[GAP Forum] unpredictable SemidirectProduct

[Alexander Hulpke]

Dear Forum, Dear Barry Monson,

>            I am puzzled by the behaviour of SemidirectProduct.
> Two similar calculations, only one successful, follow.
> Can someone explain what, if anything, I am doing wrong?

The existing (convenience) code for constructing semidirect products with vector spaces assumes that the group acts faithfully -- otherwise one cannot construct the product as a group of affine matrices. I append a modified function that in such a case constructs the vector space as a group (and forms the corresponding automorphisms), allowing to use the generic mechanism for semidirect products. I hope this resolves the issue. In this case, however, the product will not be represented any longer as a matrix group.

Best,

    Alexander Hulpke


##############################################################################
##  
#M  SemidirectProduct: with vector space
##
InstallOtherMethod( SemidirectProduct, "group with vector space: affine", true,
  [ IsGroup, IsGroupHomomorphism, IsFullRowModule and IsVectorSpace ], 0,
function( G, map, V )
local pm,F,d,b,s,t,pos,i,j,img,m,P,info,Go,bnt,N,pcgs,auts,mapi,ag,phi,imgs;
  # construction assumes faithful action. AH
  if Size(KernelOfMultiplicativeGeneralMapping(map))<>1 then
    # not faithful -- cannot simply build as matrices
    N:=ElementaryAbelianGroup(Size(V));
    pcgs:=Pcgs(N);
    auts:=[];
    mapi:=MappingGeneratorsImages(map);
    for i in mapi[2] do
      imgs:=List(i,x->PcElementByExponents(pcgs,x));
      Add(auts,GroupHomomorphismByImagesNC(N,N,pcgs,imgs));
    od;
    ag:=Group(auts,IdentityMapping(N));
    SetIsGroupOfAutomorphismsFiniteGroup(ag,true);
    phi:=GroupHomomorphismByImages(G,ag,mapi[1],auts);
    s:=SemidirectProduct(G,phi,N);
    return s;
  fi;
  G:=Image(map,G);
  F:=LeftActingDomain(V);
  d:=DimensionOfVectors(V);
  # if G is a permgroup, take permutation matrices
  Go:=G;
  if IsPermGroup(G) then
    m:=List(GeneratorsOfGroup(G),i->PermutationMat(i,d,F));
    s:=Group(m);
    pm:=GroupHomomorphismByImagesNC(G,s,GeneratorsOfGroup(G),m);
    map:=map*pm;
    G:=s;
  fi;

  if not IsMatrixGroup(G) or d<>DimensionOfMatrixGroup(G) or not
    IsSubset(F,FieldOfMatrixGroup(G)) then
    Error("the matrices do not fit with the field");
  fi;
  b:=BasisVectors(Basis(V));
  # spin up a basis
  s:=[];
  pos:=1;
  t:=[];
  while Length(s)<Length(b) do
    # skip basis vectors that give nothing new
    while Length(s)>0 and RankMat(s)=RankMat(Concatenation(s,[b[pos]])) do
      pos:=pos+1;
    od;
    Add(s,b[pos]);
    Add(t,b[pos]); # those vectors need own affine matrices
    # spin the new vector
    i:=Length(s);
    while i<=Length(s) and Length(s)<Length(b) do
      for j in GeneratorsOfGroup(G) do
        img:=s[i]*j;
        if RankMat(s)<RankMat(Concatenation(s,[img])) then
          # new dimension
          Add(s,img);
        fi;
      od;
      i:=i+1;
    od;
  od;

  # do we need to take extra vectors to extend the field?
  if FieldOfMatrixGroup(G)<>F then
    b:=BasisVectors(Basis(Field(FieldOfMatrixGroup(G),GeneratorsOfField(F))));
    s:=[];
    for i in t do
      for j in b do
        Add(s,i*j);
      od;
    od;
    t:=s;
  fi;

  m:=[];
  # build affine matrices from group generators
  for i in GeneratorsOfGroup(G) do
    b:=MutableIdentityMat(d+1,F);
    b{[1..d]}{[1..d]}:=i;
    Add(m,ImmutableMatrix(F,b));
  od;
  # and from basis vectors
  bnt:=[];
  for i in t do
    b:=MutableIdentityMat(d+1,F);
    b[d+1]{[1..d]}:=i;
    b:=ImmutableMatrix(F,b);
    Add(m,b);
    Add(bnt,b);
  od;

  P:=Group(m,One(m[1]));
  SetSize(P,Size(G)*Size(V));
  info:=rec(group:=Go,
            vectorspace:=V,
            normalsub:=bnt,
            lenlist:=[0,Length(GeneratorsOfGroup(G))],
            embeddings:=[],
            field:=F,
            dimension:=d,
            projections:=true);
  SetSemidirectProductInfo( P, info );

  return P;
end);




-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke