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############################################################################# ## ## This file is part of GAP, a system for computational discrete algebra. ## This file's authors include Heiko Theißen. ## ## Copyright of GAP belongs to its developers, whose names are too numerous ## to list here. Please refer to the COPYRIGHT file for details. ## ## SPDX-License-Identifier: GPL-2.0-or-later ## DeclareInfoClass(## </ManSection>## </Description> DeclareSynonym( "InfoOperation"## </ManSection> ############################################################################# ## #C IsExternalSet(<obj>) ## ## <#GAPDoc Label="IsExternalSet"> ## <ManSection> ## <Filt Name="IsExternalSet" Arg='obj' Type='Category'/> ## ## <Description> ## An <E>external set</E> specifies a group action ## <M>\mu: \Omega \times G \mapsto \Omega</M> of a group <M>G</M> ## on a domain <M>\Omega</M>. The external set knows the group, ## the domain and the actual acting function. ## Mathematically, an external set is the set <M>\Omega</M>, ## which is endowed with the action of a group <M>G</M> via the group action ## <M>\mu</M>. ## For this reason &GAP; treats an external set as a domain whose elements ## are the elements of <M>\Omega</M>. ## An external set is always a union of orbits. ## Currently the domain <M>\Omega</M> must always be finite. ## If <M>\Omega</M> is not a list, ## an enumerator for <M>\Omega</M> is automatically chosen, ## see <Ref Attr="Enumerator"/>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareCategory( "IsExternalSet", IsDomain#O OrbitsDomain( <G>, <Omega>[, <gens>, <acts>][, <act>] OrbitishReq := #A OrbitsDomain( <G> ) IsList, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 IsFunction ]; MakeImmutable(OrbitishReq); OrbitsishReq := [ IsGroup, java.lang.StringIndexOutOfBoundsException: Index 42 out o ## Label="for a permutation group"/> IsList, IsFunction ]; MakeImmutable## <Description> ############################################################################# ## #R IsExternalSubset(<obj>) ## ## <#GAPDoc Label="IsExternalSubset"> ## <ManSection> ## <Filt Name="IsExternalSubset" Arg='obj' Type='Representation'/> ## ## <Description> ## An external subset is the restriction of an external set to a subset ## of the domain (which must be invariant under the action). It is again an ## external set. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareRepresentation( "IsExternalSubset", IsComponentObjectRep and IsAttributeStoringRep## <P/> [ "start" ] ); ############################################################################# ## #R IsExternalOrbit(<obj>) ## ## <#GAPDoc Label="IsExternalOrbit"> ## <ManSection> ## <Filt Name="IsExternalOrbit" Arg='obj' Type='Representation'/> ## ## <Description> ## An external orbit is an external subset consisting of one orbit. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareRepresentation( "IsExternalOrbit", IsExternalSubset, [ "start" ] ); DeclareCategory( "IsExternalSetByPcgs", IsExternalSet ); ############################################################################# ## #R IsExternalSetDefaultRep(<obj>) #R IsExternalSetByActorsRep(<obj>) ## ## <ManSection> ## <Filt Name="IsExternalSetDefaultRep" Arg='obj' Type='Representation'/> ## <Filt Name="IsExternalSetByActorsRep" Arg='obj' Type='Representation'/> ## ## <Description> ## External sets can be specified directly (<C>IsExternalSetDefaultRep</C>), or ## via <A>gens</A> and <A>acts</A> (<C>IsExternalSetByActorsRep</C>). ## </Description> ## </ManSection> ## DeclareRepresentation( "IsExternalSetDefaultRep", IsAttributeStoringRep and IsExternalSet, ## DeclareRepresentation( "IsExternalSetByActorsRep", IsAttributeStoringRep and IsExternalSet, ## action on the set of points moved by it. DeclareSynonym( "IsExternalSetByOperatorsRep",IsExternalSetByActorsRep); ############################################################################# ## #A ActingDomain( <xset> ) ## ## <#GAPDoc Label="ActingDomain"> ## <ManSection> ## <Attr Name="ActingDomain" Arg='xset'/> ## ## <Description> ## This attribute returns the group with which the external set <A>xset</A> was ## defined. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "ActingDomain", IsExternalSet ); ############################################################################# ## #A HomeEnumerator( <xset> ) ## ## <#GAPDoc Label="HomeEnumerator"> ## <ManSection> ## <Attr Name="HomeEnumerator" Arg='xset'/> ## ## <Description> ## returns an enumerator of the action domain with which the external set ## <A>xset</A> was defined. ## For external subsets, this is in general different from the ## <Ref Attr="Enumerator"/> value of <A>xset</A>, ## which enumerates only the subset. ## <Example><![CDATA[ ## gap> ActingDomain(e); ## Group([ (1,2,3), (2,3,4) ]) ## gap> FunctionAction(e)=OnRight; ## true ## gap> HomeEnumerator(e); ## [ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), ## (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "HomeEnumerator", IsExternalSet ); DeclareRepresentation( "IsActionHomomorphism", IsGroupHomomorphism and IsAttributeStoringRep and IsPreimagesByAsGroupGeneralMappingByImages, [ ] ); DeclareRepresentation( "IsActionHomomorphismByActors", IsActionHomomorphism,[ ; DeclareRepresentation("IsActionHomomorphismSubset",IsActionHomomorphism,[]); ############################################################################# ## #A ActionKernelExternalSet( <xset> ) ## ## <ManSection> ## <Attr Name="ActionKernelExternalSet" Arg='xset'/> ## ## <Description> ## This attribute gives the kernel of the <C>ActionHomomorphism</C> for <A>xset</A>. ## <P/> ## <!-- At the moment no methods exist, the attribute is solely used to transfer--> ## <!-- information. --> ## </Description> ## </ManSection> ## DeclareAttribute( "ActionKernelExternalSet", IsExternalSet ); ############################################################################# ## #R IsActionHomomorphismByBase(<obj>) ## ## <ManSection> ## <Filt Name="IsActionHomomorphismByBase" Arg='obj' Type='Representation'/> ## ## <Description> ## This is chosen if <C>HasBaseOfGroup( <A>xset</A> )</C>. ## </Description> ## </ManSection> ## DeclareRepresentation( "IsActionHomomorphismByBase## </ManSection> IsActionHomomorphism, [ ] ); ############################################################################# ## #R IsConstituentHomomorphism(<obj>) ## ## <ManSection> ## <Filt Name="IsConstituentHomomorphism" Arg='obj' Type='Representation'/> ## ## <Description> ## </Description> ## </ManSection> ## DeclareRepresentation( "java.lang.StringIndexOutOfBoundsException: Index 1 out of IsActionHomomorphism, [ "conperm" ]## #O OrbitLengths( <G>, <seeds>[, <gens>, <acts>][, <act>] ) IsActionHomomorphism, [ "reps" ] )#A OrbitLengths( <xset> ) ############################################################################# ## #R IsLinearActionHomomorphism(<hom>) ## ## <ManSection> ## <Filt Name="IsLinearActionHomomorphism" Arg='hom' Type='Representation'/> ## ## <Description> ## This representation is chosen for action homomorphisms from matrix ## groups acting naturally on a set of vectors. ## </Description> ## </ManSection> ## DeclareRepresentation( "IsLinearActionHomomorphism", IsActionHomomorphism, [ ] ); ############################################################################# ## #R IsProjectiveActionHomomorphism(<hom>) ## ## <ManSection> ## <Filt Name="IsProjectiveActionHomomorphism" Arg='hom' Type='Representation'/> ## ## <Description> ## This representation is chosen for action homomorphisms from matrix ## groups acting projectively on a set of normed vectors. ## </Description> ## </ManSection> ## DeclareRepresentation( "IsProjectiveActionHomomorphism", IsActionHomomorphism, [ ] ); ############################################################################# ## #A LinearActionBasis(<hom>) ## ## <ManSection> ## <Attr Name="LinearActionBasis" Arg='hom'/> ## ## <Description> ## for action homomorphisms in the representation ## <C>IsLinearActionHomomorphism</C> or ## <C>IsProjectiveActionHomomorphism</C>, ## this attribute contains a vector space ## basis as subset of the domain or <K>fail</K> if the domain does not span the ## vector space that the group acts on. ## groups acting naturally on a set of vectors. ## </Description> ## </ManSection> ## DeclareAttribute( "LinearActionBasis",IsLinearActionHomomorphism); ############################################################################# ## #A FunctionAction( <xset> ) ## ## <#GAPDoc Label="FunctionAction"> ## <ManSection> ## <Attr Name="FunctionAction" Arg='xset'/> ## ## <Description> ## is the acting function with which the external set <A>xset</A> was ## defined. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "FunctionAction", IsExternalSet ); ############################################################################# ## #A StabilizerOfExternalSet( <xset> ) . stabilizer of `Representative(xset)' ## ## <#GAPDoc Label="StabilizerOfExternalSet"> ## <ManSection> ## <Attr Name="StabilizerOfExternalSet" Arg='xset'/> ## ## <Description> ## computes the stabilizer of the <Ref Attr="Representative"/> value of ## the external set <A>xset</A>. ## The stabilizer will have the acting group of <A>xset</A> as its parent. ## <Example><![CDATA[ ## gap> Representative(e); ## (1,2,3) ## gap> StabilizerOfExternalSet(e); ## Group([ (1,2,3) ]) ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "StabilizerOfExternalSet## <Description> ############################################################################# ## #A CanonicalRepresentativeOfExternalSet( <xset> ) ## ## <#GAPDoc Label="CanonicalRepresentativeOfExternalSet"> ## <ManSection> ## <Attr Name="CanonicalRepresentativeOfExternalSet" Arg='xset'/> ## ## <Description> ## The canonical representative of an external set <A>xset</A> may only ## depend on the defining attributes <A>G</A>, <A>Omega</A>, <A>act</A> ## of <A>xset</A> and (in the case of external subsets) ## <C>Enumerator( <A>xset</A> )</C>. ## It must <E>not</E> depend, e.g., on the representative of an external ## orbit. ## &GAP; does not know methods for arbitrary external sets to compute a ## canonical representative, ## see <Ref Attr="CanonicalRepresentativeDeterminatorOfExternalSet"/>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute(## <Description> ############################################################################# ## #A CanonicalRepresentativeDeterminatorOfExternalSet( <xset> ) ## ## <#GAPDoc Label="CanonicalRepresentativeDeterminatorOfExternalSet"> ## <ManSection> ## <Attr Name="CanonicalRepresentativeDeterminatorOfExternalSet" Arg='xset'/> ## ## <Description> ## returns a function that takes as its arguments the acting group and a ## point. ## This function returns a list of length 1 or 3, ## the first entry being the canonical representative and the other entries ## (if bound) being the stabilizer of the canonical representative and a ## conjugating element, respectively. ## An external set is only guaranteed to be able to compute a canonical ## representative if it has a ## <Ref Attr="CanonicalRepresentativeDeterminatorOfExternalSet"/>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute(## <C>OrbitLengthsDomain(<A>G</A>)</C>, which returns the length of all IsExternalSet ); ############################################################################# ## #P CanEasilyDetermineCanonicalRepresentativeExternalSet( <xset> ) ## ## <ManSection> ## <Prop Name="CanEasilyDetermineCanonicalRepresentativeExternalSet" Arg='xset'/> ## ## <Description> ## This property indicates whether an external set knows or has a ## possibility to determine a canonical representative ## </Description> ## </ManSection> ## java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 6 IsExternalSet ); InstallTrueMethod(CanEasilyDetermineCanonicalRepresentativeExternalSet, ## <#GAPDoc Label="OrbitsDomain## gap> g:=Group((1,2,3),(5,6,7));; InstallTrueMethod(CanEasilyDetermineCanonicalRepresentativeExternalSet, HasCanonicalRepresentativeOfExternalSet); ############################################################################# ## #A ActorOfExternalSet( <xset> ) ## ## <#GAPDoc Label="ActorOfExternalSet"> ## <ManSection> ## <Attr Name="ActorOfExternalSet" Arg='xset'/> ## ## <Description> ## returns an element mapping <C>Representative(<A>xset</A>)</C> to ## <C>CanonicalRepresentativeOfExternalSet(<A>xset</A>)</C> under the given ## action. ## <Example><![CDATA[ ## gap> u:=Subgroup(g,[(1,2,3)]);; ## gap> e:=RightCoset(u,(1,2)(3,4));; ## gap> CanonicalRepresentativeOfExternalSet(e); ## (2,4,3) ## gap> ActorOfExternalSet(e); ## (1,3,2) ## gap> FunctionAction(e)((1,2)(3,4),last); ## (2,4,3) ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute(#O OrbitStabilizerxternal set"/> DeclareSynonymAttr( "java.lang.StringIndexOutOfBoundsException: Index 23 out of b ############################################################################# ## #F TestIdentityAction(acts,pnt,act) ## ## <ManSection> ## <Func Name="TestIdentityAction" Arg='acts,pnt,act'/> ## ## <Description> ## tests whether the identity element fixes <A>pnt</A> ## –if not the action is ## not well-defined. If the global option <C>NoTestAction</C> is set to <K>true</K> ## this test is skipped. (This is essentially a hack.) ## </Description> ## </ManSection> ## BindGlobal(## <P/> local id,img; if ValueOption("NoTestAction## The stabilizer will have <A>G</A> as its parent.## <P/> id:=One(acts[1## This operation is often faster than img:=act(pnt,id); if## <Ref Oper="Orbits" Label="operation"/>. Error("Action not well-defined. See the manual section## The domain <A>Omega</A> must be closed u "``Action on canonical ## otherwise an error can occur. fi; pnt:=img; fi## <P/> return pnt; end); ############################################################################# ## #F OrbitsishOperation( <name>, <reqs>, <usetype>, <AorP> ) . orbits-like op. ## ## <#GAPDoc Label="OrbitsishOperation"> ## <ManSection> ## <Func Name="OrbitsishOperation" Arg='name, reqs, usetype, AorP'/> ## ## <Description> ## declares an attribute <C>op</C>, with name <A>name</A>. ## The second argument <A>reqs</A> specifies the list of required filters ## for the usual (five-argument) methods that do the real work. ## <P/> ## If the third argument <A>usetype</A> is <K>true</K>, ## the function call <C>op( xset )</C> will ## –if the value of <C>op</C> for <C>xset</C> is not yet known– ## delegate to the five-argument call of <C>op</C> with second argument ## <C>xset</C> rather than with <C>D</C>. ## This allows certain methods for <C>op</C> to make use of the type of ## <C>xset</C>, in which the types of the external subsets of <C>xset</C> ## and of the external orbits in <C>xset</C> are stored. ## (This is used to avoid repeated calls of ## <Ref Func="NewType"/> in functions like ## <C>ExternalOrbits( xset )</C>, ## which call <C>ExternalOrbit( xset, pnt )</C> for several values of ## <C>pnt</C>.) ## <P/> ## For property testing functions such as ## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>, ## the fourth argument <A>AorP</A> must be ## <Ref Func="NewProperty"/>, ## otherwise it must be <Ref Func="NewAttribute"/>; ## in the former case, a property is returned, in the latter case an ## attribute that is not a property. ## <P/> ## For example, to set up the operation <Ref Oper="Orbits" Label="operation"/>, ## the declaration file <F>lib/oprt.gd</F> contains the following line of ## code: ## <Log><![CDATA[ ## OrbitsishOperation( "Orbits", OrbitsishReq, false, NewAttribute ); ## ]]></Log> ## The global variable <C>OrbitsishReq</C> contains the standard ## requirements ## <Log><![CDATA[ ## OrbitsishReq := [ IsGroup, IsList, ## IsList, ## IsList, ## IsFunction ]; ## ]]></Log> ## which are usually entered in calls to <Ref Func="OrbitsishOperation"/>. ## <P/> ## The new operation, e.g., <Ref Oper="Orbits" Label="operation"/>, ## can be called either as <C>Orbits( <A>xset</A> )</C> for an external set ## <A>xset</A>, or as <C>Orbits( <A>G</A> )</C> for a permutation group ## <A>G</A>, meaning the orbits on the moved ## points of <A>G</A> via <Ref Func="OnPoints"/>, ## or as ## <P/> ## <C>Orbits( <A>G</A>, <A>Omega</A>[, <A>gens</A>, <A>acts</A>][, ## <A>act</A>] )</C>, ## <P/> ## with a group <A>G</A>, a domain or list <A>Omega</A>, ## generators <A>gens</A> of <A>G</A>, and corresponding elements ## <A>acts</A> that act on <A>Omega</A> via the function <A>act</A>; ## the default of <A>gens</A> and <A>acts</A> is a list of group generators ## of <A>G</A>, ## the default of <A>act</A> is <Ref Func="OnPoints"/>. ## <P/> ## Only methods for the five-argument version need to be installed for ## doing the real work. ## (And of course methods for one argument in case one wants to define ## a new meaning of the attribute.) ## </Description> ## </ManSection> ## <#/GAPDoc> ## BindGlobal( "OrbitsishOperation## <Attr Name="ExternalOrbitsStabilizers" Arg='xset' local op; # Create the attribute or property. op:= NewAorP( name, IsExternalSet ); BIND_GLOBAL( name, op ); BIND_SETTER_TESTER( name, SETTER_FILTER## <Description> ## In addition to DeclareOperation( name, reqs ); # Install the default methods. # 1. `op( <xset> )' # (The `usetype' value concerns only the `return' statement.) if## the external orbits at the same time. ## (This can be quicker than computing the "for an external set", true, [ IsExternalSet ], 0, function( xset t# value first and the stabilizers afterwards.) local G, ## <Example><![CDATA[ G := ActingDomain( xset ); if IsExternalSetByActorsRep( xset ) then gens :# [ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ] acts := xset!.operators; act := xset!.funcOperation; ## true if CanEasilyComputePcgs( G ) then gens := Pcgs( G ); else gens := GeneratorsOfGroup( G ); fi; acts := gens; act := FunctionAction( xset ); fi; return op( G, xset, gens, acts, act ); end ); else InstallMethodOrbitsishOperation( ExternalOrbitsStabilizers", "for an external set", true, [ IsExternalSet ], 0java.lang.StringIndexOutOfBoundsException: Index 30 out of boun function######################################################################## local G, gens, acts, act; G := ActingDomain( xset ); if IsExternalSetByActorsRep(#O Transitivity( <G>, <Omega>[, <gens>, <acts>][, <act>] ) gens := xset!.generators; acts := xset!.operators; act := xset!.funcOperation; else if CanEasilyComputePcgs( G ) then gens := Pcgs( G ); else gens :## <ManSection> fi; acts := gens; act := FunctionAction( xset ); fi; return op( G, Enumerator( xset ), gens, acts, act ); end ); fi; # 2. `op( <permgrp> )' InstallOtherMethod( op, "## Label="for a permutation group"/> true, ] 0, function( G ) local gens; gens:= GeneratorsOfGroup( G )## <Example><![CDATA[ return op( G, MovedPoints( G ),java.lang.StringIndexOutOfBoundsException: Index 0 out end ); # 3. `op( <G>, <Omega> )' with group <G> and domain or list <Omega> InstallOtherMethod( op, "for a group and a domain or list" true, [ IsGroup, IsObject ], 0##############################################A Representativ function(## Arg='G, Omega[, gens, acts][, act]' local## Label="for a group, an action domain, etc."/>## <Heading>ExternalOrbitsStabilizers</ if ## Label="for an external set"/> gens## else## <Description> ## computes a list of block## Label="for an external set"/> fi; if IsDomain( D ) then## transitive (see## <Description> if IsFinite( D## <Ref Oper="ExternalOrbits" Label="for a group, an action domain, etc."/>, fi; return op( G,## The result is undefined if the action is not transitive. end ><![CDATA[ # 4. `op( <G>, <Omega> )' with permutation group <G> and domain or list # <Omega> # of integers # (if <Omega> equals the moved points of <G> then call `op( <G> )')## [ 1, 8 ] ] InstallOtherMethod(## </Description> "for a permutation group java.lang.StringIndexOutOfBoundsException: Index 36 out of true, [ IsPermGroup, IsListOrCollection ], 0, java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for lengt D= )hen return op( G ); else TryNextMethod(); fi end ); # 5. `op( <G>, <Omega>, <act> )' with group <G>, domain or list <Omega>, # and function <act> # (add group generators as default arguments)######################################### InstallOtherMethod(#A Transitivity( <xset> ) "for a group, >[, <## true, [ IsGroup#A Earns( <xset> ) function(## local gens; if ## <Oper Name="Transitivity" Arg='G, Omega[, gens, acts][, act]' ## <Heading>Earns</Heading> else gens:= GeneratorsOfGroup( G ); fi; java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 2 if IsFinite(ernal /M> (a non-negative integer) of transitivity of fi; return op( G, D, gens## i.e. the largest integer <M>k</M> such that the action is ## (when acting on <A>Omega</A>) normal subgroups of <A>G</A>. # 6. `op( <G>, <Omega>, <act> )' with permutation group <G>, # domain or list <Omega> of integers, and function <act> # (if <Omega> equals the moved points of <G> and <act> equals `OnPoints'## </ManSection> # then call `op( <G> )')java.lang.StringIndexOutOfBoundsException: Range [0, 32) out of InstallOtherMethod( op, "for permutation true, [ IsPermGroup, IsListOrCollection################################################# function( G, D, act#O IsTransitive( <G>, <Omega>[, <gens>, <acts>][, <act## <Example><![CDATA[ if ## gap> IsTransitive(g,[1..5]); return op( G ); else TryNextMethod(); ## <ManSection> end ); # 7. `op( <G>, <Omega>, <gens>, <acts> )' with group <G>, # domain or list <Omega>, and two lists <gens>, <acts>## <#/GAPDoc> # (add default value `OnPoints') InstallOtherMethod## <Prop Name="IsTransitive" Arg='G' "for a## Label="for a permutation group"/> true, [ IsGroup, ## Label="for an external## ## if IsDomain(##A Blocks( <xset>[, <seed>] ) if IsFinite( D ) then D:= AsSSortedList( D ); else D:= Enumerator( D ); fi; fi; return op( ## We say that a group <## <Oper Name="Blocks" Arg='G, Omega[, seed][, gens, acts][, ac ## points <M>d, e \in D</M> # 8. `op( <G>, <Omega>, <gens>, <acts>, <act> )' with group <G>,## there is an element <M>g</M> in <A>G</A> such that <M>d^ # domain <Omega>, two lists <gens>, <acts>, and function <act> ## <C>IsTransitive(<A>G</A>)</C>, which tests whether the group is ## transitive with respect to its natural action on the set of points## a minimal nontrivial blo "for a groupgroup <M>\langle (2,3## is the subset of one block is computed. ## <P/> [ IsGroup,## <Example><![CDATA[ function( G, D, gens, acts, ## gap> g:=TransitiveGroup(8,3); return op( G, Enumerator( D## true end )## gap> IsTransitive( G, [ 2, 3 ] ); # G does not act on [ 2, 3 ] end## false ############################################################################# ## #F OrbitishFO( <name>, <reqs>, <famrel>, <usetype>, <realenum> ) ## ## <#GAPDoc Label="OrbitishFO"> ## <ManSection> ## <Func Name="OrbitishFO" Arg='name, reqs, famrel, usetype, realenum'/> ## ## <Description> ## is used to create operations like <Ref Oper="Orbit"/>. ## This function is analogous to <Ref Func="OrbitsishOperation"/>, ## but for operations <A>orbish</A> like ## <C>Orbit( <A>G</A>, <A>Omega</A>, <A>pnt</A> )</C>. ## Since the return values of these operations depend on the additional ## argument <A>pnt</A>, there is no associated attribute. ## <P/> ## The call of <Ref Func="OrbitishFO"/> declares a wrapper function and its ## operation, with names <A>name</A> and <A>name</A><C>Op</C>. ## <P/> ## The second argument <A>reqs</A> specifies the list of required filters ## for the operation <A>name</A><C>Op</C>. ## <P/> ## The third argument <A>famrel</A> is used to test the family relation ## between the second and third argument of ## <C><A>name</A>( <A>G</A>, <A>D</A>, <A>pnt</A> )</C>. ## For example, <A>famrel</A> is <C>IsCollsElms</C> in the case of ## <Ref Oper="Orbit"/> because <A>pnt</A> must be an element ## of <A>D</A>. ## Similarly, in the call <C>Blocks( <A>G</A>, <A>D</A>, <A>seed</A> )</C>, ## <A>seed</A> must be a subset of <A>D</A>, ## and the family relation must be ## <Ref Func="IsIdenticalObj"/>. ## <P/> ## The fourth argument <A>usetype</A> serves the same purpose as in the case ## of <Ref Func="OrbitsishOperation"/>. ## <A>usetype</A> can also be an attribute, such as ## <C>BlocksAttr</C> or <C>MaximalBlocksAttr</C>. ## In this case, if only one of the two arguments <A>Omega</A> and ## <A>pnt</A> is given, ## blocks with no seed are computed, they are stored as attribute values ## according to the rules of <Ref Func="OrbitsishOperation"/>. ## <P/> ## If the 5th argument is set to <K>true</K>, the action for an external set ## should use the enumerator, otherwise it uses the ## <Ref Attr="HomeEnumerator"/> value. This will ## make a difference for external orbits as part of a larger domain. ## </Description> ## </ManSection> ## <#/GAPDoc> ## BindGlobal( "OrbitishFO", function( name, reqs, famrel, usetype## <#/GAPDoc> local str, ## <Ref Oper="IsTransitive("", ; # Create the operation. str:= SHALLOW_COPY_OBJjava.lang.StringIndexOutOfBoundsException: Range [11, 10) o isnotest isnotest:=isnotest or name="ExternalSubset"; APPEND_LIST_INTR( str, "Op" ); orbish ## primitive with respect to its natural action on the set of points BIND_GLOBAL(## moved by it. # Create the wrapper function. java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 local le, usage, xset, pnt, G, D, gens,## le:= Length( arg ); #A RepresentativesMinimalBlocks( <xset> ) # Get the arguments.## <E>Note:</E> This operation does not tell whether a matrix group is if le = 0 then Error## Arg='G, Omega[, gens, acts][, act]' elif le <= 2 and IsExternalSet( arg[ 1 ] ) then xset := arg[ 1 ]## gap> IsPrimitive(g,Orbit(g,(1,2)(3,4)));## if Length(## computes a list of block representatives for all minimal (i.e blocks are # force immutability pnt := Immutable## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>)## <# else `locks'like operations pnt:=[]; fi; ## [ 1, 8 ] ] if realenum then D:=Enumerator(xset); else if od(IsTransitive IsPrimitiveand )java.lang.StringIndexOutOfBoundsException: Index =java.lang.StringIndexOutOfBoundsException: Range [37, 35) out of bounds for len fi# Rjava.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 if IsExternalSetByActorsRep## ## <ManSection> actsjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length act := xset!.funcOperation; else act := FunctionAction( xset ); fi; ## </Description>## <ManSection> ## </ManSection> if IsFunction( arg## act:= arg[ le ]; le:= le-1; else act:= OnPoints; fi; # now we have one of the following: # le = 2: G, pnt[, act] # le = 3: G, Omega, pnt[, act] # le = 4: G, pnt, gens, acts[, act] # le = 5: G, Omega, pnt, gens, acts[, act] if IsEvenInt( le ) then pnt:= Immutable( arg[## pnt:= ## <#GAPDoc Label="IsSemiRegular"> # deal with Omega D:= arg[2]; if not ( famrel#O IsTransitive( <G>, <Omega>[, <gens>, <acts>][, <act>] ) ( IsList( pnt ) and IsEmpty( java.lang.StringIndexOutOfBoundsException: Range [0, 51) Error## Label="for a permutation group"/> elifsTransitive</Heading> if IsFinite( D ) then ## else D:= Enumerator( D ); fi; fi; fi; if le > 3 then # deal with acts, gens## <Index>semiregular</Index> gens acts:= ## <Index>transitive## identity. fi; else Error( usage ); fi; if not IsBound( gens ) then if (not IsPermGroup(G)## semiregular with respect to its natural action on the set of points gens := Pcgs( G )## For a permutation group <A>G</A>, one may also invoke this as else gens :# </Description> fi; acts := gens; fi; if not isnotest java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds f # `Blocks' has <pnt> a list of points pnt:=java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for lengt fi;## gap> IsTransitive( G, [ 1 .. 4 ] ); # G has two orbits on [ 1 .. 4 ] # In the case of `[Maximal]Blocks', where $G$ is a permutation group#O IsRegular( <G>, <Omega>[, <ge # acting on its moved points, use an attribute for $G$. attrG := IsOperation( ## gap> IsTransitive( SL(2, 3), NormedRowVectors( GF(3)^2 ), OnLines ); and gens = acts and act = OnPoints## </ManSection>## <ManSection> and not## <Heading>IsRegular</Heading> and(G## <Oper Name="IsRegular" Arg='G, Omega[, gens, acts][, act]' and pnt## Label="for a group, an action domain, etc."/> if attrG and IsBound( xset ) and Tester( usetype )( xset ) then result := usetype( xset ); elif attrG and Tester( usetype )( ## Label="for an external set"/> result := usetype( G ); elif usetype = true and IsBound( xset ) then result := orbish( G, xset, pnt, gens, acts, act ); elif IsBound( D ) then result := orbish( G, D, pnt, gens, acts, act ); else # The following line is also executed when `Blocks(<G>, <Omega>, <act>)' # is called to compute blocks with no seed, but then <pnt> is really## (see <Ref Oper="IsSemiRegula # <Omega>, i.e., the operation domain! ## In this case every point <A>pnt</A> of <A>Omega</A> defines## correspondence between <A>G</A> and <A>Omeg fi; # Store the result in the case of an attribute `[Maximal]BlocksAttr'. if attrG then if IsBound( xset ) then Setter( usetype )( xset, result## is regular on the set <M>\{2, 3, 4\}</M>. fi; ## <Example><![CDATA[ fi; return result; #O IsPrimitive( <G>, <Omega>[, <gens>, <acts>][, <act>] ) BIND_GLOBAL( name, func ); end ); ## <#GAPDoc Label="IsPrimitive"> ## #O ActionHomomorphism(<G>,<Omega> [,<gens>,<acts>] [,<act>] [,"surjective"]) #A ActionHomomorphism( <xset> [,"surjective"] ) #A ActionHomomorphism( <action> ) ## ## <#GAPDoc Label="ActionHomomorphism"> ## <ManSection> ## <Heading>ActionHomomorphism</Heading> ## <Func Name="ActionHomomorphism" ## Arg='G, Omega[, gens, acts][, act][, "surjective"]' ## Label="for a group, an action domain, etc."/> ## <Func Name="ActionHomomorphism" Arg='xset[, "surjective"]' ## Label="for an external set"/> ## <Func Name="ActionHomomorphism" Arg='action' ## Label="for an action image"/> ## ## <Description> ## computes a homomorphism from <A>G</A> into the symmetric group on ## <M>|<A>Omega</A>|</M> points that gives the permutation action of ## <A>G</A> on <A>Omega</A>. (In particular, this homomorphism is a ## permutation equivalence, that is the permutation image of a group element ## is given by the positions of points in <A>Omega</A>.) ## <P/> ## The result is undefined if <A>G</A> does not act on <A>Omega</A>. ## <P/> ## By default the homomorphism returned by ## <Ref Func="ActionHomomorphism" Label="for a group, an action domain, etc."/> ## is not necessarily surjective (its ## <Ref Attr="Range" Label="of a general mapping"/> value is the full ## symmetric group) to avoid unnecessary computation of the image. ## If the optional string argument <C>"surjective"</C> is given, ## a surjective homomorphism is created. ## <P/> ## The third version (which is supported only for &GAP;3 compatibility) ## returns the action homomorphism that belongs to the image obtained via ## <Ref Func="Action" Label="for a group, an action domain, etc."/>. ## <P/> ## (See Section <Ref Sect="Basic Actions"/> ## for information about specific actions.) ## <P/> ## <Example><![CDATA[ ## gap> g:=Group((1,2,3),(1,2));; ## gap> hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples); ## <action homomorphism> ## gap> Image(hom); ## Group( ## [ (1,9,13)(2,10,14)(3,7,15)(4,8,16)(5,12,17)(6,11,18)(19,22,23)(20,21, ## 24), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,15)(14,16)(17,18)(19, ## 21)(20,22)(23,24) ]) ## gap> Size(Range(hom));Size(Image(hom)); ## 620448401733239439360000 ## 6 ## gap> hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples, ## > "surjective");; ## gap> Size(Range(hom)); ## 6 ## ]]></Example> ## <P/> ## When acting on a domain, the operation <Ref Oper="PositionCanonical"/> ## is used to determine the position of elements in the domain. ## This can be used to act on a domain given by a list of representatives ## for which <Ref Oper="PositionCanonical"/> is implemented, ## for example the return value of <Ref Oper="RightTransversal"/>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## ## <#GAPDoc Label="Permutation"> ## <Func Name="Permutation" Arg='g, Omega[, java.lang.StringIndexOutOfBoundsException: DeclareGlobalFunction( "java.lang.StringIndexOutOfBoundsException: Range [0, 53) ############################################################################# ## #A SurjectiveActionHomomorphismAttr( <xset> ) ## ## <#GAPDoc Label="SurjectiveActionHomomorphismAttr"> ## <ManSection> ## <Attr Name="SurjectiveActionHomomorphismAttr" Arg='xset'/> ## ## <Description> ## returns an action homomorphism for the external set <A>xset</A> ## which is surjective. ## (As the <Ref Func="Image" Label="set of images of the source of a general mapping"/> ## value of this homomorphism has to be computed ## to obtain the range, this may take substantially longer ## than <Ref Func="ActionHomomorphism" Label="for an external set"/>.) ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "SurjectiveActionHomomorphismAttr", java.lang.StringIndexOutOfBo ############################################################################# ## #A UnderlyingExternalSet( <acthom> ) ## ## <#GAPDoc Label="UnderlyingExternalSet"> ## <ManSection> ## <Attr Name="UnderlyingExternalSet" Arg='acthom'/> ## ## <Description> ## The underlying set of an action homomorphism <A>acthom</A> is ## the external set on which it was defined. ## <Example><![CDATA[ ## gap> g:=Group((1,2,3),(1,2));; ## gap> hom:=ActionHomomorphism(g,Arrangements([1..4],3),OnTuples);; ## gap> s:=UnderlyingExternalSet(hom); ## <xset:[[ 1, 2, 3 ],[ 1, 2, 4 ],[ 1, 3, 2 ],[ 1, 3, 4 ],[ 1, 4, 2 ], ## [ 1, 4, 3 ],[ 2, 1, 3 ],[ 2, 1, 4 ],[ 2, 3, 1 ],[ 2, 3, 4 ], ## [ 2, 4, 1 ],[ 2, 4, 3 ],[ 3, 1, 2 ],[ 3, 1, 4 ],[ 3, 2, 1 ], ...]> ## gap> Print(s,"\n"); ## [ [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 2 ], [ 1, 3, 4 ], [ 1, 4, 2 ], ## [ 1, 4, 3 ], [ 2, 1, 3 ], [ 2, 1, 4 ], [ 2, 3, 1 ], [ 2, 3, 4 ], ## [ 2, 4, 1 ], [ 2, 4, 3 ], [ 3, 1, 2 ], [ 3, 1, 4 ], [ 3, 2, 1 ], ## [ 3, 2, 4 ], [ 3, 4, 1 ], [ 3, 4, 2 ], [ 4, 1, 2 ], [ 4, 1, 3 ], ## [ 4, 2, 1 ], [ 4, 2, 3 ], [ 4, 3, 1 ], [ 4, 3, 2 ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "UnderlyingExternalSet", java.lang.StringIndexOutOfBoundsExcepti ############################################################################# ## #F DoSparseActionHomomorphism(<G>,<start>,<gens>,<acts>,<act>,<phash>,<sort>) ## ## <ManSection> ## <Func Name="DoSparseActionHomomorphism" ## Arg='G, start, gens, acts, act, phash, sort'/> ## ## <Description> ## is the function implementing the sparse action homomorphisms and syntax ## is as for these. ## <A>phash</A> must be an injective (&GAP;)-function, for ## example a perfect hash, element comparisons are done in its range. ## Unless a fast enumeration is known, <C>IdFunc</C> should be used. ## If <A>sort</A> is true, the action domain for the resulting homomorphism ## will be sorted. ## </Description> ## </ManSection> ## DeclareGlobalFunction("DoSparseActionHomomorphism"); DeclareGlobalFunction("MultiActionsHomomorphism## is regular on the set <M>\{2, 3, 4\}</M>. ############################################################################# ## #O SparseActionHomomorphism( <G>, <start> [,<gens>,<acts>] [,<act>] ) #O SortedSparseActionHomomorphism(<G>,<start>[,<gens>,<acts>] [,<act>]) ## ## <#GAPDoc Label="SparseActionHomomorphism"> ## <ManSection> ## <Oper Name="SparseActionHomomorphism" ## Arg='G, start[, gens, acts][, act]'/> ## <Oper Name="SortedSparseActionHomomorphism" ## Arg='G, start[, gens, acts][, act]'/> ## ## <Description> ## <Ref Oper="SparseActionHomomorphism"/> computes the action homomorphism ## (see <Ref Func="ActionHomomorphism" Label="for a group, an action domain, etc."/>) ## with arguments <A>G</A>, <M>D</M>, and the optional arguments given, ## where <M>D</M> is the union of the <A>G</A>-orbits of all points in ## <A>start</A>. ## In the <Ref Oper="Orbit"/> calls that are used to create <M>D</M>, ## again the optional arguments given are entered.) ## <P/> ## If <A>G</A> acts on a very large domain not surjectively ## this may yield a permutation image of ## substantially smaller degree than by action on the whole domain. ## <P/> ## The operation <Ref Oper="SparseActionHomomorphism"/> will only use ## <Ref Oper="\="/> comparisons of points in the orbit. ## Therefore it can be used even if no good <Ref Oper="\<"/> ## comparison method for these points is available. ## However the image group will depend on the ## generators <A>gens</A> of <A>G</A>. ## <P/> ## The operation <Ref Oper="SortedSparseActionHomomorphism"/> in contrast ## will sort the orbit and thus produce an image group which does not ## depend on these generators. ## <P/> ## <Example><![CDATA[ ## gap> h:=Group(Z(3)*[[[1,1],[0,1]]]); ## Group([ [ [ Z(3), Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ## gap> hom:=ActionHomomorphism(h,GF(3)^2,OnRight);; ## gap> Image(hom); ## Group([ (2,3)(4,9,6,7,5,8) ]) ## gap> hom:=SparseActionHomomorphism(h,[Z(3)*[1,0]],OnRight);; ## gap> Image(hom); ## Group([ (1,2,3,4,5,6) ]) ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## BindGlobal( "SparseActHomFO", function( name, reqs, famrel## <Func Name="Permutation" Arg local str, orbish, func; # Create the operation. str:= ## computes the permutation that corresponds to the action of <A>g</A> on APPEND_LIST_INTR( str, "Op" ); orbish := NewOperation( str, reqs ); BIND_GLOBAL( str, orbish ); # Create the wrapper function.## Note that the points of the returned permutation refer to the p func := function( arg ) local G, D, pnt, gens, acts, act, p, result,le; # Get the arguments. if 2 <= Length( arg ) then le:=Length(DeclareGlobalFunction( "Permutation" ); G := arg[ 1 ]; if IsFunction( arg[ le ] ) then t :=arg[ le ; else act := OnPoints; fi; if Length( arg ) > 2 and famrel( ## then ## <ManSection> ## <Func Name="PermutationCycle" Arg='g, Omega, pnt[, act]'/> if IsFinite( D ) ## <Description> fi; p ## else p := 2 fi; pnt := Immutable(arg[ p ]); if Length( arg ) > p + 1 then gens := arg[ p + 1 ]; acts := arg[ p + 2 ]; fi; else Error( "usage: ", name, "(<G>[## fi; if not IsBound( gens ) then if (not IsPermGroup(G)) and java.lang.StringIndexOutOfBoundsException: Index 44 out gens := Pcgs(## returns a list of the points in the cycle of <A>pnt</A> under the action else gens := GeneratorsOfGroup( G ); DeclareOperat(",[############################################################### acts :=#O Cycles( <g>, <Omega> [,<act>] ) fi; for p in pnt do TestIdentityAction(acts,p,act); od; if IsBound( D ) then result := orbish( ## <Description> else result := orbish( G, pnt, gens, acts, act ); fi; return## </ManSection> end; BIND_GLOBAL( name, func ); end ); SparseActHomFO( "SparseActionHomomorphism", OrbitishReq, IsIdenticalObj ); SparseActHomFO( "SortedSparseActionHomomorphism", OrbitishReq, IsIdenticalObj ); ############################################################################# ## #O ImageElmActionHomomorphism( <op>,<elm> ) ## ## <ManSection> ## <Oper Name="ImageElmActionHomomorphism" Arg='op,elm'/> ## ## <Description> ## computes the image of <A>elm</A> under the action homomorphism <A>op</A> and is ## guaranteed to use the action (and not the <C>AsGHBI</C>, this is required ## in some methods to bootstrap the range). ## </Description> ## </ManSection> ## DeclareGlobalFunction( "ImageElmActionHomomorphism" ); ############################################################################# ## #O Action( <G>, <Omega> [<gens>,<acts>] [,<act>] ) #A Action( <xset> ) ## ## <#GAPDoc Label="Action"> ## <ManSection> ## <Func Name="Action" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Func Name="Action" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## returns the image group of ## <Ref Func="ActionHomomorphism" Label="for a group, an action domain, etc."/> ## called with the same parameters. ## <P/> ## Note that (for compatibility reasons to be able to get the ## action homomorphism) this image group internally stores the action ## homomorphism. ## If <A>G</A> or <A>Omega</A> are extremely big, this can cause memory ## problems. In this case compute only generator images and form the image ## group yourself. ## <P/> ## (See Section <Ref Sect="Basic Actions"/> ## for information about specific actions.) ## <P/> ## <Index>regular action</Index> ## The following code shows for example how to create the regular action of a ## group. ## <P/> ## <Example><![CDATA[ ## gap> g:=Group((1,2,3),(1,2));; ## gap> Action(g,AsList(g),OnRight); ## Group([ (1,5,3)(2,6,4), (1,6)(2,5)(3,4) ]) ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "Action" ); ############################################################################# ## #O ExternalSet( <G>, <Omega>[, <gens>, <acts>][, <act>] ) ## ## <#GAPDoc Label="ExternalSet"> ## <ManSection> ## <Oper Name="ExternalSet" Arg='G, Omega[, gens, acts][, act]'/> ## ## <Description> ## creates the external set for the action <A>act</A> of <A>G</A> on <A>Omega</A>. ## <A>Omega</A> can be either a proper set, or a domain which is represented as ## described in <Ref Sect="Domains"/> and <Ref Chap="Collections"/>, or (to use ## less memory but with a slower performance) an enumerator ## (see <Ref Attr="Enumerator"/> ) of this domain. ## <P/> ## The result is undefined if <A>G</A> does not act on <A>Omega</A>. ## <Example><![CDATA[ ## gap> g:=Group((1,2,3),(2,3,4));; ## gap> e:=ExternalSet(g,[1..4]); ## <xset:[ 1, 2, 3, 4 ]> ## gap> e:=ExternalSet(g,g,OnRight); ## <xset:[ (), (2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), ## (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3) ]> ## gap> Orbits(e); ## [ [ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,4,2), ## (1,2,3), (1,3,4), (2,3,4), (1,3,2), (1,4,3), (1,2,4) ] ] ## ]]></Example> ## ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "ExternalSet", OrbitsishReq, false, NewAttribute ); DeclareGlobalFunction( "ExternalSetByFilterConstructor" ); DeclareGlobalFunction( "ExternalSetByTypeConstructor" ); ############################################################################# ## #O RestrictedExternalSet( <xset>, <U> ) ## ## <ManSection> ## <Oper Name="RestrictedExternalSet" Arg='xset, U'/> ## ## <Description> ## If <A>U</A> is a subgroup of the <C>ActingDomain</C> of <A>xset</A> this operation ## returns an external set for the same action which has the ## <C>ActingDomain</C> <A>U</A>. ## </Description> ## </ManSection> ## DeclareOperation("RestrictedExternalSet",[IsExternalSet,IsGroup]); ############################################################################# ## #O ExternalSubset(<G>,<Omega>,<start>,[<gens>,<acts>,]<act>) ## ## <#GAPDoc Label="ExternalSubset"> ## <ManSection> ## <Oper Name="ExternalSubset" Arg='G,Omega,start,[gens,acts,]act'/> ## ## <Description> ## constructs the external subset of <A>Omega</A> on the union of orbits of ## the points in <A>start</A>. ## <P/> ## The result is undefined if <A>G</A> does not act on <A>Omega</A>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitishFO( "ExternalSubset", [ IsGroup, IsList, IsList, IsList, IsList, IsFunction ], IsIdenticalObj, true, false ); ############################################################################# ## #O ExternalOrbit( <G>, <Omega>, <pnt>, [<gens>,<acts>,] <act> ) ## ## <#GAPDoc Label="ExternalOrbit"> ## <ManSection> ## <Oper Name="ExternalOrbit" Arg='G, Omega, pnt, [gens,acts,] act'/> ## ## <Description> ## constructs the external subset on the orbit of <A>pnt</A>. The ## <Ref Attr="Representative"/> value of this external set is <A>pnt</A>. ## <P/> ## The result is undefined if <A>G</A> does not act on <A>Omega</A>. ## <Example><![CDATA[ ## gap> e:=ExternalOrbit(g,g,(1,2,3)); ## (1,2,3)^G ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitishFO( "ExternalOrbit", ############################################################################# ## #O Orbit( <G>[, <Omega>], <pnt>[, <gens>, <acts>][, <act>] ) ## ## <#GAPDoc Label="Orbit"> ## <ManSection> ## <Oper Name="Orbit" Arg='G[, Omega], pnt[, gens, acts][, act]'/> ## ## <Description> ## The orbit of the point <A>pnt</A> is the list of all images of <A>pnt</A> ## under the action of the group <A>G</A> w.r.t. the action function ## <A>act</A> or <Ref Func="OnPoints"/> if no action function is given. ## <P/> ## (Note that the arrangement of points in this list is not defined by the ## operation.) ## <P/> ## The orbit of <A>pnt</A> will always contain one element that is ## <E>equal</E> to <A>pnt</A>, however for performance reasons ## this element is not necessarily <E>identical</E> to <A>pnt</A>, ## in particular if <A>pnt</A> is mutable. ## <Example><![CDATA[ ## gap> g:=Group((1,3,2),(2,4,3));; ## gap> Orbit(g,1); ## [ 1, 3, 2, 4 ] ## gap> Orbit(g,[1,2],OnSets); ## [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ], [ 2, 4 ] ] ## ]]></Example> ## <P/> ## (See Section <Ref Sect="Basic Actions"/> ## for information about specific actions.) ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitishFO( "Orbit", OrbitishReq, IsCollsElms, false, false ); ############################################################################# ## #O Orbits( <G>, <seeds>[, <gens>, <acts>][, <act>] ) #A Orbits( <G> ) #A Orbits( <xset> ) ## ## <#GAPDoc Label="Orbits"> ## <ManSection> ## <Oper Name="Orbits" Arg='G, seeds[, gens, acts][, act]' Label="operation"/> ## <Attr Name="Orbits" Arg='G' ## Label="for a permutation group"/> ## <Attr Name="Orbits" Arg='xset' Label="attribute"/> ## ## <Description> ## returns a duplicate-free list of the orbits of the elements in ## <A>seeds</A> under the action <A>act</A> of <A>G</A> or under ## <Ref Func="OnPoints"/> if no action function is given. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>Orbits(<A>G</A>)</C>, which returns all the orbits of its natural ## action on the set of points moved by it. ## For example the group <M>\langle (1,2,3), (4,5) \rangle</M> ## has the orbits <M>\{1,2,3\}</M> and <M>\{4,5\}</M>. ## <P/> ## (Note that the arrangement of orbits or of points within one orbit is ## not defined by the operation.) ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "Orbits", OrbitsishReq, ## ############################################################################# ## #O OrbitsDomain( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #A OrbitsDomain( <G> ) #A OrbitsDomain( <xset> ) ## ## <#GAPDoc Label="OrbitsDomain"> ## <ManSection> ## <Heading>OrbitsDomain</Heading> ## <Oper Name="OrbitsDomain" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group and an action domain"/> ## <Attr Name="OrbitsDomain" Arg='G' ## Label="for a permutation group"/> ## <Attr Name="OrbitsDomain" Arg='xset' ## Label="of an external set"/> ## ## <Description> ## returns a list of the orbits of <A>G</A> on the domain <A>Omega</A> ## (given as lists) under the action <A>act</A> or under ## <Ref Func="OnPoints"/> if no action function is given. ## <P/> ## This operation is often faster than ## <Ref Oper="Orbits" Label="operation"/>. ## The domain <A>Omega</A> must be closed under the action of <A>G</A>, ## otherwise an error can occur. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>OrbitsDomain(<A>G</A>)</C>, which returns all the orbits of its natural ## action on the set of points moved by it. ## <P/> ## (Note that the arrangement of orbits or of points within one orbit is ## not defined by the operation.) ## <Example><![CDATA[ ## gap> g:=Group((1,3,2),(2,4,3));; ## gap> Orbits(g,[1..5]); ## [ [ 1, 3, 2, 4 ], [ 5 ] ] ## gap> OrbitsDomain(g,Arrangements([1..4],3),OnTuples); ## [ [ [ 1, 2, 3 ], [ 3, 1, 2 ], [ 1, 4, 2 ], [ 2, 3, 1 ], [ 2, 1, 4 ], ## [ 3, 4, 1 ], [ 1, 3, 4 ], [ 4, 2, 1 ], [ 4, 1, 3 ], ## [ 2, 4, 3 ], [ 3, 2, 4 ], [ 4, 3, 2 ] ], ## [ [ 1, 2, 4 ], [ 3, 1, 4 ], [ 1, 4, 3 ], [ 2, 3, 4 ], [ 2, 1, 3 ], ## [ 3, 4, 2 ], [ 1, 3, 2 ], [ 4, 2, 3 ], [ 4, 1, 2 ], ## [ 2, 4, 1 ], [ 3, 2, 1 ], [ 4, 3, 1 ] ] ] ## gap> OrbitsDomain(g,GF(2)^2,[(1,2,3),(1,4)(2,3)], ## > [[[Z(2)^0,Z(2)^0],[Z(2)^0,0*Z(2)]],[[Z(2)^0,0*Z(2)],[0*Z(2),Z(2)^0]]]); ## [ [ <an immutable GF2 vector of length 2> ], ## [ <an immutable GF2 vector of length 2>, ## <an immutable GF2 vector of length 2>, ## <an immutable GF2 vector of length 2> ] ] ## ]]></Example> ## <P/> ## (See Section <Ref Sect="Basic Actions"/> ## for information about specific actions.) ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation(## that is the subgroup of those elements of <A>G</A> that fix <A>pnt</A>. ############################################################################# ## #O OrbitLength( <G>[, <Omega>], <pnt> [, <gens>, <acts>][, <act>] ) ## ## <#GAPDoc Label="OrbitLength"> ## <ManSection> ## <Oper Name="OrbitLength" Arg='G[, Omega], pnt[, gens, acts][, act]'/> ## ## <Description> ## computes the length of the orbit of <A>pnt</A> under ## the action function <A>act</A> or <Ref Func="OnPoints"/> ## if no action function is given. ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitishFO( "OrbitLength", OrbitishReq, IsCollsElms, false, false ); ############################################################################# ## #O OrbitLengths( <G>, <seeds>[, <gens>, <acts>][, <act>] ) #A OrbitLengths( <G> ) #A OrbitLengths( <xset> ) ## ## <#GAPDoc Label="OrbitLengths"> ## <ManSection> ## <Heading>OrbitLengths</Heading> ## <Oper Name="OrbitLengths" Arg='G, seeds[, gens, acts][, act]' ## Label="for a group, a set of seeds, etc."/> ## <Attr Name="OrbitLengths" Arg='G' Label="for a permutation group"/> ## <Attr Name="OrbitLengths" Arg='xset' Label="for an external set"/> ## ## <Description> ## computes the lengths of all the orbits of the elements in <A>seeds</A> ## under the action <A>act</A> of <A>G</A>. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>OrbitLengths(<A>G</A>)</C>, which returns the lengths of all the ## orbits of its natural action on the set of points moved by it. ## For example the group <M>\langle (1,2,3), (5,6) \rangle</M> ## has the orbit lengths 2 and 3. ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation(## <P/> ############################################################################# ## <ManSection> #O OrbitLengthsDomain( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #A OrbitLengthsDomain( <G> ) #A OrbitLengthsDomain( <xset> ) ## ## <#GAPDoc Label="OrbitLengthsDomain"> ## <ManSection> ## <Heading>OrbitLengthsDomain</Heading> ## <Oper Name="OrbitLengthsDomain" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group and a set of seeds"/> ## <Attr Name="OrbitLengthsDomain" Arg='G' ## Label="for a permutation group"/> ## <Attr Name="OrbitLengthsDomain" Arg='xset' Label="of an external set"/> ## ## <Description> ## computes the lengths of all the orbits of <A>G</A> on <A>Omega</A>. ## <P/> ## This operation is often faster than ## <Ref Oper="OrbitLengths" Label="for a group, a set of seeds, etc."/>. ## The domain <A>Omega</A> must be closed under the action of <A>G</A>, ## otherwise an error can occur. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>OrbitLengthsDomain(<A>G</A>)</C>, which returns the length of all ## the orbits of its natural action on the set of points moved by it. ## <Example><![CDATA[ ## gap> g:=Group((1,3,2),(2,4,3));; ## gap> OrbitLength(g,[1,2,3,4],OnTuples); ## 12 ## gap> OrbitLengths(g,Arrangements([1..4],4),OnTuples); ## [ 12, 12 ] ## gap> g:=Group((1,2,3),(5,6,7));; ## gap> OrbitLengthsDomain(g,[1,2,3]); ## [ 3 ] ## gap> OrbitLengthsDomain(g); ## [ 3, 3 ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation(## <Description> ############################################################################# ## #O OrbitStabilizer( <G>[, <Omega>], <pnt>[, <gens>,<acts>][, <act>] ) ## ## <#GAPDoc Label="OrbitStabilizer"> ## <ManSection> ## <Oper Name="OrbitStabilizer" Arg='G[, Omega], pnt[, gens, acts][, act]'/> ## ## <Description> ## computes the orbit and the stabilizer of <A>pnt</A> simultaneously in a ## single orbit-stabilizer algorithm. ## <P/> ## The stabilizer will have <A>G</A> as its parent. ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitishFO( "OrbitStabilizer", OrbitishReq, ## gap> RepresentativeAction(g,[1,2,3],[1, ############################################################################# ## #O ExternalOrbits( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #A ExternalOrbits( <xset> ) ## ## <#GAPDoc Label="ExternalOrbits"> ## <ManSection> ## <Heading>ExternalOrbits</Heading> ## <Oper Name="ExternalOrbits" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Attr Name="ExternalOrbits" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## computes a list of external orbits that give the orbits of <A>G</A>. ## <Example><![CDATA[ ## gap> ExternalOrbits(g,AsList(g)); ## [ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation(############################################################# ############################################################################# ## #O ExternalOrbitsStabilizers( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #A ExternalOrbitsStabilizers( <xset> ) ## ## <#GAPDoc Label="ExternalOrbitsStabilizers"> ## <ManSection> ## <Heading>ExternalOrbitsStabilizers</Heading> ## <Oper Name="ExternalOrbitsStabilizers" ## Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Attr Name="ExternalOrbitsStabilizers" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## In addition to ## <Ref Oper="ExternalOrbits" Label="for a group, an action domain, etc."/>, ## this operation also computes the stabilizers of the representatives of ## the external orbits at the same time. ## (This can be quicker than computing the ## <Ref Oper="ExternalOrbits" Label="for a group, an action domain, etc."/> ## value first and the stabilizers afterwards.) ## <Example><![CDATA[ ## gap> e:=ExternalOrbitsStabilizers(g,AsList(g)); ## [ ()^G, (2,3,4)^G, (2,4,3)^G, (1,2)(3,4)^G ] ## gap> HasStabilizerOfExternalSet(e[3]); ## true ## gap> StabilizerOfExternalSet(e[3]); ## Group([ (2,4,3) ]) ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "ExternalOrbitsStabilizers", OrbitsishReq, true, NewAttribute ); ############################################################################# ## #O Transitivity( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #A Transitivity( <G> ) #A Transitivity( <xset> ) ## ## <#GAPDoc Label="Transitivity:oprt"> ## <ManSection> ## <Heading>Transitivity</Heading> ## <Oper Name="Transitivity" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group and an action domain"/> ## <Attr Name="Transitivity" Arg='G' ## Label="for a permutation group"/> ## <Attr Name="Transitivity" Arg='xset' Label="for an external set"/> ## ## <Description> ## returns the degree <M>k</M> (a non-negative integer) of transitivity of ## the action implied by the arguments, ## i.e. the largest integer <M>k</M> such that the action is ## <M>k</M>-transitive. ## If the action is not transitive <C>0</C> is returned. ## <P/> ## An action is <E><M>k</M>-transitive</E> if every <M>k</M>-tuple of points ## can be mapped simultaneously to every other <M>k</M>-tuple. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>Transitivity(<A>G</A>)</C>, which returns the degree of transitivity ## of the group with respect to its natural action on the set of points ## moved by it. ## For example the group <M>\langle (2,3,4),(2,3) \rangle</M> ## is 3-transitive on the set <M>\{2, 3, 4\}</M>. ## <Example><![CDATA[ ## gap> g:=Group((1,3,2),(2,4,3));; ## gap> IsTransitive(g,[1..5]); ## false ## gap> Transitivity(g,[1..4]); ## 2 ## gap> Transitivity(g); ## 2 ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## #F OrbitStabilizerAlgorithm( <G>, <Omega>, <blist>, <gens>,<acts>, <pntact> ) ############################################################################# ## #O Blocks( <G>, <Omega>[, <seed>][, <gens>, <acts>][, <act>] ) #A Blocks( <xset>[, <seed>] ) ## ## <#GAPDoc Label="Blocks"> ## <ManSection> ## <Heading>Blocks</Heading> ## <Oper Name="Blocks" Arg='G, Omega[, seed][, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Attr Name="Blocks" Arg='xset[, seed]' ## Label="for an external set"/> ## ## <Description> ## computes a block system for the transitive (see ## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>) ## action of <A>G</A> on <A>Omega</A>. ## If <A>seed</A> is not given and the action is imprimitive, ## a minimal nontrivial block system will be found. ## If <A>seed</A> is given, a block system in which <A>seed</A> ## is the subset of one block is computed. ## <P/> ## The result is undefined if the action is not transitive. ## <Example><![CDATA[ ## gap> g:=TransitiveGroup(8,3); ## E(8)=2[x]2[x]2 ## gap> Blocks(g,[1..8]); ## [ [ 1, 8 ], [ 2, 3 ], [ 4, 5 ], [ 6, 7 ] ] ## gap> Blocks(g,[1..8],[1,4]); ## [ [ 1, 4 ], [ 2, 7 ], [ 3, 6 ], [ 5, 8 ] ] ## ]]></Example> ## <P/> ## (See Section <Ref Sect="Basic Actions"/> ## for information about specific actions.) ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( the group <A>G</A> acting with the generators <A>gens<## a bit list corresponding OrbitishFO( "Blocks", [ IsGroup, IsList, IsList, IsList, IsList, IsFunction ], IsIdenticalObj, BlocksAttr, true ); ############################################################################# ## #O MaximalBlocks( <G>, <Omega>[, <seed>][, <gens>, <acts>][, <act>] ) #A MaximalBlocks( <xset>[, <seed>] ) ## ## <#GAPDoc Label="MaximalBlocks"> ## <ManSection> ## <Heading>MaximalBlocks</Heading> ## <Oper Name="MaximalBlocks" Arg='G, Omega[, seed][, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Attr Name="MaximalBlocks" Arg='xset[, seed]' ## Label="for an external set"/> ## ## <Description> ## for the orbit of <A>pnt</A> still must be all set to <K>false</K>). Again the ## respect to inclusion) for the transitive (see ## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>) ## action of <A>G</A> on <A>Omega</A>. ## If <A>seed</A> is given, a block system is computed in which <A>seed</A> ## is a subset of one block. ## <P/> ## The result is undefined if the action is not transitive. ## <Example><![CDATA[ ## gap> MaximalBlocks(g,[1..8]); ## [ [ 1, 2, 3, 8 ], [ 4 .. 7 ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "MaximalBlocksAttr", IsExternalSet ); OrbitishFO( "MaximalBlocks", [ DeclareOperation "rbitStabilizerAlgorithm", IsList, IsList, IsFunction ], IsIdenticalObj, MaximalBlocksAttr,true ); #T the following syntax would be nice for consistency as well: ## RepresentativesMinimalBlocks(<G>,<Omega>[,<seed>][,<gens>,<acts>][,<act>]) ## RepresentativesMinimalBlocks( <xset>, <seed> ) ############################################################################# ## #O RepresentativesMinimalBlocks(<G>,<Omega>[,<gens>,<acts>][,<act>]) #A RepresentativesMinimalBlocks( <xset> ) ## ## <#GAPDoc Label="RepresentativesMinimalBlocks"> ## <ManSection> ## <Heading>RepresentativesMinimalBlocks</Heading> ## <Oper Name="RepresentativesMinimalBlocks" ## Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Attr Name="RepresentativesMinimalBlocks" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## computes a list of block representatives for all minimal (i.e blocks are ## minimal with respect to inclusion) nontrivial block systems for the ## transitive (see ## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>) ## action of <A>G</A> on <A>Omega</A>. ## <P/> ## The result is undefined if the action is not transitive. ## <Example><![CDATA[ ## gap> RepresentativesMinimalBlocks(g,[1..8]); ## [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 1, 5 ], [ 1, 6 ], [ 1, 7 ], ## [ 1, 8 ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareAttribute( "RepresentativesMinimalBlocksAttr"## 3 OrbitishFO( "RepresentativesMinimalBlocks", [ IsGroup, IsList, IsList, IsList,## false IsList, IsFunction ], IsIdenticalObj, RepresentativesMinimalBlocksAttr,true ); ############################################################################# ## #O Earns( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #A Earns( <xset> ) ## ## <#GAPDoc Label="Earns"> ## <ManSection> ## <Heading>Earns</Heading> ## <Oper Name="Earns" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Attr Name="Earns" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## returns a list of the elementary abelian regular ## (when acting on <A>Omega</A>) normal subgroups of <A>G</A>. ## <P/> ## At the moment only methods for a primitive group <A>G</A> are implemented. ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "## true ############################################################################# ## #O IsTransitive( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #P IsTransitive( <permgroup> ) #P IsTransitive( <xset> ) ## ## <#GAPDoc Label="IsTransitive:oprt"> ## <ManSection> ## <Heading>IsTransitive</Heading> ## <Oper Name="IsTransitive" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Prop Name="IsTransitive" Arg='G' ## Label="for a permutation group"/> ## <Prop Name="IsTransitive" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## returns <K>true</K> if the action implied by the arguments is transitive, ## or <K>false</K> otherwise. ## <P/> ## <Index>transitive</Index> ## We say that a group <A>G</A> acts <E>transitively</E> on a domain ## <M>D</M> if and only if <A>G</A> acts on <M>D</M> and for every pair of ## points <M>d, e \in D</M> ## there is an element <M>g</M> in <A>G</A> such that <M>d^g = e</M>. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>IsTransitive(<A>G</A>)</C>, which tests whether the group is ## transitive with respect to its natural action on the set of points ## moved by it. ## For example the group <M>\langle (2,3,4),(2,3) \rangle</M> ## is transitive on the set <M>\{2, 3, 4\}</M>. ## <Example><![CDATA[ ## gap> G:= Group( (2,3,4), (2,3) );; ## gap> IsTransitive( G, [ 2 .. 4 ] ); ## true ## gap> IsTransitive( G, [ 2, 3 ] ); # G does not act on [ 2, 3 ] ## false ## gap> IsTransitive( G, [ 1 .. 4 ] ); # G has two orbits on [ 1 .. 4 ] ## false ## gap> IsTransitive( G ); # G is transitive on [ 2 .. 4 ] ## true ## gap> IsTransitive( SL(2, 3), NormedRowVectors( GF(3)^2 ) ); ## false ## gap> IsTransitive( SL(2, 3), NormedRowVectors( GF(3)^2 ), OnLines ); ## true ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "IsTransitive", OrbitsishReq, false, ## as in &GAP; groups always act from th ############################################################################# ## #O IsPrimitive( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #P IsPrimitive( <G> ) #P IsPrimitive( <xset> ) ## ## <#GAPDoc Label="IsPrimitive"> ## <ManSection> ## <Heading>IsPrimitive</Heading> ## <Oper Name="IsPrimitive" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Prop Name="IsPrimitive" Arg='G' ## Label="for a permutation group"/> ## <Prop Name="IsPrimitive" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## returns <K>true</K> if the action implied by the arguments is primitive, ## or <K>false</K> otherwise. ## <P/> ## <Index>primitive</Index> ## An action is <E>primitive</E> if it is transitive (see ## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>) ## and the action admits ## no nontrivial block systems. See <Ref Sect="Block Systems"/> for ## the definition of block systems. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>IsPrimitive(<A>G</A>)</C>, which tests whether the group is ## primitive with respect to its natural action on the set of points ## moved by it. ## For example the group <M>\langle (2,3,4),(2,3) \rangle</M> ## is primitive on the set <M>\{2, 3, 4\}</M>. ## <P/> ## For an explanation of the meaning of all the inputs, please refer to ## <Ref Sect="About Group Actions"/>. ## <P/> ## <E>Note:</E> This operation does not tell whether a matrix group is ## primitive in the sense of preserving a direct sum of vector spaces. ## To do this use <C>IsPrimitiveMatrixGroup</C> or ## <C>IsPrimitive</C> from the package <Package>IRREDSOL</Package>. ## ## <Example><![CDATA[ ## gap> IsPrimitive(g,Orbit(g,(1,2)(3,4))); ## true ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "IsPrimitive", OrbitsishReq, false, NewProperty ); InstallTrueMethod( IsPrimitive, IsNaturalSymmetricGroup ); InstallTrueMethod( IsPrimitive, IsNaturalAlternatingGroup ); InstallTrueMethod( IsTransitive,## of entries.) ############################################################################# ## #O IsPrimitiveAffine( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #P IsPrimitiveAffine( <xset> ) ## ## <ManSection> ## <Oper Name="IsPrimitiveAffine" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Prop Name="IsPrimitiveAffine" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## </Description> ## </ManSection> ## OrbitsishOperation( "IsPrimitiveAffine", OrbitsishReq, false ############################################################################# ## #O IsSemiRegular( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #P IsSemiRegular( <G> ) #P IsSemiRegular( <xset> ) ## ## <#GAPDoc Label="IsSemiRegular"> ## <ManSection> ## <Heading>IsSemiRegular</Heading> ## <Oper Name="IsSemiRegular" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Prop Name="IsSemiRegular" Arg='G' ## Label="for a permutation group"/> ## <Prop Name="IsSemiRegular" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## returns <K>true</K> if the action implied by the arguments is ## semiregular, or <K>false</K> otherwise. ## <P/> ## <Index>semiregular</Index> ## An action is <E>semiregular</E> if the stabilizer of each point is the ## identity. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>IsSemiRegular(<A>G</A>)</C>, which tests whether the group is ## semiregular with respect to its natural action on the set of points ## moved by it. ## For example the group <M>\langle (2,3,4) (5,6,7) \rangle</M> ## is semiregular on the set <M>\{2, 3, 4, 5, 6, 7\}</M>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation(## its first nonzero entry is normed to the identity of the relevant field ############################################################################# ## #O IsRegular( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #P IsRegular( <G> ) #P IsRegular( <xset> ) ## ## <#GAPDoc Label="IsRegular"> ## <ManSection> ## <Heading>IsRegular</Heading> ## <Oper Name="IsRegular" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Prop Name="IsRegular" Arg='G' ## Label="for a permutation group"/> ## <Prop Name="IsRegular" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## returns <K>true</K> if the action implied by the arguments is regular, or ## <K>false</K> otherwise. ## <P/> ## <Index>regular</Index> ## An action is <E>regular</E> if it is both semiregular ## (see <Ref Oper="IsSemiRegular" Label="for a group, an action domain, etc."/>) ## and transitive ## (see <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>). ## In this case every point <A>pnt</A> of <A>Omega</A> defines a one-to-one ## correspondence between <A>G</A> and <A>Omega</A>. ## <P/> ## For a permutation group <A>G</A>, one may also invoke this as ## <C>IsRegular(<A>G</A>)</C>, which tests whether the group is ## regular with respect to its natural action on the set of points moved by it. ## For example the group <M>\langle (2,3,4) \rangle</M> ## is regular on the set <M>\{2, 3, 4\}</M>. ## ## <Example><![CDATA[ ## gap> IsSemiRegular(g,Arrangements([1..4],3),OnTuples); ## true ## gap> IsRegular(g,Arrangements([1..4],3),OnTuples); ## false ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "IsRegular", OrbitsishReq, false, NewProperty ); InstallTrueMethod( IsRegular, IsTransitive and IsSemiRegular ); ############################################################################# ## #O RankAction( <G>, <Omega>[, <gens>, <acts>][, <act>] ) #A RankAction( <xset> ) ## ## <#GAPDoc Label="RankAction"> ## <ManSection> ## <Heading>RankAction</Heading> ## <Oper Name="RankAction" Arg='G, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Attr Name="RankAction" Arg='xset' ## Label="for an external set"/> ## ## <Description> ## returns the rank of the transitive (see ## <Ref Oper="IsTransitive" Label="for a group, an action domain, etc."/>) ## action of <A>G</A> on <A>Omega</A>, i. e., the number of orbits of ## any point stabilizer. ## <Example><![CDATA[ ## gap> RankAction(g,Combinations([1..4],2),OnSets); ## 4 ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## OrbitsishOperation( "RankAction", OrbitsishReq, false, NewAttribute ); ############################################################################# ## #F Permutation( <g>, <Omega>[, <gens>, <acts>][, <act>] ) #F Permutation( <g>, <xset> ) ## ## <#GAPDoc Label="Permutation"> ## <ManSection> ## <Heading>Permutation</Heading> ## <Func Name="Permutation" Arg='g, Omega[, gens, acts][, act]' ## Label="for a group, an action domain, etc."/> ## <Func Name="Permutation" Arg='g, xset' Label="for an external set"/> ## ## <Description> ## computes the permutation that corresponds to the action of <A>g</A> on ## the permutation domain <A>Omega</A> ## (a list of objects that are permuted). ## If an external set <A>xset</A> is given, ## the permutation domain is the <Ref Attr="HomeEnumerator"/> value ## of this external set (see Section <Ref Sect="External Sets"/>). ## Note that the points of the returned permutation refer to the positions ## in <A>Omega</A>, even if <A>Omega</A> itself consists of integers. ## <P/> ## If <A>g</A> does not leave the domain invariant, or does not map the ## </ManSection> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "Permutation" ); DeclareOperation( "PermutationOp", [ IsObject, IsListjava.lang.StringIndexOutOfBound ############################################################################# ## #O PermutationCycle( <g>, <Omega>, <pnt>[, <act>] ) ## ## <#GAPDoc Label="PermutationCycle"> ## <ManSection> ## <Func Name="PermutationCycle" Arg='g, Omega, pnt[, act]'/> ## ## <Description> ## computes the permutation that represents the cycle of <A>pnt</A> under ## the action of the element <A>g</A>. ## <Example><![CDATA[ ## gap> Permutation([[Z(3),-Z(3)],[Z(3),0*Z(3)]],AsList(GF(3)^2)); ## (2,7,6)(3,4,8) ## gap> Permutation((1,2,3)(4,5)(6,7),[4..7]); ## (1,2)(3,4) ## gap> PermutationCycle((1,2,3)(4,5)(6,7),[4..7],4); ## (1,2) ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "PermutationCycle" ); DeclareOperation( "PermutationCycleOp", [ IsObject, IsList, IsObject, IsFunction ]## </Description> ############################################################################# ## #O Cycle( <g>, <Omega>, <pnt> [,<act>] ) ## ## <#GAPDoc Label="Cycle"> ## <ManSection> ## <Func Name="Cycle" Arg='g, Omega, pnt[, act]'/> ## ## <Description> ## returns a list of the points in the cycle of <A>pnt</A> under the action ## of the element <A>g</A>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction(# of length 2. DeclareOperation( "CycleOp", [ IsObject, IsList, IsObject, IsFunction ] ); ############################################################################# ## #O Cycles( <g>, <Omega> [,<act>] ) ## ## <#GAPDoc Label="Cycles"> ## <ManSection> ## <Func Name="Cycles" Arg='g, Omega[, act]'/> ## ## <Description> ## returns a list of the cycles (as lists of points) of the action of the ## element <A>g</A>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "Cycles" ); DeclareOperation( "CyclesOp", [ IsObject, IsList, IsFunction ] ); ############################################################################# ## #O CycleLength( <g>, <Omega>, <pnt> [,<act>] ) ## ## <#GAPDoc Label="CycleLength"> ## <ManSection> ## <Func Name="CycleLength" Arg='g, Omega, pnt[, act]'/> ## ## <Description> ## returns the length of the cycle of <A>pnt</A> under the action of the element ## <A>g</A>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "CycleLength" ); DeclareOperation( "CycleLengthOp", [ IsObject, IsList, IsObject, IsFunction ] ); ############################################################################# ## #O CycleLengths( <g>, <Omega>[, <act>] ) ## ## <#GAPDoc Label="CycleLengths"> ## <ManSection> ## <Oper Name="CycleLengths" Arg='g, Omega[, act]'/> ## ## <Description> ## returns the lengths of all the cycles under the action of the element ## <A>g</A> on <A>Omega</A>. ## <Example><![CDATA[ ## gap> Cycle((1,2,3)(4,5)(6,7),[4..7],4); ## [ 4, 5 ] ## gap> CycleLength((1,2,3)(4,5)(6,7),[4..7],4); ## 2 ## gap> Cycles((1,2,3)(4,5)(6,7),[4..7]); ## [ [ 4, 5 ], [ 6, 7 ] ] ## gap> CycleLengths((1,2,3)(4,5)(6,7),[4..7]); ## [ 2, 2 ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "CycleLengths## ]]></Example> DeclareOperation## </ManSection> [ IsObject, IsList, IsFunction ] ); ############################################################################# ## #F CycleIndex( <g>, <Omega>[, <act>] ) #F CycleIndex( <G>, <Omega>[, <act>] ) ## ## <#GAPDoc Label="CycleIndex"> ## <ManSection> ## <Heading>CycleIndex</Heading> ## <Func Name="CycleIndex" Arg='g, Omega[, act]' ## Label="for a permutation and an action domain"/> ## <Func Name="CycleIndex" Arg='G, Omega[, act]' ## Label="for a permutation group and an action domain"/> ## ## <Description> ## The <E>cycle index</E> of a permutation <A>g</A> acting on <A>Omega</A> ## is defined as ## <Display Mode="M"> ## z(<A>g</A>) = s_1^{{c_1}} s_2^{{c_2}} \cdots s_n^{{c_n}} ## </Display> ## where <M>c_k</M> is the number of <M>k</M>-cycles in the cycle ## decomposition of <A>g</A> and the <M>s_i</M> are indeterminates. ## <P/> ## The <E>cycle index</E> of a group <A>G</A> is defined as ## <Display Mode="M"> ## Z(<A>G</A>) = \left( \sum_{{g \in <A>G</A>}} z(g) \right) / |<A>G</A>| . ## </Display> ## <P/> ## The indeterminates used by ## <Ref Func="CycleIndex" Label="for a permutation and an action domain"/> ## are the indeterminates <M>1</M> to <M>n</M> over the rationals ## (see <Ref Oper="Indeterminate" Label="for a ring (and a number)"/>). ## <P/> ## <Example><![CDATA[ ## gap> g:=TransitiveGroup(6,8); ## S_4(6c) = 1/2[2^3]S(3) ## gap> CycleIndex(g); ## 1/24*x_1^6+1/8*x_1^2*x_2^2+1/4*x_1^2*x_4+1/4*x_2^3+1/3*x_3^2 ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "CycleIndex" )## DeclareOperation( "CycleIndexOp", [ IsObject, IsListOrCollection, IsFunction ] ); ############################################################################# ## #O RepresentativeAction( <G> [,<Omega>], <d>, <e> [,<gens>,<acts>] [,<act>] ) ## ## <#GAPDoc Label="RepresentativeAction"> ## <ManSection> ## <Func Name="RepresentativeAction" ## Arg='G[, Omega], d, e[, gens, acts][, act]'/> ## ## <Description> ## computes an element of <A>G</A> that maps <A>d</A> to <A>e</A> under the ## given action and returns <K>fail</K> if no such element exists. ## <Example><![CDATA[ ## gap> g:=Group((1,3,2),(2,4,3));; ## gap> RepresentativeAction(g,1,3); ## (1,3)(2,4) ## gap> RepresentativeAction(g,1,3,OnPoints); ## (1,3)(2,4) ## gap> RepresentativeAction(g,(1,2,3),(2,4,3)); ## (1,2,4) ## gap> RepresentativeAction(g,(1,2,3),(2,3,4)); ## fail ## gap> RepresentativeAction(g,Group((1,2,3)),Group((2,3,4))); ## (1,2,4) ## gap> RepresentativeAction(g,[1,2,3],[1,2,4],OnSets); ## (2,4,3) ## gap> RepresentativeAction(g,[1,2,3],[1,2,4],OnTuples); ## fail ## ]]></Example> ## <P/> ## (See Section <Ref Sect="Basic Actions"/> ## for information about specific actions.) ## <P/> ## Again the standard method for <Ref Func="RepresentativeAction"/> is ## an orbit-stabilizer algorithm, ## for permutation groups and standard actions a backtrack algorithm is used. ## </Description> ## </ManSection> ## <#/GAPDoc> ## ## gap> OnSetsTuples( [ [ 1, 2 ], [ 3, 4 ] ], (1,2,3) ); ## [ [ 1, 4 ], [ 2, 3 ] ] [ IsGroup, IsList, IsObject, IsObject, IsFunction ] ); ############################################################################# ## #F Stabilizer( <G> [,<Omega>], <pnt> [,<gens>,<acts>] [,<act>] ) ## ## <#GAPDoc Label="Stabilizer"> ## <ManSection> ## <Func Name="Stabilizer" Arg='G[, Omega], pnt[, gens, acts][, act]'/> ## ## <Description> ## computes the stabilizer in <A>G</A> of the point <A>pnt</A>, ## that is the subgroup of those elements of <A>G</A> that fix <A>pnt</A>. ## The stabilizer will have <A>G</A> as its parent. ## <Example><![CDATA[ ## gap> g:=Group((1,3,2),(2,4,3));; ## gap> stab:=Stabilizer(g,4); ## Group([ (1,3,2) ]) ## gap> Parent(stab); ## Group([ (1,3,2), (2,4,3) ]) ## ]]></Example> ## <P/> ## The stabilizer of a set or tuple of points can be computed by specifying ## an action of sets or tuples of points. ## <Example><![CDATA[ ## gap> Stabilizer(g,[1,2],OnSets); ## Group([ (1,2)(3,4) ]) ## gap> Stabilizer(g,[1,2],OnTuples); ## Group(()) ## gap> orbstab:=OrbitStabilizer(g,[1,2],OnSets); ## rec( ## orbit := [ [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 3 ], [ 3, 4 ], ## [ 2, 4 ] ], stabilizer := Group([ (1,2)(3,4) ]) ) ## gap> Parent(orbstab.stabilizer); ## Group([ (1,3,2), (2,4,3) ]) ## ]]></Example> ## <P/> ## (See Section <Ref Sect="Basic Actions"/> ## for information about specific actions.) ## <P/> ## The standard methods for all these actions are an orbit-stabilizer ## algorithm. For permutation groups backtrack algorithms are used. For ## solvable groups an orbit-stabilizer algorithm for solvable groups, which ## uses the fact that the orbits of a normal subgroup form a block system ## (see <Cite Key="SOGOS"/>) is used. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "Stabilizer" ); OrbitishFO( "StabilizerFunc", OrbitishReq, IsCollsElms, false,false ); BindGlobal( "StabilizerOp", StabilizerFuncOp ); ############################################################################# ## #F StabilizerPcgs( <pcgs>, <pnt> [,<acts>] [,<act>] ) ## ## <#GAPDoc Label="StabilizerPcgs"> ## <ManSection> ## <Func Name="StabilizerPcgs" Arg='pcgs, pnt[, acts][, act]'/> ## ## <Description> ## computes the stabilizer in the group generated by <A>pcgs</A> of the ## point <A>pnt</A>. ## If given, <A>acts</A> are elements by which <A>pcgs</A> acts, ## <A>act</A> is the acting function. ## This function returns a pcgs for the stabilizer which is induced by the ## <C>ParentPcgs</C> of <A>pcgs</A>, that is it is compatible ## with <A>pcgs</A>. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction"") ############################################################################# ## #F OrbitStabilizerAlgorithm( <G>, <Omega>, <blist>, <gens>,<acts>, <pntact> ) ## ## <#GAPDoc Label="OrbitStabilizerAlgorithm"> ## <ManSection> ## <Oper Name="OrbitStabilizerAlgorithm" ## Arg='G, Omega, blist, gens, acts, pntact'/> ## ## <Description> ## This operation should not be called by a user. It is documented however ## for purposes to extend or maintain the group actions package ## (the word <Q>package</Q> here refers to the &GAP; functionality for ## group actions, not to a &GAP; package). ## <P/> ## <Ref Oper="OrbitStabilizerAlgorithm"/> performs an orbit stabilizer ## algorithm for the group <A>G</A> acting with the generators <A>gens</A> ## via the generator images <A>gens</A> and the group action <A>act</A> on ## the element <A>pnt</A>. ## (For technical reasons <A>pnt</A> and <A>act</A> are put in one record ## with components <C>pnt</C> and <C>act</C> respectively.) ## <P/> ## The <A>pntact</A> record may carry a component <A>stabsub</A>. ## If given, this must be a subgroup stabilizing <E>all</E> points in the ## domain and can be used to abbreviate stabilizer calculations. ## <P/> ## The <A>pntact</A> component also may contain the boolean entry <C>onlystab</C> set ## to <K>true</K>. In this case the <C>orbit</C> component may be omitted from the ## result. ## <P/> ## The argument <A>Omega</A> (which may be replaced by <K>false</K> to be ignored) is ## the set within which the orbit is computed (once the orbit is the full ## domain, the orbit calculation may stop). If <A>blist</A> is given it must be ## a bit list corresponding to <A>Omega</A> in which elements which have been found ## already will be <Q>ticked off</Q> with <K>true</K>. (In particular, the entries ## for the orbit of <A>pnt</A> still must be all set to <K>false</K>). Again the ## remaining action domain (the bits set initially to <K>false</K>) can be ## used to stop if the orbit cannot grow any longer. ## Another use of the bit list is if <A>Omega</A> is an enumerator which can ## determine <Ref Oper="PositionCanonical"/> values very quickly. ## In this situation it can be ## worth to search images not in the orbit found so far, but via their ## position in <A>Omega</A> and use a the bit list to keep track whether the ## element is in the orbit found so far. ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareOperation( "java.lang.StringIndexOutOfBoundsException: Index 39 out of bou [IsGroup,IsObject,IsObject## generated by <A>acts</A>. (Such a domain can be helpful for obtain DeclareGlobalFunction( "OrbitByPosOp" ); # AH, 5-feb-99 This function is neither documented not used. #DeclareGlobalFunction( "OrbitStabilizerListByGenerators" ); DeclareGlobalFunction( "## </ManSection> DeclareGlobalFunction( "StabilizerOfBlockNCDeclareOperation("",[,IsListOrCollecti ############################################################################# ## #O AbelianSubfactorAction(<G>,<M>,<N>) ## ## <#GAPDoc Label="AbelianSubfactorAction"> ## <ManSection> ## <Oper Name="AbelianSubfactorAction" Arg='G, M, N'/> ## ## <Description> ## Let <A>G</A> be a group and <M><A>M</A> \geq <A>N</A></M> be subgroups ## of a common parent that are normal under <A>G</A>, such that ## the subfactor <M><A>M</A>/<A>N</A></M> is elementary abelian. ## The operation <Ref Oper="AbelianSubfactorAction"/> returns a list ## <C>[ <A>phi</A>, <A>alpha</A>, <A>bas</A> ]</C> where ## <A>bas</A> is a list of elements of <A>M</A> which are representatives ## for a basis of <M><A>M</A>/<A>N</A></M>, ## <A>alpha</A> is a map from <A>M</A> into a <M>n</M>-dimensional row space ## over <M>GF(p)</M> where <M>[<A>M</A>:<A>N</A>] = p^n</M> that is the ## natural homomorphism of <A>M</A> by <A>N</A> with the quotient ## represented as an additive group. ## Finally <A>phi</A> is a homomorphism from <A>G</A> ## into <M>GL_n(p)</M> that represents the action of <A>G</A> on the factor ## <M><A>M</A>/<A>N</A></M>. ## <P/> ## Note: If only matrices for the action are needed, ## <Ref Func="LinearActionLayer"/> might be faster. ## <Example><![CDATA[ ## gap> g:=Group((1,8,10,7,3,5)(2,4,12,9,11,6), ## > (1,9,5,6,3,10)(2,11,12,8,4,7));; ## gap> c:=ChiefSeries(g);;List(c,Size); ## [ 96, 48, 16, 4, 1 ] ## gap> HasElementaryAbelianFactorGroup(c[3],c[4]); ## true ## gap> SetName(c[3],"my_group");; ## gap> a:=AbelianSubfactorAction(g,c[3],c[4]); ## [ [ (1,8,10,7,3,5)(2,4,12,9,11,6), (1,9,5,6,3,10)(2,11,12,8,4,7) ] -> ## [ <an immutable 2x2 matrix over GF2>, ## <an immutable 2x2 matrix over GF2> ], ## MappingByFunction( my_group, ( GF(2)^ ## 2 ), function( e ) ... end, function( r ) ... end ), ## Pcgs([ (2,9,3,8)(4,11,5,10), (1,6,12,7)(4,10,5,11) ]) ] ## gap> mat:=Image(a[1],g); ## Group([ <an immutable 2x2 matrix over GF2>, ## <an immutable 2x2 matrix over GF2> ]) ## gap> Size(mat); ## 3 ## gap> e:=PreImagesRepresentative(a[2],[Z(2),0*Z(2)]); ## (2,9,3,8)(4,11,5,10) ## gap> e in c[3];e in c[4]; ## true ## false ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareOperation( "AbelianSubfactorAction",[IsGroup,IsGroup,IsGroup] ); ############################################################################# ## #F OnPoints( <pnt>, <g> ) ## ## <#GAPDoc Label="OnPoints"> ## <ManSection> ## <Func Name="OnPoints" Arg='pnt, g'/> ## ## <Description> ## <Index>conjugation</Index> ## <Index Subkey="by conjugation">action</Index> ## returns <C><A>pnt</A> ^ <A>g</A></C>. ## This is for example the action of a permutation group on points, ## or the action of a group on its elements via conjugation, ## that is, if both <A>pnt</A> and <A>g</A> are elements from a common group ## then <C><A>pnt</A> ^ <A>g</A></C> is equal to ## <A>g</A><M>^{{-1}}</M><C>*</C><A>pnt</A><C>*</C><A>g</A>. ## The action of a matrix group on vectors from the right is described by ## both <Ref Func="OnPoints"/> and <Ref Func="OnRight"/>. ## <Example><![CDATA[ ## gap> OnPoints( 1, (1,2,3) ); ## 2 ## gap> OnPoints( (1,2), (1,2,3) ); ## (2,3) ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, 1, OnPoints ); ## [ 1, 2, 3, 4 ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## ############################################################################# ## #F OnRight( <pnt>, <g> ) ## ## <#GAPDoc Label="OnRight"> ## <ManSection> ## <Func Name="OnRight" Arg='pnt, g'/> ## ## <Description> ## returns <C><A>pnt</A> * <A>g</A></C>. ## This is for example the action of a group on its elements via right ## multiplication, ## or the action of a group on the cosets of a subgroup. ## The action of a matrix group on vectors from the right is described by ## both <Ref Func="OnPoints"/> and <Ref Func="OnRight"/>. ## <Example><![CDATA[ ## gap> OnRight( [ 1, 2 ], [ [ 1, 2 ], [ 3, 4 ] ] ); ## [ 7, 10 ] ## gap> OnRight( (1,2,3), (2,3,4) ); ## (1,3)(2,4) ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, (), OnRight ); ## [ (), (1,2,3), (2,3,4), (1,3,2), (1,3)(2,4), (1,2)(3,4), (2,4,3), ## (1,4,2), (1,4,3), (1,3,4), (1,2,4), (1,4)(2,3) ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## ############################################################################# ## #F OnLeftInverse( <pnt>, <g> ) ## ## <#GAPDoc Label="OnLeftInverse"> ## <ManSection> ## <Func Name="OnLeftInverse" Arg='pnt, g'/> ## ## <Description> ## returns <M><A>g</A>^{{-1}}</M> <C>* <A>pnt</A></C>. ## Forming the inverse is necessary to make this a proper action, ## as in &GAP; groups always act from the right. ## <P/> ## <Ref Func="OnLeftInverse"/> is used for example in the representation ## of a right coset as an external set ## (see <Ref Sect="External Sets"/>), ## that is, a right coset <M>Ug</M> is an external set for the group ## <M>U</M> acting on it via <Ref Func="OnLeftInverse"/>.) ## <Example><![CDATA[ ## gap> OnLeftInverse( [ 1, 2 ], [ [ 1, 2 ], [ 3, 4 ] ] ); ## [ 0, 1/2 ] ## gap> OnLeftInverse( (1,2,3), (2,3,4) ); ## (1,2,4) ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, (), OnLeftInverse ); ## [ (), (1,3,2), (2,4,3), (1,2,3), (1,3)(2,4), (1,2)(3,4), (2,3,4), ## (1,2,4), (1,3,4), (1,4,3), (1,4,2), (1,4)(2,3) ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## ############################################################################# ## #F OnSets( <set>, <g> ) ## ## <#GAPDoc Label="OnSets"> ## <ManSection> ## <Func Name="OnSets" Arg='set, g'/> ## ## <Description> ## <Index Subkey="on sets">action</Index> ## <Index Subkey="on blocks">action</Index> ## Let <A>set</A> be a proper set ## (see <Ref Sect="Sorted Lists and Sets"/>). ## <Ref Func="OnSets"/> returns the proper set formed by the images ## of all points <M>x</M> of <A>set</A> via the action function ## <Ref Func="OnPoints"/>, applied to <M>x</M> and <A>g</A>. ## <P/> ## <Ref Func="OnSets"/> is for example used to compute the action of ## a permutation group on blocks. ## <P/> ## (<Ref Func="OnTuples"/> is an action on lists that preserves the ordering ## of entries.) ## <Example><![CDATA[ ## gap> OnSets( [ 1, 3 ], (1,2,3) ); ## [ 1, 2 ] ## gap> OnSets( [ (2,3), (1,2) ], (1,2,3) ); ## [ (2,3), (1,3) ] ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, [ 1, 2 ], OnSets ); ## [ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 4 ], [ 1, 4 ], [ 2, 4 ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## ############################################################################# ## #F OnTuples( <tup>, <g> ) ## ## <#GAPDoc Label="OnTuples"> ## <ManSection> ## <Func Name="OnTuples" Arg='tup, g'/> ## ## <Description> ## Let <A>tup</A> be a list. ## <Ref Func="OnTuples"/> returns the list formed by the images ## of all points <M>x</M> of <A>tup</A> via the action function ## <Ref Func="OnPoints"/>, applied to <M>x</M> and <A>g</A>. ## <P/> ## (<Ref Func="OnSets"/> is an action on lists that additionally sorts ## the entries of the result.) ## <Example><![CDATA[ ## gap> OnTuples( [ 1, 3 ], (1,2,3) ); ## [ 2, 1 ] ## gap> OnTuples( [ (2,3), (1,2) ], (1,2,3) ); ## [ (1,3), (2,3) ] ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, [ 1, 2 ], OnTuples ); ## [ [ 1, 2 ], [ 2, 3 ], [ 1, 3 ], [ 3, 1 ], [ 3, 4 ], [ 2, 1 ], ## [ 1, 4 ], [ 4, 1 ], [ 4, 2 ], [ 3, 2 ], [ 2, 4 ], [ 4, 3 ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## ############################################################################# ## #F OnPairs( <tup>, <g> ) ## ## <#GAPDoc Label="OnPairs"> ## <ManSection> ## <Func Name="OnPairs" Arg='tup, g'/> ## ## <Description> ## is a special case of <Ref Func="OnTuples"/> for lists <A>tup</A> ## of length 2. ## </Description> ## </ManSection> ## <#/GAPDoc> ## ############################################################################# ## #F OnLines( <vec>, <g> ) ## ## <#GAPDoc Label="OnLines"> ## <ManSection> ## <Func Name="OnLines" Arg='vec, g'/> ## ## <Description> ## Let <A>vec</A> be a <E>normed</E> row vector, that is, ## its first nonzero entry is normed to the identity of the relevant field, ## see <Ref Attr="NormedRowVector"/>. ## The function <Ref Func="OnLines"/> returns the row vector obtained from ## first multiplying <A>vec</A> from the right with <A>g</A> ## (via <Ref Func="OnRight"/>) and then normalizing the resulting row vector ## by scalar multiplication from the left. ## <P/> ## This action corresponds to the projective action of a matrix group ## on one-dimensional subspaces. ## <P/> ## If <A>vec</A> is a zero vector or is not normed then ## an error is triggered ## (see <Ref Sect="Action on canonical representatives"/>). ## <P/> ## <Example><![CDATA[ ## gap> OnLines( [ 1, 2 ], [ [ 1, 2 ], [ 3, 4 ] ] ); ## [ 1, 10/7 ] ## gap> gl:=GL(2,5);;v:=[1,0]*Z(5)^0; ## [ Z(5)^0, 0*Z(5) ] ## gap> h:=Action(gl,Orbit(gl,v,OnLines),OnLines); ## Group([ (2,3,5,6), (1,2,4)(3,6,5) ]) ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction("OnLines"); ############################################################################# ## #F OnSetsSets( <set>, <g> ) ## ## <#GAPDoc Label="OnSetsSets"> ## <ManSection> ## <Func Name="OnSetsSets" Arg='set, g'/> ## ## <Description> ## implements the action on sets of sets. ## For the special case that the sets are pairwise disjoint, ## it is possible to use <Ref Func="OnSetsDisjointSets"/>. ## <A>set</A> must be a sorted list whose entries are again sorted lists, ## otherwise an error is triggered ## (see <Ref Sect="Action on canonical representatives"/>). ## <Example><![CDATA[ ## gap> OnSetsSets( [ [ 1, 2 ], [ 3, 4 ] ], (1,2,3) ); ## [ [ 1, 4 ], [ 2, 3 ] ] ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, [ [ 1, 2 ], [ 3, 4 ] ], OnSetsSets ); ## [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], ## [ [ 1, 3 ], [ 2, 4 ] ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "OnSetsSets" ); ############################################################################# ## #F OnSetsDisjointSets( <set>, <g> ) ## ## <#GAPDoc Label="OnSetsDisjointSets"> ## <ManSection> ## <Func Name="OnSetsDisjointSets" Arg='set, g'/> ## ## <Description> ## implements the action on sets of pairwise disjoint sets ## (see also <Ref Func="OnSetsSets"/>). ## <A>set</A> must be a sorted list whose entries are again sorted lists, ## otherwise an error is triggered ## (see <Ref Sect="Action on canonical representatives"/>). ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction( "OnSetsDisjointSets" ); ############################################################################# ## #F OnSetsTuples( <set>, <g> ) ## ## <#GAPDoc Label="OnSetsTuples"> ## <ManSection> ## <Func Name="OnSetsTuples" Arg='set, g'/> ## ## <Description> ## implements the action on sets of tuples. ## <A>set</A> must be a sorted list, ## otherwise an error is triggered ## (see <Ref Sect="Action on canonical representatives"/>). ## <Example><![CDATA[ ## gap> OnSetsTuples( [ [ 1, 2 ], [ 3, 4 ] ], (1,2,3) ); ## [ [ 1, 4 ], [ 2, 3 ] ] ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, [ [ 1, 2 ], [ 3, 4 ] ], OnSetsTuples ); ## [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 1, 4 ], [ 2, 3 ] ], ## [ [ 1, 3 ], [ 4, 2 ] ], [ [ 2, 4 ], [ 3, 1 ] ], ## [ [ 2, 1 ], [ 4, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction("OnSetsTuples"); ############################################################################# ## #F OnTuplesSets( <set>, <g> ) ## ## <#GAPDoc Label="OnTuplesSets"> ## <ManSection> ## <Func Name="OnTuplesSets" Arg='set, g'/> ## ## <Description> ## implements the action on tuples of sets. ## <A>set</A> must be a list whose entries are again sorted lists, ## otherwise an error is triggered ## (see <Ref Sect="Action on canonical representatives"/>). ## <Example><![CDATA[ ## gap> OnTuplesSets( [ [ 2, 3 ], [ 3, 4 ] ], (1,2,3) ); ## [ [ 1, 3 ], [ 1, 4 ] ] ## gap> g:= Group( (1,2,3), (2,3,4) );; ## gap> Orbit( g, [ [ 1, 2 ], [ 3, 4 ] ], OnTuplesSets ); ## [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], ## [ [ 1, 3 ], [ 2, 4 ] ], [ [ 3, 4 ], [ 1, 2 ] ], ## [ [ 1, 4 ], [ 2, 3 ] ], [ [ 2, 4 ], [ 1, 3 ] ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction("OnTuplesSets"); ############################################################################# ## #F OnTuplesTuples( <set>, <g> ) ## ## <#GAPDoc Label="OnTuplesTuples"> ## <ManSection> ## <Func Name="OnTuplesTuples" Arg='set, g'/> ## ## <Description> ## implements the action on tuples of tuples. ## <Example><![CDATA[ ## gap> OnTuplesTuples( [ [ 2, 3 ], [ 3, 4 ] ], (1,2,3) ); ## [ [ 3, 1 ], [ 1, 4 ] ] ## gap> g:=Group((1,2,3),(2,3,4));; ## gap> Orbit(g,[[1,2],[3,4]],OnTuplesTuples); ## [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 2, 3 ], [ 1, 4 ] ], ## [ [ 1, 3 ], [ 4, 2 ] ], [ [ 3, 1 ], [ 2, 4 ] ], ## [ [ 3, 4 ], [ 1, 2 ] ], [ [ 2, 1 ], [ 4, 3 ] ], ## [ [ 1, 4 ], [ 2, 3 ] ], [ [ 4, 1 ], [ 3, 2 ] ], ## [ [ 4, 2 ], [ 1, 3 ] ], [ [ 3, 2 ], [ 4, 1 ] ], ## [ [ 2, 4 ], [ 3, 1 ] ], [ [ 4, 3 ], [ 2, 1 ] ] ] ## ]]></Example> ## </Description> ## </ManSection> ## <#/GAPDoc> ## DeclareGlobalFunction("OnTuplesTuples"); ############################################################################# ## #O DomainForAction( <pnt>, <acts>, <act> ) ## ## <ManSection> ## <Oper Name="DomainForAction" Arg='pnt, acts, act'/> ## ## <Description> ## returns a domain which will contain the orbit of <A>pnt</A> under the action ## <A>act</A> of the group ## generated by <A>acts</A>. (Such a domain can be helpful for obtaining ## a dictionary.) ## The default method returns <K>fail</K> to indicate that no special domain is ## defined, a special method exists for matrix groups over finite fields. ## </Description> ## </ManSection> ## DeclareOperation("DomainForAction",[IsObject,IsListOrCollection,IsFunction]);
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