% This file was created automatically from grpprod.msk. % DO NOT EDIT! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A grpprod.msk GAP documentation Alexander Hulpke %% %A @(#)$Id: grpprod.msk,v 1.29.2.6 2006/09/16 19:02:49 jjm Exp $ %% %Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland %Y Copyright (C) 2002 The GAP Group %% \Chapter{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"); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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. 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. \atindex{Embedding!example for direct products}% {@\noexpand`Embedding'!example for direct products} \atindex{Projection!example for direct products}% {@\noexpand`Projection'!example for direct products} 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"). \beginexample 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) \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Semidirect Products} The semidirect product of a group <N> with a group <G> acting on <N> via a homomorphism $\alpha$ from <G> into the automorphism group of <N> is the cartesian product $<G>\times<N>$ with the multiplication $(g,n)\cdot(h,m)=(gh,n^{(h\alpha)}m)$. \>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. \beginexample 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 \endexample \beginexample 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 Error, no 2nd choice method found for `IsomorphismPcGroup' on 1 arguments gap> IsGroupOfAutomorphisms(au); true gap> SemidirectProduct(au,n); <pc group with 3 generators> \endexample \beginexample 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))); [ f1, f2, f3, f4 ] -> [ (2,3), (2,3,4), (1,4)(2,3), (1,2)(3,4) ] \endexample \beginexample 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> \endexample \beginexample 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> \endexample \atindex{Embedding!example for semidirect products}% {@\noexpand`Embedding'!example for semidirect products} \atindex{Projection!example for semidirect products}% {@\noexpand`Projection'!example for semidirect products} 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"). \beginexample 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 ... ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Subdirect Products} The subdirect product of the groups $G$ and $H$ with respect to the epimorphisms $\varphi\colon G\to A$ and $\psi\colon H\to A$ (for a common group $A$) is the subgroup of the direct product $G\times H$ consisting of the elements $(g,h)$ for which $g\varphi=h\psi$. It is the pull-back of the diagram: %display{tex} $$ \matrix{ &&G&\cr &&\Big\downarrow&\varphi\cr H&\mathop{\longrightarrow}\limits^{\psi}&A&\cr } $$ %display{html} %<PRE> % G % | phi % psi V % H ---> A %</PRE> %display{text} % G % | phi % psi V % H ---> A %enddisplay \>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>. \atindex{Projection!example for subdirect products}% {@\noexpand`Projection'!example for subdirect products} 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.) \beginexample 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) \endexample \>SubdirectProducts( <G>, <H> ) F this function computes all subdirect products of <G> and <H> up to conjugacy in Parent(<G>) x Parent(<H>). The subdirect products are returned as subgroups of this direct product. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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>). \atindex{Embedding!example for wreath products}% {@\noexpand`Embedding'!example for wreath products} \atindex{Projection!example for wreath products}% {@\noexpand`Projection'!example for wreath products} For a wreath product <W> of <G> with a permutation group <P> of degree <n> and $1\le <nr>\le <n>$ 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"). \beginexample 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) \endexample \>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 $l\cdot m$ 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. \beginexample gap> w:=WreathProductImprimitiveAction(g,p);; gap> LargestMovedPoint(w); 9 \endexample \>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 $l^m$ points. The operations `Embedding' and `Projection' operate on this product as described for general wreath products. \beginexample gap> w:=WreathProductProductAction(g,p); <permutation group of size 648 with 7 generators> gap> LargestMovedPoint(w); 27 \endexample \>KuKGenerators( <G>, <beta>, <alpha> ) F \atindex{Krasner-Kaloujnine theorem}{@Krasner-Kaloujnine theorem} \index{Wreath product embedding} 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.) \beginexample 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) ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Free Products} Let $G$ and $H$ be groups with presentations $\langle X\mid R\rangle$ and $\langle Y\mid S\rangle$ respectively. Then the free product $G*H$ is the group with presentation $\langle X\cup Y\mid R\cup S\rangle$. This construction can be generalized to an arbitrary number of groups. \>FreeProduct( <G> \{, <H>\} ) F \>FreeProduct( list ) F 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' must be defined for them. The operation `Embedding' operates on this product. \beginexample 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 ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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>)!{for group products} 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>])!{for group products} 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.