[ref] 47 Group Products

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# 47 Group Products

### Sections

1. Direct Products
2. Semidirect Products
3. Subdirect Products
4. Wreath Products
5. Embeddings and Projections for Group Products

This chapter describes the various group product constructions that are possible in GAP.

At the moment for some of the products methods are available only if both factors are given in the same representation or only for certain types of groups such as permutation groups and pc groups when the product can be naturally represented as a group of the same kind.

GAP does not guarantee that a product of two groups will be in a particular representation. (Exceptions are `WreathProductImprimitiveAction` and `WreathProductProductAction` which are construction that makes sense only for permutation groups, see WreathProduct).

GAP however will try to choose an efficient representation, so products of permutation groups or pc groups often will be represented as a group of the same kind again.

Therefore the only guaranteed way to relate a product to its factors is via the embedding and projection homomorphisms (see Embeddings and Projections for Group Products);

## 47.1 Direct Products

The direct product of groups is the cartesian product of the groups (considered as element sets) with component-wise multiplication.

- `DirectProduct(` G`,` H`) F`
- `DirectProductOp(` list`,` expl`) O`

  These functions construct the direct product of the groups given as arguments. `DirectProduct` takes an arbitrary positive number of arguments and calls the operation `DirectProductOp`, which takes exactly two arguments, namely a nonempty list of groups and one of these groups. (This somewhat strange syntax allows the method selection to choose a reasonable method for special cases, e.g., if all groups are permutation groups or pc groups.)

  GAP will try to choose an efficient representation for the direct product. For example the direct product of permutation groups will be a permutation group again and the direct product of pc groups will be a pc group.

  For a product P the operation `Embedding(`P`,`nr`)` returns the homomorphism embedding the nr-th factor into P. The operation `Projection(`P`,`nr`)` gives the projection of P onto the nr-th factor (see Embeddings and Projections for Group Products).

      gap> g:=Group((1,2,3),(1,2));;
      gap> d:=DirectProduct(g,g,g);
      Group([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ])
      gap> Size(d);
      216
      gap> e:=Embedding(d,2);
      2nd embedding into Group([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8) ])
      gap> Image(e,(1,2));
      (4,5)
      gap> Image(Projection(d,2),(1,2,3)(4,5)(8,9));
      (1,2)

  ## 47.2 Semidirect Products

  The semidirect product of a group N with a group G acting on N via a homomorphism a from G into the automorphism group of N is the cartesian product G ×N with the multiplication (g,n)·(h,m)=(gh,n(ha)m).

- `SemidirectProduct(` G`,` alpha`,` N`) O`
- `SemidirectProduct(` autgp`,` N`) O`

  constructs the semidirect product of N with G acting via alpha. alpha must be a homomorphism from G into a group of automorphisms of N.

  If N is a group, alpha must be a homomorphism from G into a group of automorphisms of N.

  If N is a full row space over a field F, alpha must be a homomorphism from G into a matrix group of the right dimension over a subfield of F, or into a permutation group (in this case permutation matrices are taken).

  In the second variant, autgp must be a group of automorphism of N, it is a shorthand for `SemidirectProduct(`autgp`,IdentityMapping(`autgp`),`N`)`. Note that (unless autgrp has been obtained by the operation `AutomorphismGroup`) you have to test `IsGroupOfAutomorphisms(`autgrp`)` to ensure that GAP knows that autgrp consists of group automorphisms.

      gap> n:=AbelianGroup(IsPcGroup,[5,5]);
      <pc group of size 25 with 2 generators>
      gap> au:=DerivedSubgroup(AutomorphismGroup(n));;
      gap> Size(au);
      120
      gap> p:=SemidirectProduct(au,n);
      <permutation group with 4 generators>
      gap> Size(p);
      3000

      gap> n:=Group((1,2),(3,4));;
      gap> au:=AutomorphismGroup(n);;
      gap> au:=First(Elements(au),i->Order(i)=3);;
      gap> au:=Group(au);
      <group with 1 generators>

      gap> SemidirectProduct(au,n);
      Error, no method found! For debugging hints type ?Recovery from NoMethodFound

      gap> IsGroupOfAutomorphisms(au);
      true
      gap> SemidirectProduct(au,n);
      <pc group with 3 generators>

      gap> n:=AbelianGroup(IsPcGroup,[2,2]);
      <pc group of size 4 with 2 generators>
      gap> au:=AutomorphismGroup(n);
      <group with 4 generators>
      gap> apc:=IsomorphismPcGroup(au);
      CompositionMapping( Pcgs([ (2,3), (1,3,2) ]) ->
      [ f1, f2 ], <action isomorphism> )
      gap> g:=Image(apc);
      Group([ f1, f2 ])
      gap> apci:=InverseGeneralMapping(apc);
      [ f1*f2^2, f1*f2^2, f1*f2, <identity> of ... ] ->
      [ [ f1, f2 ] -> [ f2, f1 ], [ f1, f2 ] -> [ f2, f1 ],
        [ f1, f2 ] -> [ f1*f2, f2 ], [ f1, f2 ] -> [ f1, f2 ] ]
      gap> IsGroupHomomorphism(apci);
      true
      gap> p:=SemidirectProduct(g,apci,n);
      <pc group of size 24 with 4 generators>
      gap> IsomorphismGroups(p,Group((1,2,3,4),(1,2)));
      [ f1, f2, f3, f4 ] -> [ (1,2), (1,2,4), (1,2)(3,4), (1,4)(2,3) ]

      gap> SemidirectProduct(SU(3,3),GF(9)^3);
      <matrix group of size 4408992 with 3 generators>
      gap> SemidirectProduct(Group((1,2,3),(2,3,4)),GF(5)^4);
      <matrix group of size 7500 with 3 generators>

      gap> g:=Group((3,4,5),(1,2,3));;
      gap> mats:=[[[Z(2^2),0*Z(2)],[0*Z(2),Z(2^2)^2]],
      >          [[Z(2)^0,Z(2)^0], [Z(2)^0,0*Z(2)]]];;
      gap> hom:=GroupHomomorphismByImages(g,Group(mats),[g.1,g.2],mats);;
      gap> SemidirectProduct(g,hom,GF(4)^2);
      <matrix group of size 960 with 3 generators>
      gap> SemidirectProduct(g,hom,GF(16)^2);
      <matrix group of size 15360 with 4 generators>

  For the semidirect product P of G with N, `Embedding(`P`,1)` embeds G, `Embedding(`P`,2)` embeds N. The operation `Projection(`P`)` returns the projection of P onto G (see Embeddings and Projections for Group Products).

      gap> Size(Image(Embedding(p,1)));
      6
      gap> Embedding(p,2);
      [ f1, f2 ] -> [ f3, f4 ]
      gap> Projection(p);
      [ f1, f2, f3, f4 ] -> [ f1, f2, <identity> of ..., <identity> of ... ]

  ## 47.3 Subdirect Products

  The subdirect product of the groups G and H with respect to the epimorphisms j:G® A and y:H® A (for a common group A) is the subgroup of the direct product G×H consisting of the elements (g,h) for which gj = hy. It is the pull-back of the diagram:

                         G
                         | phi
                   psi   V
                H  --->  A

- `SubdirectProduct(` G`,` H`,` Ghom`,` Hhom`) O`

  constructs the subdirect product of G and H with respect to the epimorphisms Ghom from G onto a group A and Hhom from H onto the same group A.

  For a subdirect product P, the operation `Projection(`P`,`nr`` returns the projections on the nr-th factor. (In general the factors do not embed in a subdirect product.)

      gap> g:=Group((1,2,3),(1,2));
      Group([ (1,2,3), (1,2) ])
      gap> hom:=GroupHomomorphismByImagesNC(g,g,[(1,2,3),(1,2)],[(),(1,2)]);
      [ (1,2,3), (1,2) ] -> [ (), (1,2) ]
      gap> s:=SubdirectProduct(g,g,hom,hom);
      Group([ (1,2,3), (1,2)(4,5), (4,5,6) ])
      gap> Size(s);
      18
      gap> p:=Projection(s,2);
      2nd projection of Group([ (1,2,3), (1,2)(4,5), (4,5,6) ])
      gap> Image(p,(1,3,2)(4,5,6));
      (1,2,3)

  ## 47.4 Wreath Products

  The wreath product of a group G with a permutation group P acting on n points is the semidirect product of the normal subgroup G n with the group P which acts on G n by permuting the components.

- `WreathProduct(` G`,` P`) O`
- `WreathProduct(` G`,` H`[,` hom`] ) O`

  constructs the wreath product of the group G with the permutation group P (acting on its `MovedPoints`).

  The second usage constructs the wreath product of the group G with the image of the group H under hom where hom must be a homomorphism from H into a permutation group. (If hom is not given, and P is not a permutation group the result of `IsomorphismPermGroup(P)` -- whose degree may be dependent on the method and thus is not well-defined! -- is taken for hom).

  For a wreath product W of G with a permutation group P of degree n and 1 £ nr £ n the operation `Embedding(`W`,`nr`)` provides the embedding of G in the nr-th component of the direct product of the base group G n of W. `Embedding(`W`,`n`+1)` is the embedding of P into W. The operation `Projection(`W`)` provides the projection onto the acting group P (see Embeddings and Projections for Group Products).

      gap> g:=Group((1,2,3),(1,2));
      Group([ (1,2,3), (1,2) ])
      gap> p:=Group((1,2,3));
      Group([ (1,2,3) ])
      gap> w:=WreathProduct(g,p);
      Group([ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8), (1,4,7)(2,5,8)(3,6,9)
       ])
      gap> Size(w);
      648
      gap> Embedding(w,1);
      1st embedding into Group( [ (1,2,3), (1,2), (4,5,6), (4,5), (7,8,9), (7,8),
        (1,4,7)(2,5,8)(3,6,9) ] )
      gap> Image(Embedding(w,3));
      Group([ (7,8,9), (7,8) ])
      gap> Image(Embedding(w,4));
      Group([ (1,4,7)(2,5,8)(3,6,9) ])
      gap> Image(Projection(w),(1,4,8,2,6,7,3,5,9));
      (1,2,3)

- `WreathProductImprimitiveAction(` G`,` H`) F`

  for two permutation groups G and H this function constructs the wreath product of G and H in the imprimitive action. If G acts on l points and H on m points this action will be on l·m points, it will be imprimitive with m blocks of size l each.

  The operations `Embedding` and `Projection` operate on this product as described for general wreath products.

      gap> w:=WreathProductImprimitiveAction(g,p);;
      gap> LargestMovedPoint(w);
      9

- `WreathProductProductAction(` G`,` H`) F`

  for two permutation groups G and H this function constructs the wreath product in product action. If G acts on l points and H on m points this action will be on lm points.

  The operations `Embedding` and `Projection` operate on this product as described for general wreath products.

      gap> w:=WreathProductProductAction(g,p);
      <permutation group of size 648 with 7 generators>
      gap> LargestMovedPoint(w);
      27

- `KuKGenerators(` G`,` beta`,` alpha`) F`

  If beta is a homomorphism from G in a transitive permutation group, U the full preimage of the point stabilizer and and alpha a homomorphism defined on (a superset) of U, this function returns images of the generators of G when mapping to the wreath product (U alpha) wr (G beta). (This is the Krasner-Kaloujnine embedding theorem.)

      gap> g:=Group((1,2,3,4),(1,2));;
      gap> hom:=GroupHomomorphismByImages(g,Group((1,2)),
      > GeneratorsOfGroup(g),[(1,2),(1,2)]);;
      gap> u:=PreImage(hom,Stabilizer(Image(hom),1));
      Group([ (2,3,4), (1,2,4) ])
      gap> hom2:=GroupHomomorphismByImages(u,Group((1,2,3)),
      > GeneratorsOfGroup(u),[ (1,2,3), (1,2,3) ]);;
      gap> KuKGenerators(g,hom,hom2);
      [ (1,4)(2,5)(3,6), (1,6)(2,4)(3,5) ]

  ## 47.5 Embeddings and Projections for Group Products

  The relation between a group product and its factors is provided via homomorphisms, the embeddings in the product and the projections from the product. Depending on the kind of product only some of these are defined.

- `Embedding(`P`,`nr`) O`

  returns the nr-th embedding in the group product P. The actual meaning of this embedding is described in the section for the appropriate product.

- `Projection(`P`[,`nr`]) O`

  returns the (nr-th) projection of the group product P. The actual meaning of the projection returned is described in the section for the appropriate product.

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  GAP 4 manual
  May 2002