Quelle goutergroup.gi Sprache: unbekannt

#(C) 2008 Graham Ellis

#############################################################################
##
#W  goutergroup.gi                 HAP

#############################################################################
##
##  Method for viewing GOuterGroups.
##
InstallMethod( ViewObj,
        "for GOuterGroups",
        [IsGOuterGroup],
    function(N)
if not IsAbelian(ActedGroup(N)) then
    Print("G-outer group ", ActedGroup(N), " with actor ", ActingGroup(N), "\n");
else
    Print("ZG-module with abelian invariants ", AbelianInvariants(ActedGroup(N)), " and G= ", ActingGroup(N), "\n");
fi;
end);

InstallMethod( PrintObj,
        "for GOuterGroups",
        [IsGOuterGroup],
    function(N)
if not IsAbelian(ActedGroup(N)) then
    Print("G-outer group ", ActedGroup(N), " with actor ", ActingGroup(N), "\n");
else
    Print("ZG-module with abelian invariants ", AbelianInvariants(ActedGroup(N)), " and G= ", ActingGroup(N), "\n");
fi;
end);

InstallMethod( ViewObj,
        "for GOuterGroupHomomorphisms",
        [IsGOuterGroupHomomorphism],
    function(phi)
    Print("G-outer group homomorphism A ----> B \n");
end);

InstallMethod( PrintObj,
        "for GOuterGroupHomomorphisms",
        [IsGOuterGroupHomomorphism],
    function(phi)
    Print("G-outer group homomorphism A ----> B \n");
end);

#############################################################################
##
##  Creation empty G-Outer group. Attributes must be set later. There
##  is no method for printing an empty G-Outer group (nor should there be).
##
InstallMethod( GOuterGroup,
    "method for creating a GOuterGroup with no attributes set",
    [ ],

    function( )
        local N, type;
   N:= rec();
         type:= NewType(NewFamily("gog"),
                       IsGOuterGroup and
                       IsComponentObjectRep and IsAttributeStoringRep);

        ObjectifyWithAttributes(N,type);
        return N;
     end);

#############################################################################
##
##  Creation of a G-Outer A group from a group G assumed to be acting
##  trivially on an abelian group A.
##
##
InstallMethod( TrivialGModuleAsGOuterGroup,
    "basic method for creating a GOuterGroup",
    [ IsGroup, IsGroup ],

    function( G, A )
        local
              N,
              alpha;     ## Action of G on A


        if not (IsGroup(G) and IsGroup(A) and IsAbelian(A)) then
            Error("A must be abelian");
        fi;

       ######################################################
        alpha := function(g,a);
            return a;
        end;
       ######################################################

        N:=GOuterGroup();
               SetActingGroup(N,G);
               SetActedGroup(N,A);
               SetOuterAction(N,alpha);
        return N;
    end);

#############################################################################
##
##  Creation of a G-Outer A group from a group G assumed to be acting
##  via a function alpha(g,a) on an abelian group A.
##
##
InstallMethod( GModuleAsGOuterGroup,
    "basic method for creating a GOuterGroup",
    [ IsGroup, IsGroup, IsFunction ],

    function( G, A, alpha )
        local
              N;

        if not (IsGroup(G) and IsGroup(A) and IsAbelian(A)) then
            Error("A must be abelian");
        fi;

        N:=GOuterGroup();
               SetActingGroup(N,G);
               SetActedGroup(N,A);
               SetOuterAction(N,alpha);
        return N;
    end);

#############################################################################
##
##  Creation of a G-Outer N group from a group E (Extension) with
##  normal subgroup A.
##
##      A --> E --> G
##
InstallMethod( GOuterGroup,
    "basic method for creating a GOuterGroup",
    [ IsGroup, IsGroup ],

    function( E, A )
        local
       N, type, G,
              alpha,     ## Action of G on A
              nat,       ## Natural homomorphism from E to G
       alphaRec,p,q,bool, coc;

        if not IsNormal(E,A) then
            Error("A must be a normal subgroup of E");
        fi;

        nat := NaturalHomomorphismByNormalSubgroup(E,A);
        G:=Image(nat);

bool:=true;
        if IsFinite(G) and IsFinite(A) then
           if  Order(G)*Size(A) > 2000*2000 then bool:=false; fi;
        fi;
if bool then
######################################################
        alpha := function(g,a);
            return
            Representative(PreImages(nat,g))
                           *a*
            Representative(PreImages(nat,g))^-1;
            end;
######################################################
        else
        alphaRec:=List([1..Order(G)],i->List([1..Size(A)],j->0));
        ######################################################
        alpha := function(g,a);
            p:=Position(Elements(G),g);
            q:=Position(Elements(A),a);
            if alphaRec[p][q]=0 then
            alphaRec[p][q]:=
            Representative(PreImages(nat,g))
                           *a*
            Representative(PreImages(nat,g))^-1;
     fi;
            return alphaRec[p][q];
            end;
        ######################################################
fi;

  N:=GOuterGroup();
               SetActingGroup(N,G);
               SetActedGroup(N,A);
               SetOuterAction(N,alpha);

###################
if not IsAbelian(A) then
coc:=function(x,y);
return
Representative(PreImages(nat,x))*
Representative(PreImages(nat,y))*
Representative(PreImages(nat,x*y))^-1;
end;
N!.nonabeliancocycle:=coc;
fi;
###################
        return N;
    end);

#############################################################################
##
##  Creation of a G-Outer E group from a group E (with G trivial)
##
InstallMethod( GOuterGroup,
    "basic method for creating a GOuterGroup from a group",
    [ IsGroup ],

    function( E )
        local
              A, N, type, G, ##E=A, G=1
              alpha,     ## Action of G on G
              nat;       ## Natural homomorphism from G to G

        nat := NaturalHomomorphismByNormalSubgroup(E,E);
        A:=E;
        G:=Group(Identity(E));

        ######################################################
        alpha := function(g,a);
            return g*a*g^-1;
            end;
        ######################################################

        N:=GOuterGroup();
               SetActingGroup(N,G);
               SetActedGroup(N,A);
               SetOuterAction(N,alpha);
        return N;
    end);

#############################################################################
##
##  Creation of a G-Outer group homomorphism from a grouphomomorphism
##
InstallMethod( GOuterGroup,
    "basic method for creating a GOuterGroup homomorphism from a group homomorphism",
    [ IsGroupHomomorphism ],

    function( phi )
    local PHI,S, T;

    S:=GOuterGroup(Source(phi));
    T:=GOuterGroup(Range(phi));
    return GOuterGroupHomomorphism(S,T,phi);
    end);

#############################################################################
##
##  The centre of the acted group of a G-outer group is a G-module. We
##  return this centre as a G-outer group.
##
##
InstallOtherMethod( Center,
    "method for returning the centre of a G-outer group as a G-outer group.",
    [ IsGOuterGroup ],

    function( N )
    local C, type;

  C:=GOuterGroup();
               SetActingGroup(C,ActingGroup(N));
               SetActedGroup(C,Center(ActedGroup(N)));
               SetOuterAction(C,OuterAction(N));
        return C;
    end );


#############################################################################
##
##  Test to see if a group homomorphism is a G-outer group homomorphism.
##  The test is clumsy, only treats finite G, and the
##  definition of homomorphism used here may not be the most appropriate in
##  the case when the G-outer group B is non-commutative.
##
InstallMethod( GOuterHomomorphismTester,
    "basic method for creating a GOuterGroup",
    [ IsGOuterGroup, IsGOuterGroup, IsGroupHomomorphism ],

    function(A,B, phi )
     local OA,OB,G,alpha,beta,a,g;

        if not
        (HasActingGroup(A) and HasActingGroup(B))
        then return false; fi;
        if not ActingGroup(A)=ActingGroup(B)  then
        return false;
        fi;
        OA:=Source(phi);
        OB:=Target(phi);
        if not (ActedGroup(A)=OA and ActedGroup(B)=OB) then
        return false; fi;
        G:=ActingGroup(A);
        alpha:=OuterAction(A);
        beta:=OuterAction(B);

        if (not IsFinite(G))
        then TryNextMethod(); fi;

        for a in GeneratorsOfGroup(OA) do
        for g in G do
        if not
        Image(phi,alpha(g,a)) = beta(g,Image(phi,a))
        then return false; fi;
        od;
        od;
      return true;
     end);

#########################################################################
##
##  Constructor for an empty GOuterGroup homomorphism. No method to print
##  such an empty homomorphism (nor should there be).
##
InstallMethod( GOuterGroupHomomorphism,
    "method for constructing an empty GOuterGroup homomorphism",
    [ ],

    function()
    local PHI, type;
        PHI:=rec();
        type:= NewType(NewFamily("gogh"),
               IsGOuterGroupHomomorphism and
               IsComponentObjectRep and IsAttributeStoringRep);
ObjectifyWithAttributes( PHI,type);
        return PHI;
    end);

#########################################################################
##
##  Constructor for a GOuterGroup homomorphism.
##
##
InstallMethod( GOuterGroupHomomorphism,
    "method for constructing GOuterGroup homomorphisms",
    [ IsGOuterGroup, IsGOuterGroup, IsGroupHomomorphism ],

    function( A, B, phi  )
    local PHI, type;

        PHI:=GOuterGroupHomomorphism();
               SetSource(PHI,A);
               SetTarget(PHI, B);
               SetMapping(PHI, phi);
        return PHI;
    end );

#########################################################################
##
##  Install method of arrow composition on *
##
InstallOtherMethod( \*,
    "basic method for composing GOuterGroup Homomorphisms",
    [ IsGOuterGroupHomomorphism, IsGOuterGroupHomomorphism ],

  function(PHI,THETA)
     local phi, theta, thetaphi;

        if not (Source(PHI)=Target(THETA)) then
        Print("Arrows and not composable. \n");
        return fail;
        fi;
        phi:=Mapping(PHI);
        theta:=Mapping(THETA);
        thetaphi:=GroupHomomorphismByFunction(
        Source(theta), Target(phi),
        x-> Image(phi,Image(theta,x)) );

        return
        GOuterGroupHomomorphism(Source(THETA),Target(PHI),thetaphi);
        end);

######################################################################
##
##  Install method of arrow addition on +. This won't be a homomorphism
##  unless the images of PHI and THETA are abelian groups.
##
InstallOtherMethod( \+,
    "method for adding GOuterGroup Homomorphisms",
    [ IsGOuterGroupHomomorphism, IsGOuterGroupHomomorphism ],

  function(PHI,THETA)
     local phi, theta, thetaphi;

        if not (
        Source(PHI)=Target(THETA)
        and
        Target(PHI)=Target(THETA)
        ) then
        Print("Arrows can not be added \n");
        return fail;
        fi;
        phi:=Mapping(PHI);
        theta:=Mapping(THETA);
        thetaphi:=GroupHomomorphismByFunction(
        Source(theta), Target(phi),
        x-> Image(phi,x) * Image(theta,x) );

        return
        GOuterGroupHomomorphism(Source(THETA),Target(PHI),thetaphi);
        end);

######################################################################
##
##  Install method of arrow subtraction on \-. This won't be a homomorphism
##  unless the images of PHI and THETA are abelian groups.
##
InstallOtherMethod( \-,
    "method for subtracting GOuterGroup Homomorphisms",
    [ IsGOuterGroupHomomorphism, IsGOuterGroupHomomorphism ],

  function(PHI,THETA)
     local phi, theta, thetaphi;

        if not (
        Source(PHI)=Target(THETA)
        and
        Target(PHI)=Target(THETA)
        ) then
        Print("Arrows can not be added \n");
        return fail;
        fi;
        phi:=Mapping(PHI);
        theta:=Mapping(THETA);
        thetaphi:=GroupHomomorphismByFunction(
        Source(theta), Target(phi),
        x-> Image(phi,x) * Image(theta,x)^-1 );

        return
        GOuterGroupHomomorphism(Source(THETA),Target(PHI),thetaphi);
        end);

######################################################################
##
##  Install method for DirectProduct of two GOuter groups, with common
##  acting group and diagonal action
##
InstallOtherMethod( DirectProductGog,
    "method for direct product of two GOuterGroups",
    [ IsGOuterGroup, IsGOuterGroup ],

  function(M,N)
     local A,B,C,G,MN,alpha,beta,gamma,
           i1,i2,p1,p2;

if not ActingGroup(M)=ActingGroup(N) then
TryNextMethod(); fi;

A:=ActedGroup(M);
B:=ActedGroup(N);
G:=ActingGroup(M);
alpha:=OuterAction(M);
beta:=OuterAction(N);

C:=DirectProduct(A,B);
i1:=Embedding(C,1);
i2:=Embedding(C,2);
p1:=Projection(C,1);
p2:=Projection(C,2);

gamma:=function(g,x)
return
         Image(i1,alpha(g,Image(p1,x)))
  *
  Image(i2,beta(g,Image(p2,x)));
end;

MN:=GOuterGroup();
  SetActedGroup(MN,C);
SetActingGroup(MN,G);
SetOuterAction(MN,gamma);

#We'll later construct the two embeddings and two projections.

#MN!.e1:=GOuterGroupHomomorphism(M,MN,Embedding(C,1));
#MN!.e2:=GOuterGroupHomomorphism(M,MN,Embedding(C,2));
#MN!.p1:=GOuterGroupHomomorphism(M,MN,Projection(C,1));
        #MN!.p2:=GOuterGroupHomomorphism(M,MN,Projection(C,2));

return MN;
end);

######################################################################
##
##  Install method for DirectProduct embeddings for GOuter groups.
##
InstallOtherMethod( Embedding,
    "method for direct product embeddings",
    [ IsGOuterGroup, IsInt ],

  function(D,n)
local A,e,p,beta;
e:=Embedding(ActedGroup(D),n);
p:=Projection(ActedGroup(D),n);
A:=GOuterGroup();
SetActedGroup(A,Source(e));
SetActingGroup(A,ActingGroup(D));
beta:=function(g,a);
       return Image(p,OuterAction(g,Image(e,a)));
end;
SetOuterAction(A,beta);

return GOuterGroupHomomorphism(A,D,e);
  end);

######################################################################
##
##  Install method for DirectProduct projections for GOuter groups.
##
InstallOtherMethod( Projection,
    "method for direct product Projections",
    [ IsGOuterGroup, IsInt ],

  function(D,n)
        local A,e,p,beta;
        e:=Embedding(ActedGroup(D),n);
        p:=Projection(ActedGroup(D),n);
        A:=GOuterGroup();
        SetActedGroup(A,Source(e));
        SetActingGroup(A,ActingGroup(D));
        beta:=function(g,a);
              return Image(p,OuterAction(g,Image(e,a)));
        end;
        SetOuterAction(A,beta);

        return GOuterGroupHomomorphism(D,A,p);
  end);

######################################################################
##
##  Install method for DirectProductGog of a list of GOuter groups.
##   It inputs a list Lst of GOuter groups.
##
InstallOtherMethod( DirectProductGog,
    "method for direct product of list of G-outer groups",
    [ IsList ],

  function(Lst)
        local D,UD,G,alpha,beta;

if Length(Lst)=0 then TryNextMethod(); fi;
if not IsGOuterGroup(Lst[1]) then
Print("Must be a list of G-outer groups.\n");
return fail; fi;

        #Are all acting groups identical. If not, try another method.
if not Length(SSortedList(List(Lst,ActingGroup)))=1 then
TryNextMethod(); fi;

G:=ActingGroup(Lst[1]);
        #UD:=DirectProductOp(List(Lst,ActedGroup), ActedGroup(Lst[1]));
   UD:=DirectProduct(List(Lst,ActedGroup));

beta:=function(g,a)
local answer;
answer:=
List([1..Length(Lst)], i->Image(Projection(UD,i),a));
answer:=
List([1..Length(Lst)], i-> OuterAction(Lst[i])(g,answer[i]));
answer:=
List([1..Length(Lst)], i-> Image(Embedding(UD,i),answer[i]));
answer:=Product(answer);
return answer;
end;

D:=GOuterGroup();
SetActingGroup(D,G);
SetActedGroup(D,UD);
SetOuterAction(D,beta);

return D;
  end);

#############################################################################
##
##
InstallOtherMethod( GDerivedSubgroup,
    "method for returning the G-derived subgroup of a G-outer group as a G-outer group.",
    [ IsGOuterGroup ],

    function( N )
    local C, type, A, G, phi, x, g, a, GD;

        C:=GOuterGroup();
               GD:=[];
               A:=Center(ActedGroup(N));
               G:=ActingGroup(N);
               phi:=OuterAction(N);
               for a in GeneratorsOfGroup(A) do   #
               for g in GeneratorsOfGroup(G) do   #
               x:=phi(g,a)*a^-1;                  #
               Add(GD,x);                         #
               od;                                #
               od;                                #
               if Length(GD)=0 then GD:=[One(A)]; fi;
               GD:=Group(GD);
               GD:=NormalClosure(A,GD);
               SetActingGroup(C,G);
               SetActedGroup(C,GD);
               SetOuterAction(C,phi);
        return C;
    end );

#############################################################################

#############################################################################
##
##
InstallOtherMethod( LowerGCentralSeries,
    "method for returning the G-central series of a G-outer group.",
    [ IsGOuterGroup ],

    function( N )
    local L, D, bool, M;

                L:=[N];
                bool:=true;

                while bool do
                M:=L[Length(L)];
                D:=GDerivedSubgroup(M);
                bool:= not Size(M!.ActedGroup)=Size(D!.ActedGroup);
                if bool then Add(L,D); fi;
                od;
    return L;
    end);

#############################################################################

##################################################
##################################################
InstallGlobalFunction(ImageOfGOuterGroupHomomorphism,
function(phi)
local A, B, G, h,f, L;

A:=phi!.Source;
B:=phi!.Target;
G:=B!.ActingGroup;
h:=phi!.Mapping;
f:=Image(h);
L:=GOuterGroup();
SetActingGroup(L,G);
SetActedGroup(L,f);
SetOuterAction(L,B!.OuterAction);

return L;

end);
##################################################
##################################################

##################################################
##################################################

##################################################
##################################################
InstallGlobalFunction(KernelOfGOuterGroupHomomorphism,
function(phi)
local h,k,K,A,G;

A:=phi!.Source;
G:=A!.ActingGroup;
h:=phi!.Mapping;
k:=Kernel(h);

K:=GOuterGroup();
SetActingGroup(K,G);
SetActedGroup(K,k);
SetOuterAction(K,A!.OuterAction);

return K;
end);
##################################################
##################################################

##################################################
##################################################
InstallOtherMethod(Size,
        "for GOuterGroups",
        [IsGOuterGroup],
    function(N)
    return Size( ActedGroup(N));
end);
##################################################
##################################################

##################################################
##################################################
InstallOtherMethod(Source,
        "for GOuterGroups",
        [IsGOuterGroupHomomorphism],
    function(N)
    return N!.Source;
end);
##################################################
##################################################

##################################################
##################################################
InstallOtherMethod(Target,
        "for GOuterGroups",
        [IsGOuterGroupHomomorphism],
    function(N)
    return N!.Target;
end);
##################################################
##################################################

Quelle goutergroup.gi Sprache: unbekannt

[ Dauer der Verarbeitung: 0.41 Sekunden (vorverarbeitet) ]