#############################################################################
##
## 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
##
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##
#V InfoBckt
##
## <#GAPDoc Label="InfoBckt">
## <ManSection>
## <InfoClass Name="InfoBckt"/>
##
## <Description>
## is the info class for the partition backtrack routines.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareInfoClass( "InfoBckt" );
DeclareGlobalFunction( "UnslicedPerm@" );
DeclareGlobalFunction( "PreImageWord" );
DeclareGlobalFunction( "ExtendedT" );
DeclareGlobalFunction( "MeetPartitionStrat" );
DeclareGlobalFunction( "MeetPartitionStratCell" );
DeclareGlobalFunction( "StratMeetPartition" );
DeclareGlobalFunction( "Suborbits" );
DeclareGlobalFunction( "OrbitalPartition" );
DeclareGlobalFunction( "EmptyRBase" );
DeclareGlobalFunction( "IsTrivialRBase" );
DeclareGlobalFunction( "AddRefinement" );
DeclareGlobalFunction( "ProcessFixpoint" );
DeclareGlobalFunction( "RegisterRBasePoint" );
DeclareGlobalFunction( "NextRBasePoint" );
DeclareGlobalFunction( "RRefine" );
DeclareGlobalFunction( "PBIsMinimal" );
DeclareGlobalFunction( "SubtractBlistOrbitStabChain" );
DeclareGlobalFunction( "PartitionBacktrack" );
DeclareGlobalFunction("SuboLiBli");
DeclareGlobalFunction("SuboSiBli");
DeclareGlobalFunction("SuboTruePos");
DeclareGlobalFunction("SuboUniteBlist");
DeclareGlobalFunction("ConcatSubos");
DeclareGlobalFunction("Refinements_ProcessFixpoint");
DeclareGlobalFunction("Refinements_Intersection");
DeclareGlobalFunction("Refinements_Centralizer");
DeclareGlobalFunction("Refinements__MakeBlox");
DeclareGlobalFunction("Refinements_SplitOffBlock");
DeclareGlobalFunction("Refinements__RegularOrbit1");
DeclareGlobalFunction("Refinements_RegularOrbit2");
DeclareGlobalFunction("Refinements_RegularOrbit3");
DeclareGlobalFunction("Refinements_Suborbits0");
DeclareGlobalFunction("Refinements_Suborbits1");
DeclareGlobalFunction("Refinements_Suborbits2");
DeclareGlobalFunction("Refinements_Suborbits3");
DeclareGlobalFunction("Refinements_TwoClosure");
DeclareGlobalFunction( "NextLevelRegularGroups" );
DeclareGlobalFunction( "RBaseGroupsBloxPermGroup" );
DeclareGlobalFunction( "RepOpSetsPermGroup" );
DeclareGlobalFunction( "RepOpElmTuplesPermGroup" );
DeclareGlobalFunction( "ConjugatorPermGroup" );
DeclareGlobalFunction( "NormalizerPermGroup" );
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##
#F ElementProperty( <G>, <Pr>[, <L>[, <R>]] ) one element with property
##
## <#GAPDoc Label="ElementProperty">
## <ManSection>
## <Func Name="ElementProperty" Arg='G, Pr[, L[, R]]'/>
##
## <Description>
## <Ref Func="ElementProperty"/> returns an element <M>\pi</M> of the
## permutation group <A>G</A> such that the one-argument function <A>Pr</A>
## returns <K>true</K> for <M>\pi</M>.
## It returns <K>fail</K> if no such element exists in <A>G</A>.
## The optional arguments <A>L</A> and <A>R</A> are subgroups of <A>G</A>
## such that the property <A>Pr</A> has the same value for all elements in
## the cosets <A>L</A> <M>g</M> and <M>g</M> <A>R</A>, respectively,
## with <M>g \in <A>G</A></M>.
## <P/>
## A typical example of using the optional subgroups <A>L</A> and <A>R</A>
## is the conjugacy test for elements <M>a</M> and <M>b</M> for which one
## can set <A>L</A><M>:= C_{<A>G</A>}(a)</M> and
## <A>R</A><M>:= C_{<A>G</A>}(b)</M>.
## <P/>
## <Example><![CDATA[
## gap> propfun:= el -> (1,2,3)^el in [ (1,2,3), (1,3,2) ];;
## gap> SubgroupProperty( g, propfun, Subgroup( g, [ (1,2,3) ] ) );
## Group([ (1,2,3), (2,3) ])
## gap> ElementProperty( g, el -> Order( el ) = 2 );
## (2,4)
## ]]></Example>
## <P/>
## Chapter <Ref Chap="Permutations"/> describes special operations to
## construct permutations in the symmetric group without using backtrack
## constructions.
## <P/>
## Backtrack routines are also called by the methods for permutation groups
## that compute centralizers, normalizers, intersections,
## conjugating elements as well as stabilizers for the operations of a
## permutation group via <Ref Func="OnPoints"/>, <Ref Func="OnSets"/>,
## <Ref Func="OnTuples"/> and <Ref Func="OnSetsSets"/>.
## Some of these methods use more specific refinements than
## <Ref Func="SubgroupProperty"/> or <Ref Func="ElementProperty"/>.
## For the definition of refinements, and how one can define refinements,
## see Section <Ref Sect="The General Backtrack Algorithm with Ordered Partitions"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ElementProperty" );
#############################################################################
##
#F SubgroupProperty( <G>, <Pr>[, <L> ] ) . . . . . . . . fulfilling subgroup
##
## <#GAPDoc Label="SubgroupProperty">
## <ManSection>
## <Func Name="SubgroupProperty" Arg='G, Pr[, L ]'/>
##
## <Description>
## <A>Pr</A> must be a one-argument function that returns <K>true</K> or
## <K>false</K> for elements of the permutation group <A>G</A>,
## and the subset of elements of <A>G</A> that fulfill <A>Pr</A> must
## be a subgroup. (<E>If the latter is not true the result of this operation
## is unpredictable!</E>) This command computes this subgroup.
## The optional argument <A>L</A> must be a subgroup of the set of all
## elements in <A>G</A> fulfilling <A>Pr</A> and can be given if known
## in order to speed up the calculation.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SubgroupProperty" );
#############################################################################
##
#O PartitionStabilizerPermGroup( <G>, <part> )
##
## <ManSection>
## <Oper Name="PartitionStabilizerPermGroup" Arg='G, part'/>
##
## <Description>
## <A>part</A> must be a list of pairwise disjoint sets of points
## on which the permutation group <A>G</A> acts via <C>OnPoints</C>.
## This function computes the stabilizer in <A>G</A> of <A>part</A>, that is,
## the subgroup of all those elements in <A>G</A> that map each set in <A>part</A>
## onto a set in <A>part</A>, via <C>OnSets</C>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "PartitionStabilizerPermGroup" );
#############################################################################
##
#A TwoClosure( <G> )
##
## <#GAPDoc Label="TwoClosure">
## <ManSection>
## <Attr Name="TwoClosure" Arg='G'/>
##
## <Description>
## The <E>2-closure</E> of a transitive permutation group <A>G</A> on
## <M>n</M> points is the largest subgroup of the symmetric group <M>S_n</M>
## which has the same orbits on sets of ordered pairs of points as the group
## <A>G</A> has.
## It also can be interpreted as the stabilizer of the orbital graphs of
## <A>G</A>.
## <Example><![CDATA[
## gap> TwoClosure(Group((1,2,3),(2,3,4)));
## Sym( [ 1 .. 4 ] )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "TwoClosure", IsPermGroup );
