| Current Path : /usr/share/gap/lib/ |
Linux ift1.ift-informatik.de 5.4.0-216-generic #236-Ubuntu SMP Fri Apr 11 19:53:21 UTC 2025 x86_64 |
| Current File : //usr/share/gap/lib/gprd.gd |
#############################################################################
##
#W gprd.gd GAP library Heiko Theißen
##
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
#############################################################################
##
#F DirectProduct( <G>{, <H>} )
#O DirectProductOp( <list>, <expl> )
##
## <#GAPDoc Label="DirectProduct">
## <ManSection>
## <Func Name="DirectProduct" Arg='G[, H, ...]'/>
## <Oper Name="DirectProductOp" Arg='list, expl'/>
##
## <Description>
## These functions construct the direct product of the groups given as
## arguments.
## <Ref Func="DirectProduct"/> takes an arbitrary positive number of
## arguments and calls the operation <Ref Func="DirectProductOp"/>,
## which takes exactly two arguments,
## namely a nonempty list <A>list</A> of groups and one of these groups,
## <A>expl</A>.
## (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.)
## <P/>
## &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.
## <P/>
## 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.
## <P/>
## <Index Key="Embedding" Subkey="example for direct products">
## <C>Embedding</C></Index>
## <Index Key="Projection" Subkey="example for direct products">
## <C>Projection</C></Index>
## For a direct product <M>P</M>, calling
## <Ref Func="Embedding" Label="for a domain and a positive integer"/> with
## <M>P</M> and <M>n</M> yields the homomorphism embedding the <M>n</M>-th
## factor into <M>P</M>; calling
## <Ref Func="Projection" Label="for a domain and a positive integer"/> with
## <A>P</A> and <A>n</A> yields the projection of <M>P</M> onto the
## <M>n</M>-th factor,
## see <Ref Sect="Embeddings and Projections for Group Products"/>.
## <P/>
## <Example><![CDATA[
## 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)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "DirectProduct" );
DeclareOperation( "DirectProductOp", [ IsList, IsGroup ] );
#############################################################################
##
#F PcgsDirectProduct( <D>, <pcgsop>, <indsop>, <filter> )
##
## <ManSection>
## <Func Name="PcgsDirectProduct" Arg='D, pcgsop, indsop, filter'/>
##
## <Description>
## constructs a new pcgs from pcgses of the components of D, setting
## the necessary indices for the new pcgs and sets the property
## specified by filter.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "PcgsDirectProduct" );
#############################################################################
##
#O SubdirectProduct(<G>, <H>, <Ghom>, <Hhom> )
##
## <#GAPDoc Label="SubdirectProduct">
## <ManSection>
## <Func Name="SubdirectProduct" Arg='G, H, Ghom, Hhom'/>
##
## <Description>
## constructs the subdirect product of <A>G</A> and <A>H</A> with respect to
## the epimorphisms <A>Ghom</A> from <A>G</A> onto a group <M>A</M> and
## <A>Hhom</A> from <A>H</A> onto the same group <M>A</M>.
## <P/>
## <Index Key="Projection" Subkey="example for subdirect products">
## <C>Projection</C></Index>
## For a subdirect product <M>P</M>, calling
## <Ref Func="Projection" Label="for a domain and a positive integer"/> with
## <M>P</M> and <M>n</M> yields the projection on the <M>n</M>-th factor.
## (In general the factors do not embed into a subdirect product.)
## <P/>
## <Example><![CDATA[
## 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)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("SubdirectProduct");
DeclareOperation( "SubdirectProductOp",
[ IsGroup, IsGroup, IsGroupHomomorphism, IsGroupHomomorphism ] );
#############################################################################
##
#F SubdirectDiagonalPerms(<l>,<m>)
##
## <ManSection>
## <Func Name="SubdirectDiagonalPerms" Arg='l,m'/>
##
## <Description>
## Let <A>l</A> and <A>m</A> be lists of permutations that are the images of
## the same generating set <A>gens</A>.
## This function returns permutations for the images
## of <A>gens</A> under the subdirect product of the homomorphisms.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("SubdirectDiagonalPerms");
#############################################################################
##
#O SemidirectProduct(<G>, <alpha>, <N> )
#O SemidirectProduct(<autgp>, <N> )
##
## <#GAPDoc Label="SemidirectProduct">
## <ManSection>
## <Heading>SemidirectProduct</Heading>
## <Oper Name="SemidirectProduct" Arg='G, alpha, N'
## Label="for acting group, action, and a group"/>
## <Oper Name="SemidirectProduct" Arg='autgp, N'
## Label="for a group of automorphisms and a group"/>
##
## <Description>
## constructs the semidirect product of <A>N</A> with <A>G</A> acting via
## <A>alpha</A>, which must be a homomorphism from <A>G</A> into a group of
## automorphisms of <A>N</A>.
## <P/>
## If <A>N</A> is a group, <A>alpha</A> must be a homomorphism from <A>G</A>
## into a group of automorphisms of <A>N</A>.
## <P/>
## If <A>N</A> is a full row space over a field <A>F</A>, <A>alpha</A> must
## be a homomorphism from <A>G</A> into a matrix group of the right
## dimension over a subfield of <A>F</A>, or into a permutation group
## (in this case permutation matrices are taken).
## <P/>
## In the second variant, <A>autgp</A> must be a group of automorphism of
## <A>N</A>, it is a shorthand for
## <C>SemidirectProduct(<A>autgp</A>,IdentityMapping(<A>autgp</A>),<A>N</A>)</C>.
## Note that (unless <A>autgrp</A> has been obtained by the operation
## <Ref Func="AutomorphismGroup"/>)
## you have to test <Ref Func="IsGroupOfAutomorphisms"/> for <A>autgrp</A>
## to ensure that &GAP; knows that <A>autgrp</A> consists of
## group automorphisms.
## <Example><![CDATA[
## 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);;
## gap> Size(p);
## 3000
## gap> n:=Group((1,2),(3,4));;
## gap> au:=AutomorphismGroup(n);;
## gap> au:=First(AsSet(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);;
## gap> apc:=IsomorphismPcGroup(au);;
## gap> g:=Image(apc);
## Group([ f1, f2 ])
## gap> apci:=InverseGeneralMapping(apc);;
## 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>
## ]]></Example>
## <P/>
## <Index Key="Embedding" Subkey="example for semidirect products">
## <C>Embedding</C></Index>
## <Index Key="Projection" Subkey="example for semidirect products">
## <C>Projection</C></Index>
## For a semidirect product <M>P</M> of <A>G</A> with <A>N</A>, calling
## <Ref Func="Embedding" Label="for a domain and a positive integer"/> with
## <M>P</M> and <C>1</C> yields the embedding of <A>G</A>, calling
## <Ref Func="Embedding" Label="for a domain and a positive integer"/> with
## <M>P</M> and <C>2</C> yields the embedding of <A>N</A>; calling
## <Ref Func="Projection" Label="for a domain and a positive integer"/> with
## <A>P</A> yields the projection of <M>P</M> onto <A>G</A>,
## see <Ref Sect="Embeddings and Projections for Group Products"/>.
## <P/>
## <Example><![CDATA[
## 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 ... ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "SemidirectProduct",
[ IsGroup, IsGroupHomomorphism, IsObject ] );
#############################################################################
##
#O WreathProduct(<G>, <H>[, <hom>] )
##
## <#GAPDoc Label="WreathProduct">
## <ManSection>
## <Oper Name="WreathProduct" Arg='G, H[, hom]'/>
## <Oper Name="StandardWreathProduct" Arg='G, H'/>
##
## <Description>
## <C>WreathProduct</C>
## constructs the wreath product of the group <A>G</A> with the group
## <A>H</A>, acting as a permutation group.
## <P/>
## If a third argument <A>hom</A> is given, it must be
## a homomorphism from <A>H</A> into a permutation group,
## and the action of this group on its moved points is considered.
## <P/>
## If only two arguments are given, <A>H</A> must be a permutation group.
## <P/>
## <C>StandardWreathProduct</C> returns the wreath product for the (right
## regular) permutation action of <A>H</A> on its elements.
## <P/>
## <Index Key="Embedding" Subkey="example for wreath products">
## <C>Embedding</C></Index>
## <Index Key="Projection" Subkey="example for wreath products">
## <C>Projection</C></Index>
## For a wreath product <M>W</M> of <A>G</A> with a permutation group
## <M>P</M> of degree <M>n</M> and <M>1 \leq i \leq n</M> calling
## <Ref Func="Embedding" Label="for a domain and a positive integer"/> with
## <M>W</M> and <M>i</M> yields the embedding of <A>G</A> in the <M>i</M>-th
## component of the direct product of the base group <M><A>G</A>^n</M> of
## <M>W</M>.
## For <M>i = n+1</M>,
## <Ref Func="Embedding" Label="for a domain and a positive integer"/>
## yields the embedding of <M>P</M> into <M>W</M>. Calling
## <Ref Func="Projection" Label="for a domain and a positive integer"/> with
## <M>W</M> yields the projection onto the acting group <M>P</M>,
## see <Ref Sect="Embeddings and Projections for Group Products"/>.
## <P/>
## <Example><![CDATA[
## 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)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "WreathProduct", [ IsGroup, IsGroup ] );
DeclareOperation( "StandardWreathProduct", [ IsGroup, IsGroup ] );
#############################################################################
##
#F WreathProductImprimitiveAction(<G>, <H> )
##
## <#GAPDoc Label="WreathProductImprimitiveAction">
## <ManSection>
## <Func Name="WreathProductImprimitiveAction" Arg='G, H'/>
##
## <Description>
## For two permutation groups <A>G</A> and <A>H</A>,
## this function constructs the wreath product of <A>G</A> and <A>H</A>
## in the imprimitive action.
## If <A>G</A> acts on <M>l</M> points and <A>H</A> on <M>m</M> points
## this action will be on <M>l \cdot m</M> points,
## it will be imprimitive with <M>m</M> blocks of size <M>l</M> each.
## <P/>
## The operations <Ref Func="Embedding" Label="for two domains"/>
## and <Ref Func="Projection" Label="for two domains"/>
## operate on this product as described for general wreath products.
## <P/>
## <Example><![CDATA[
## gap> w:=WreathProductImprimitiveAction(g,p);;
## gap> LargestMovedPoint(w);
## 9
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "WreathProductImprimitiveAction" );
#############################################################################
##
#F WreathProductProductAction(<G>, <H> )
##
## <#GAPDoc Label="WreathProductProductAction">
## <ManSection>
## <Func Name="WreathProductProductAction" Arg='G, H'/>
##
## <Description>
## For two permutation groups <A>G</A> and <A>H</A>,
## this function constructs the wreath product in product action.
## If <A>G</A> acts on <M>l</M> points and <A>H</A> on
## <M>m</M> points this action will be on <M>l^m</M> points.
## <P/>
## The operations <Ref Func="Embedding" Label="for two domains"/>
## and <Ref Func="Projection" Label="for two domains"/>
## operate on this product as described for general wreath products.
## <Example><![CDATA[
## gap> w:=WreathProductProductAction(g,p);
## <permutation group of size 648 with 7 generators>
## gap> LargestMovedPoint(w);
## 27
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "WreathProductProductAction" );
#############################################################################
##
#F SubdirectProducts( <G>, <H> )
##
## <#GAPDoc Label="SubdirectProducts">
## <ManSection>
## <Func Name="SubdirectProducts" Arg='G, H'/>
##
## <Description>
## this function computes all subdirect products of <A>G</A> and <A>H</A> up
## to conjugacy in the direct product of Parent(<A>G</A>) and
## Parent(<A>H</A>).
## The subdirect products are returned as subgroups of this direct product.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "InnerSubdirectProducts" );
DeclareGlobalFunction( "InnerSubdirectProducts2" );
DeclareGlobalFunction( "SubdirectProducts" );
#############################################################################
##
#F FreeProduct( <G>{, <H>} )
#F FreeProduct( list )
##
## <#GAPDoc Label="FreeProduct">
## <ManSection>
## <Heading>FreeProduct</Heading>
## <Func Name="FreeProduct" Arg='G[, H, ...]' Label="for several groups"/>
## <Func Name="FreeProduct" Arg='list' Label="for a list"/>
##
## <Description>
## 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 <Ref Func="IsomorphismFpGroup"/> must be defined for them.
## <P/>
## The operation <Ref Func="Embedding" Label="for two domains"/>
## operates on this product.
## <Example><![CDATA[
## 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 ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("FreeProduct");
DeclareOperation( "FreeProductOp", [ IsList, IsGroup ] );
#############################################################################
##
#A DirectProductInfo( <G> )
##
## <ManSection>
## <Attr Name="DirectProductInfo" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "DirectProductInfo", IsGroup, "mutable" );
#############################################################################
##
#A SubdirectProductInfo( <G> )
##
## <ManSection>
## <Attr Name="SubdirectProductInfo" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "SubdirectProductInfo", IsGroup, "mutable" );
#############################################################################
##
#A SemidirectProductInfo( <G> )
##
## <ManSection>
## <Attr Name="SemidirectProductInfo" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "SemidirectProductInfo", IsGroup, "mutable" );
#############################################################################
##
#A WreathProductInfo( <G> )
##
## <ManSection>
## <Attr Name="WreathProductInfo" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "WreathProductInfo", IsGroup, "mutable" );
#############################################################################
##
#A FreeProductInfo( <G> )
##
## <ManSection>
## <Attr Name="FreeProductInfo" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "FreeProductInfo", IsGroup, "mutable" );
#############################################################################
##
#F SubdirProdPcGroups( <G>,<gi>,<H>,<hi> )
##
## <ManSection>
## <Func Name="SubdirProdPcGroups" Arg='G,gi,H,hi'/>
##
## <Description>
## Let <A>G</A> and <A>H</A> be two pc groups which are both projections of a
## subdirect product with generator images <A>gi</A> and <A>hi</A>. the function
## returns a list <A>l</A> with <A>l</A>[1] a new pc group and <A>l</A>[2] a corresponding
## generator images list.
## <P/>
## No parameter checking is done.
## (This function is used in a variant of the SQ.)
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SubdirProdPcGroups" );
#############################################################################
##
#C IsWreathProductElement
#C IsWreathProductElementCollection
##
## <ManSection>
## <Filt Name="IsWreathProductElement" Arg='obj' Type='Category'/>
## <Filt Name="IsWreathProductElementCollection" Arg='obj' Type='Category'/>
##
## <Description>
## categories for elements of generic wreath products: elements are stored
## as list of base components and permutation.
## </Description>
## </ManSection>
##
DeclareCategory("IsWreathProductElement",
IsMultiplicativeElementWithInverse and IsAssociativeElement);
DeclareCategoryCollections("IsWreathProductElement");
InstallTrueMethod(IsGeneratorsOfMagmaWithInverses,
IsWreathProductElementCollection);
DeclareRepresentation("IsWreathProductElementDefaultRep",
IsWreathProductElement and IsPositionalObjectRep,[]);
#############################################################################
##
#E