return ProjectorFromExtendedBoundaryFunction(
            G,
            rec(
                dfunc := function(gpcgs, npcgs, p, data)
                    if Length(npcgs) > 1 then
                        return true;
                    else
                        return fail;
                    fi;
                end,
                cfunc := function(gpcgs, npcgs, p, cent, data)
                    return cent > 1;
                end),
            false);
    end);

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#M  NilpotentProjector(<grp>)
##
CRISP_RedispatchOnCondition(NilpotentProjector,
    "redispatch if group is finite or soluble",
    true,
    [IsGroup],
    [IsFinite and IsSolvableGroup], 0);

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#V  SupersolubleGroups
##
BindGlobal("SupersolubleGroups", SaturatedFormation( rec(
    res := SupersolvableResiduum,
    proj := SupersolubleProjector,
    locdef := function(G, p)
        local gens, res, i, j;
        gens := GeneratorsOfGroup(G);
        res := List(gens, x -> x^(p-1));
        for i in [1..Length(gens)] do
            for j in [i+1..Length(gens)] do
                Add(res, Comm(gens[i], gens[j]));
            od;
        od;
        return res;
    end,
    char := AllPrimes,
    bound := G -> not IsPrime(Size(Socle(G)))
    )));
SetIsSubgroupClosed(SupersolubleGroups, true);
SetName(SupersolubleGroups, "<class of all supersoluble groups>");

DeclareSynonym("SupersolvableGroups", SupersolubleGroups);

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#M  SupersolubleProjector(<grp>)
##
InstallMethod(SupersolubleProjector, "for finite soluble groups", true,
    [IsFinite and IsGroup and IsSolvableGroup], 0,
    function(G)
        return ProjectorFromExtendedBoundaryFunction(
            G,
            rec(
                dfunc := function(gpcgs, npcgs, p, data)
                    return Length(npcgs) > 1;
                end),
            false);
    end);

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#M  SupersolubleProjector(<grp>)
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CRISP_RedispatchOnCondition(SupersolubleProjector,
    "redispatch if group is finite or soluble",
    true,
    [IsGroup],
    [IsFinite and IsSolvableGroup],
    0);

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#V  AbelianGroups
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BindGlobal("AbelianGroups", OrdinaryFormation( rec(
    res := DerivedSubgroup,
    char := AllPrimes
    )));
SetIsSubgroupClosed(AbelianGroups, true);
SetName(AbelianGroups, "<class of all abelian groups>");

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#F  AbelianGroupsOfExponent(<exp>)
##
InstallGlobalFunction("AbelianGroupsOfExponent", function(exp)
    local form;
    if not IsPosInt(exp) then
        Error("<exp> must be a positive integer");
    fi;
    form := OrdinaryFormation( rec(
        res := G -> ClosureGroup(DerivedSubgroup(G),
                List(GeneratorsOfGroup(G), x -> x^exp)),
        char := PrimeDivisors(exp)
        ));
    SetIsSubgroupClosed(form, true);
    SetName(form, Concatenation(
        "<class of all abelian groups of exponent dividing ",
        String(exp), ">"));
        return form;
end);

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#F  PiGroups(<list>)
##
InstallGlobalFunction("PiGroups",
    function(pi)
        local class;
        if not IsListOrCollection(pi) then
            Error("<pi> must be a list or collection or class");
        fi;
        if HasIsEmpty(pi) and IsEmpty(pi) then
            return TrivialGroups; # this also avoids a problem with String
                                         # which returns "" for an empty list
        fi;
        class := SaturatedFittingFormation( rec(
            \in := G -> ForAll(PrimeDivisors(Size(G)), p -> p in pi),
            proj := G -> HallSubgroup(G, Filtered(PrimeDivisors(Size(G)), p -> p in pi)),
            inj  := G -> HallSubgroup(G, Filtered(PrimeDivisors(Size(G)), p -> p in pi)),
            locdef := function(G, p)
                if p in pi then
                    return [];
                else
                    return fail;
                fi;
            end,
            char := pi,
            bound := G -> not SmallestRootInt(Size(Socle(G))) in pi
            ));
        SetIsSubgroupClosed(class, true);

        # the following may cause trouble if there is no String method for pi
        SetName(class, Concatenation(
            "<class of all ",String(pi), "-groups>"));
        return class;
    end);

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#F  PGroups(<p>)
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InstallGlobalFunction("PGroups",
    function(p)
        local class;
        if not IsInt(p) or not IsPrimeInt(p) then
            Error("<p> must be a prime integer");
        fi;
        class := SaturatedFittingFormation( rec(
            \in := G -> IsTrivial(G)
             or SmallestRootInt(Size(G)) = p,
            proj := G -> SylowSubgroup(G, p),
            inj := G -> SylowSubgroup(G, p),
            locdef := function(G, q)
                if p = q then
                    return [];
                else
                    return fail;
                fi;
            end,
            char := [p],
            bound := G -> Factors(Size(Socle(G)))[1] <> p
            ));
        SetIsSubgroupClosed(class, true);
        SetName(class, Concatenation(
            "<class of all ",String(p), "-groups"));
        return class;
    end);


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#M  HallSubgroupOp(<grp>, <pi>)
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##  make sure that HallSubgroupOp works for arbitrary soluble groups
##
CRISP_RedispatchOnCondition(HallSubgroupOp,
    "redispatch if group is finite or soluble",
    true,
    [IsGroup,IsList],
    [IsSolvableGroup and IsFinite,],
    1);

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#M  SylowComplementOp(<grp>, <p>)
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CRISP_RedispatchOnCondition(SylowComplementOp,
    "redispatch if group is finite or soluble",
    true,
    [IsGroup,IsPosInt],
    [IsSolvableGroup and IsFinite,
    IsPosInt],
    1);


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#E
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Quelle samples.gi Sprache: unbekannt

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