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 (49.4-2) and WreathProductProductAction (49.4-3) which are construction that makes sense only for permutation groups, see WreathProduct (49.4-1)).
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 49.6.
The direct product of groups is the cartesian product of the groups (considered as element sets) with component-wise multiplication.
‣ DirectProduct( G[, H, ...] ) | ( function ) |
‣ DirectProductOp( list, expl ) | ( operation ) |
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 list of groups and one of these groups, expl. (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.
If the groups are in different representations a generic direct product will be formed which may not be particularly efficient for many calculations. Instead it may be worth to convert all factors to a common representation first, before forming the product.
For a direct product P, calling Embedding (32.2-10) with P and n yields the homomorphism embedding the n-th factor into P; calling Projection (32.2-11) with P and n yields the projection of P onto the n-th factor, see 49.6.
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)
The semidirect product of a group N with a group G acting on N via a homomorphism α from G into the automorphism group of N is the cartesian product G × N with the multiplication (g, n) ⋅ (h, m) = (gh, n^{h^α}m).
‣ SemidirectProduct( G, alpha, N ) | ( operation ) |
‣ SemidirectProduct( autgp, N ) | ( operation ) |
constructs the semidirect product of N with G acting via alpha, which 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 (40.7-1)) you have to test IsGroupOfAutomorphisms (40.7-2) for 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 5 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> 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 of size 6 with 2 generators> gap> apc:=IsomorphismPcGroup(au); CompositionMapping( Pcgs([ (2,3), (1,2,3) ]) -> [ f1, f2 ], <action isomorphism> ) gap> g:=Image(apc); Group([ f1, f2 ]) gap> apci:=InverseGeneralMapping(apc); [ f1*f2^2, f1*f2 ] -> [ Pcgs([ f1, f2 ]) -> [ f1*f2, f2 ], Pcgs([ f1, f2 ]) -> [ f2, f1 ] ] 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))) <> fail; true 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 a semidirect product P of G with N, calling Embedding (32.2-10) with P and 1 yields the embedding of G, calling Embedding (32.2-10) with P and 2 yields the embedding of N; calling Projection (32.2-11) with P yields the projection of P onto G, see 49.6.
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 ... ]
The subdirect product of the groups G and H with respect to the epimorphisms φ: G → A and ψ: H → A (for a common group A) is the subgroup of the direct product G × H consisting of the elements (g,h) for which g^φ = h^ψ. It is the pull-back of the following diagram.
G
| phi
psi V
H ---> A
‣ SubdirectProduct( G, H, Ghom, Hhom ) | ( operation ) |
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, calling Projection (32.2-11) with P and n yields the projection on the n-th factor. (In general the factors do not embed into 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)
‣ SubdirectProducts( G, H ) | ( function ) |
this function computes all subdirect products of G and H up to conjugacy in the direct product of Parent(G) and Parent(H). The subdirect products are returned as subgroups of this direct product.
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.
Note that GAP always considers the domain of a permutation group to be the points moved by elements of the group as returned by MovedPoints (42.3-3), i.e. it is not possible to have a domain to include fixed points, I.e. P = ⟨ (1,2,3) ⟩ and P = ⟨ (1,3,5) ⟩ result in isomorphic wreath products. (If fixed points are desired the wreath product G ≀ T has to be formed with a transitive overgroup T of P and then the pre-image of P under the projection G ≀ T → T has to be taken.)
‣ WreathProduct( G, H[, hom] ) | ( operation ) |
‣ StandardWreathProduct( G, H ) | ( operation ) |
WreathProduct constructs the wreath product of the group G with the group H, acting as a permutation group.
If a third argument hom is given, it must be a homomorphism from H into a permutation group, and the action of this group on its moved points is considered.
If only two arguments are given, H must be a permutation group.
StandardWreathProduct returns the wreath product for the (right regular) permutation action of H on its elements.
For a wreath product W of G with a permutation group P of degree n and 1 ≤ i ≤ n calling Embedding (32.2-10) with W and i yields the embedding of G in the i-th component of the direct product of the base group G^n of W. For i = n+1, Embedding (32.2-10) yields the embedding of P into W. Calling Projection (32.2-11) with W yields the projection onto the acting group P, see 49.6.
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 ) | ( function ) |
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 (32.2-10) and Projection (32.2-11) operate on this product as described for general wreath products.
gap> w:=WreathProductImprimitiveAction(g,p);; gap> LargestMovedPoint(w); 9
‣ WreathProductProductAction( G, H ) | ( function ) |
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 l^m points.
The operations Embedding (32.2-10) and Projection (32.2-11) 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 ) | ( function ) |
If beta is a homomorphism from G into a transitive permutation group, U the full preimage of the point stabilizer 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) ≀ (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) ]
Let G and H be groups with presentations ⟨ X ∣ R ⟩ and ⟨ Y ∣ S ⟩, respectively. Then the free product G*H is the group with presentation ⟨ X ∪ Y ∣ R ∪ S ⟩. This construction can be generalized to an arbitrary number of groups.
‣ FreeProduct( G[, H, ...] ) | ( function ) |
‣ FreeProduct( list ) | ( function ) |
constructs a finitely presented group which is the free product of the groups given as arguments. If the group arguments are not finitely presented groups, then IsomorphismFpGroup (47.11-1) must be defined for them.
The operation Embedding (32.2-10) operates on this product.
gap> g := DihedralGroup(8);; gap> h := CyclicGroup(5);; gap> fp := FreeProduct(g,h,h); <fp group on the generators [ f1, f2, f3, f4, f5 ]> gap> fp := FreeProduct([g,h,h]); <fp group on the generators [ f1, f2, f3, f4, f5 ]> gap> Embedding(fp,2); [ f1 ] -> [ f4 ]
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 ) | ( operation ) |
returns the nr-th embedding in the group product P. The actual meaning of this embedding is described in the manual section for the appropriate product.
‣ Projection( P, nr ) | ( operation ) |
returns the (nr-th) projection of the group product P. The actual meaning of the projection returned is described in the manual section for the appropriate product.
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