|
#############################################################################
##
## 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( "InfoAction" );
DeclareSynonym( "InfoOperation",InfoAction );
#############################################################################
##
#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 );
OrbitishReq := [ IsGroup, IsListOrCollection, IsObject,
IsList,
IsList,
IsFunction ];
MakeImmutable(OrbitishReq);
OrbitsishReq := [ IsGroup, IsListOrCollection,
IsList,
IsList,
IsFunction ];
MakeImmutable(OrbitsishReq);
#############################################################################
##
#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 and IsExternalSet,
[ "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,
[ "generators", "operators", "funcOperation" ] );
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",
IsActionHomomorphism, [ ] );
#############################################################################
##
#R IsConstituentHomomorphism(<obj>)
##
## <ManSection>
## <Filt Name="IsConstituentHomomorphism" Arg='obj' Type='Representation'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareRepresentation( "IsConstituentHomomorphism",
IsActionHomomorphism, [ "conperm" ] );
DeclareRepresentation( "IsBlocksHomomorphism",
IsActionHomomorphism, [ "reps" ] );
#############################################################################
##
#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", IsExternalSet );
#############################################################################
##
#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( "CanonicalRepresentativeOfExternalSet", IsExternalSet );
#############################################################################
##
#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( "CanonicalRepresentativeDeterminatorOfExternalSet",
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>
##
DeclareProperty( "CanEasilyDetermineCanonicalRepresentativeExternalSet",
IsExternalSet );
InstallTrueMethod(CanEasilyDetermineCanonicalRepresentativeExternalSet,
HasCanonicalRepresentativeDeterminatorOfExternalSet);
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( "ActorOfExternalSet", IsExternalSet );
DeclareSynonymAttr( "OperatorOfExternalSet", ActorOfExternalSet );
#############################################################################
##
#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("TestIdentityAction",function(acts,pnt,act)
local id,img;
if ValueOption("NoTestAction")<>true and Length(acts)>0 then
id:=One(acts[1]);
img:=act(pnt,id);
if img<>pnt then
Error("Action not well-defined. See the manual section\n",
"``Action on canonical representatives''.");
fi;
pnt:=img;
fi;
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", function( name, reqs, usetype, NewAorP )
local op;
# Create the attribute or property.
op:= NewAorP( name, IsExternalSet );
BIND_GLOBAL( name, op );
BIND_SETTER_TESTER( name, SETTER_FILTER( op ), TESTER_FILTER( op ) );
# Make a declaration for non-default methods.
DeclareOperation( name, reqs );
# Install the default methods.
# 1. `op( <xset> )'
# (The `usetype' value concerns only the `return' statement.)
if usetype then
InstallMethod( op,
"for an external set",
true,
[ IsExternalSet ], 0,
function( xset )
local G, gens, acts, act;
G := ActingDomain( xset );
if IsExternalSetByActorsRep( xset ) then
gens := xset!.generators;
acts := xset!.operators;
act := xset!.funcOperation;
else
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
InstallMethod( op,
"for an external set",
true,
[ IsExternalSet ], 0,
function( xset )
local G, gens, acts, act;
G := ActingDomain( xset );
if IsExternalSetByActorsRep( xset ) then
gens := xset!.generators;
acts := xset!.operators;
act := xset!.funcOperation;
else
if CanEasilyComputePcgs( G ) then
gens := Pcgs( G );
else
gens := GeneratorsOfGroup( G );
fi;
acts := gens;
act := FunctionAction( xset );
fi;
return op( G, Enumerator( xset ), gens, acts, act );
end );
fi;
# 2. `op( <permgrp> )'
InstallOtherMethod( op,
"for a permutation group",
true,
[ IsPermGroup ], 0,
function( G )
local gens;
gens:= GeneratorsOfGroup( G );
return op( G, MovedPoints( G ), gens, gens, OnPoints );
end );
# 3. `op( <G>, <Omega> )' with group <G> and domain or list <Omega>
# (add group generators and `OnPoints' as default arguments)
InstallOtherMethod( op,
"for a group and a domain or list",
true,
[ IsGroup, IsObject ], 0,
function( G, D )
local gens;
if CanEasilyComputePcgs( G ) then
gens:= Pcgs( G );
else
gens:= GeneratorsOfGroup( G );
fi;
if IsDomain( D ) then
if IsFinite( D ) then D:= AsSSortedList( D ); else D:= Enumerator( D ); fi;
fi;
return op( G, D, gens, gens, OnPoints );
end );
# 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> )')
InstallOtherMethod( op,
"for a permutation group and a domain or list of integers",
true,
[ IsPermGroup, IsListOrCollection ], 0,
function( G, D )
if D = MovedPoints( G ) then
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( op,
"for a group, a domain or list, and a function",
true,
[ IsGroup, IsObject, IsFunction ], 0,
function( G, D, act )
local gens;
if CanEasilyComputePcgs( G ) then
gens:= Pcgs( G );
else
gens:= GeneratorsOfGroup( G );
fi;
if IsDomain( D ) then
if IsFinite( D ) then D:= AsSSortedList( D ); else D:= Enumerator( D ); fi;
fi;
return op( G, D, gens, gens, act );
end );
# 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'
# then call `op( <G> )')
InstallOtherMethod( op,
"for permutation group, domain or list of integers, and function",
true,
[ IsPermGroup, IsListOrCollection, IsFunction ], 0,
function( G, D, act )
if D = MovedPoints( G ) and IsIdenticalObj( act, OnPoints ) then
return op( G );
else
TryNextMethod();
fi;
end );
# 7. `op( <G>, <Omega>, <gens>, <acts> )' with group <G>,
# domain or list <Omega>, and two lists <gens>, <acts>
# (add default value `OnPoints')
InstallOtherMethod( op,
"for a group, a domain or list, and two lists",
true,
[ IsGroup, IsObject, IsList, IsList ], 0,
function( G, D, gens, acts )
if IsDomain( D ) then
if IsFinite( D ) then D:= AsSSortedList( D ); else D:= Enumerator( D ); fi;
fi;
return op( G, D, gens, acts, OnPoints );
end );
# 8. `op( <G>, <Omega>, <gens>, <acts>, <act> )' with group <G>,
# domain <Omega>, two lists <gens>, <acts>, and function <act>
# (delegate to a (non-default!) method with <Omega> a list)
InstallOtherMethod( op,
"for a group, a domain, two lists, and a function",
true,
[ IsGroup, IsDomain, IsList, IsList, IsFunction ], 0,
function( G, D, gens, acts, act )
return op( G, Enumerator( D ), gens, acts, act );
end );
end );
#############################################################################
##
#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,realenum )
local str, orbish, func,isnotest;
# Create the operation.
str:= SHALLOW_COPY_OBJ( name );
isnotest:=Length(name)>5 and name{[Length(name)-5..Length(name)]}="Blocks";
isnotest:=isnotest or name="ExternalSubset";
APPEND_LIST_INTR( str, "Op" );
orbish := NewOperation( str, reqs );
BIND_GLOBAL( str, orbish );
# Create the wrapper function.
func := function( arg )
local le, usage, xset, pnt, G, D, gens, acts, act, attrG, result;
le:= Length( arg );
usage:= Concatenation( "usage: ", name, "(<xset>,<pnt>)\n",
"or ", name, "(<G>[,<Omega>],<pnt>[,<gens>,<acts>][,<act>])" );
# Get the arguments.
if le = 0 then
Error( usage );
elif le <= 2 and IsExternalSet( arg[ 1 ] ) then
xset := arg[ 1 ];
if Length(arg)>1 then
# force immutability
pnt := Immutable(arg[ 2 ]);
else
# `Blocks' like operations
pnt:=[];
fi;
G := ActingDomain( xset );
if realenum then
D:=Enumerator(xset);
else
if HasHomeEnumerator( xset ) then
D := HomeEnumerator( xset );
fi;
fi;
if IsExternalSetByActorsRep( xset ) then
gens := xset!.generators;
acts := xset!.operators;
act := xset!.funcOperation;
else
act := FunctionAction( xset );
fi;
elif 2 <= le and le <= 6 then
G:= arg[ 1 ];
if IsFunction( arg[ le ] ) then
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[2] );
else
pnt:= Immutable( arg[3] );
# deal with Omega
D:= arg[2];
if not ( famrel( FamilyObj( D ), FamilyObj( pnt ) ) or
( IsList( pnt ) and IsEmpty( pnt ) ) ) then
Error( "wrong relation between <D> and <pnt>" );
elif IsDomain( D ) then
if IsFinite( D ) then
D:= AsSSortedList( D );
else
D:= Enumerator( D );
fi;
fi;
fi;
if le > 3 then
# deal with acts, gens
gens:= arg[ le - 1 ];
acts:= arg[ le ];
fi;
else
Error( usage );
fi;
if not IsBound( gens ) then
if (not IsPermGroup(G)) and CanEasilyComputePcgs( G ) then
gens := Pcgs( G );
else
gens := GeneratorsOfGroup( G );
fi;
acts := gens;
fi;
if not isnotest then
# `Blocks' has <pnt> a list of points
pnt:=TestIdentityAction(acts,pnt,act);
fi;
# In the case of `[Maximal]Blocks', where $G$ is a permutation group
# acting on its moved points, use an attribute for $G$.
attrG := IsOperation( usetype )
and gens = acts
and act = OnPoints
and not IsBound( D )
and HasMovedPoints( G )
and pnt = MovedPoints( G );
if attrG and IsBound( xset ) and Tester( usetype )( xset ) then
result := usetype( xset );
elif attrG and Tester( usetype )( G ) then
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
# <Omega>, i.e., the operation domain!
result := orbish( G, pnt, gens, acts, act );
fi;
# Store the result in the case of an attribute `[Maximal]BlocksAttr'.
if attrG then
if IsBound( xset ) then
Setter( usetype )( xset, result );
fi;
Setter( usetype )( G, result );
fi;
return result;
end;
BIND_GLOBAL( name, func );
end );
#############################################################################
##
#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>
##
DeclareGlobalFunction( "ActionHomomorphism" );
DeclareAttribute( "ActionHomomorphismAttr", IsExternalSet );
DeclareGlobalFunction( "ActionHomomorphismConstructor" );
#############################################################################
##
#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", IsExternalSet );
#############################################################################
##
#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", IsActionHomomorphism );
#############################################################################
##
#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");
#############################################################################
##
#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 )
local str, orbish, func;
# Create the operation.
str:= SHALLOW_COPY_OBJ( name );
APPEND_LIST_INTR( str, "Op" );
orbish := NewOperation( str, reqs );
BIND_GLOBAL( str, orbish );
# Create the wrapper function.
func := function( arg )
local G, D, pnt, gens, acts, act, p, result,le;
# Get the arguments.
if 2 <= Length( arg ) then
le:=Length(arg);
G := arg[ 1 ];
if IsFunction( arg[ le ] ) then
act := arg[ le ];
else
act := OnPoints;
fi;
if Length( arg ) > 2
and famrel( FamilyObj( arg[ 2 ] ), FamilyObj( arg[ 3 ] ) )
then
D := arg[ 2 ];
if IsDomain( D ) then
if IsFinite( D ) then D:= AsSSortedList( D ); else D:= Enumerator( D ); fi;
fi;
p := 3;
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>[,<Omega>],<pnts>[,<gens>,<acts>][,<act>])" );
fi;
if not IsBound( gens ) then
if (not IsPermGroup(G)) and CanEasilyComputePcgs( G ) then
gens := Pcgs( G );
else
gens := GeneratorsOfGroup( G );
fi;
acts := gens;
fi;
for p in pnt do
TestIdentityAction(acts,p,act);
od;
if IsBound( D ) then
result := orbish( G, D, pnt, gens, acts, act );
else
result := orbish( G, pnt, gens, acts, act );
fi;
return result;
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", OrbitishReq, IsCollsElms, true, false );
#############################################################################
##
#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, false, NewAttribute );
#############################################################################
##
#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( "OrbitsDomain", OrbitsishReq, false, NewAttribute );
#############################################################################
##
#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( "OrbitLengths", OrbitsishReq, false, NewAttribute );
#############################################################################
##
#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( "OrbitLengthsDomain", OrbitsishReq, false, NewAttribute );
#############################################################################
##
#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, IsCollsElms, false,false );
#############################################################################
##
#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( "ExternalOrbits", OrbitsishReq, true, NewAttribute );
#############################################################################
##
#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>
##
OrbitsishOperation( "Transitivity", OrbitsishReq, false, NewAttribute );
#############################################################################
##
#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( "BlocksAttr", IsExternalSet );
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>
## returns a block system that is maximal (i.e., blocks are maximal with
## 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",
[ IsGroup, IsList, IsList,
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", IsExternalSet );
OrbitishFO( "RepresentativesMinimalBlocks",
[ IsGroup, IsList, IsList,
IsList,
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( "Earns", OrbitsishReq, false, NewAttribute );
#############################################################################
##
#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, NewProperty );
#############################################################################
##
#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/>
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