#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Volkmar Felsch.
##
## 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
##
## This file contains the declarations for finitely presented groups
## (fp groups).
##
#############################################################################
##
#V CosetTableDefaultLimit
##
## <#GAPDoc Label="CosetTableDefaultLimit">
## <ManSection>
## <Var Name="CosetTableDefaultLimit"/>
##
## <Description>
## is the default number of cosets with which any coset table is
## initialized before doing a coset enumeration.
## <P/>
## The function performing this coset enumeration will automatically extend
## the table whenever necessary (as long as the number of cosets does not
## exceed the value of <Ref Var="CosetTableDefaultMaxLimit"/>),
## but this is an expensive operation. Thus, if you change the value of
## <Ref Var="CosetTableDefaultLimit"/>, you should set it to a number of
## cosets that you expect to be sufficient for your subsequent
## coset enumerations.
## On the other hand, if you make it too large, your job will unnecessarily
## waste a lot of space.
## <P/>
## The default value of <Ref Var="CosetTableDefaultLimit"/> is <M>1000</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
CosetTableDefaultLimit := 1000;
if IsHPCGAP then
MakeThreadLocal("CosetTableDefaultLimit");
fi;
#############################################################################
##
#V CosetTableDefaultMaxLimit
##
## <#GAPDoc Label="CosetTableDefaultMaxLimit">
## <ManSection>
## <Var Name="CosetTableDefaultMaxLimit"/>
##
## <Description>
## is the default limit for the number of cosets allowed in a coset
## enumeration.
## <P/>
## A coset enumeration will not finish if the subgroup does not have finite
## index, and even if it has it may take many more intermediate cosets than
## the actual index of the subgroup is. To avoid a coset enumeration
## <Q>running away</Q> therefore &GAP; has a <Q>safety stop</Q> built in.
## This is controlled by the global variable
## <Ref Var="CosetTableDefaultMaxLimit"/>.
## <P/>
## If this number of cosets is reached, &GAP; will issue an error message
## and prompt the user to either continue the calculation or to stop it.
## The default value is <M>4096000</M>.
## <P/>
## See also the description of the options to
## <Ref Func="CosetTableFromGensAndRels"/>.
## <P/>
## <Log><![CDATA[
## gap> f := FreeGroup( "a", "b" );;
## gap> u := Subgroup( f, [ f.2 ] );
## Group([ b ])
## gap> Index( f, u );
## Error, the coset enumeration has defined more than 4096000 cosets
## called from
## TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
## CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
## TryCosetTableInWholeGroup( H ) called from
## CosetTableInWholeGroup( H ) called from
## IndexInWholeGroup( H ) called from
## ...
## Entering break read-eval-print loop ...
## type 'return;' if you want to continue with a new limit of 8192000 cosets,
## type 'quit;' if you want to quit the coset enumeration,
## type 'maxlimit := 0; return;' in order to continue without a limit
## brk> quit;
## ]]></Log>
## <P/>
## At this point, a <K>break</K>-loop
## (see Section <Ref Sect="Break Loops"/>) has been entered.
## The line beginning <C>Error</C> tells you why this occurred.
## The next seven lines occur if <Ref Func="OnBreak"/> has its default value
## <Ref Func="Where"/>.
## They explain, in this case,
## how &GAP; came to be doing a coset enumeration.
## Then you are given a number of options of how to escape the
## <K>break</K>-loop:
## you can either continue the calculation with a larger
## number of permitted cosets, stop the calculation if you don't
## expect the enumeration to finish (like in the example above), or continue
## without a limit on the number of cosets. (Choosing the first option will,
## of course, land you back in a <K>break</K>-loop. Try it!)
## <P/>
## Setting <Ref Var="CosetTableDefaultMaxLimit"/>
## (or the <C>max</C> option value, for any function that invokes a coset
## enumeration) to <Ref Var="infinity"/> (or to <M>0</M>) will force all
## coset enumerations to continue until
## they either get a result or exhaust the whole available space.
## For example, each of the following two inputs
## <P/>
## <Listing><![CDATA[
## gap> CosetTableDefaultMaxLimit := 0;;
## gap> Index( f, u );
## ]]></Listing>
## <P/>
## or
## <P/>
## <Listing><![CDATA[
## gap> Index( f, u : max := 0 );
## ]]></Listing>
## <P/>
## have essentially the same effect as choosing the third option
## (typing: <C>maxlimit := 0; return;</C>) at the <C>brk></C> prompt above
## (instead of <C>quit;</C>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
CosetTableDefaultMaxLimit := 2^12*1000;
if IsHPCGAP then
MakeThreadLocal("CosetTableDefaultMaxLimit");
fi;
#############################################################################
##
#V CosetTableStandard
##
## <#GAPDoc Label="CosetTableStandard">
## <ManSection>
## <Var Name="CosetTableStandard"/>
##
## <Description>
## specifies the definition of a <E>standard coset table</E>. It is used
## whenever coset tables or augmented coset tables are created. Its value
## may be <C>"lenlex"</C> or <C>"semilenlex"</C>.
## If it is <C>"lenlex"</C> coset tables will be standardized using
## all their columns as defined in Charles Sims' book
## (this is the new default standard of &GAP;). If it is <C>"semilenlex"</C>
## they will be standardized using only their generator columns (this was
## the original &GAP; standard).
## The default value of <Ref Var="CosetTableStandard"/> is <C>"lenlex"</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
if IsHPCGAP then
MakeThreadLocal("CosetTableStandard");
BindThreadLocal("CosetTableStandard", MakeImmutable("lenlex"));
else
CosetTableStandard := MakeImmutable("lenlex");
fi;
#############################################################################
##
#V InfoFpGroup
##
## <#GAPDoc Label="InfoFpGroup">
## <ManSection>
## <InfoClass Name="InfoFpGroup"/>
##
## <Description>
## The info class for functions dealing with finitely presented groups is
## <Ref InfoClass="InfoFpGroup"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareInfoClass( "InfoFpGroup" );
#############################################################################
##
#C IsSubgroupFgGroup( <H> )
##
## <ManSection>
## <Filt Name="IsSubgroupFgGroup" Arg='H' Type='Category'/>
##
## <Description>
## This category (intended for future extensions) represents (subgroups of)
## a finitely generated group, whose elements are represented as words in
## the generators. However we do not necessarily have a set or relators.
## </Description>
## </ManSection>
##
DeclareCategory( "IsSubgroupFgGroup", IsGroup );
#############################################################################
##
#C IsSubgroupFpGroup( <H> )
##
## <#GAPDoc Label="IsSubgroupFpGroup">
## <ManSection>
## <Filt Name="IsSubgroupFpGroup" Arg='H' Type='Category'/>
##
## <Description>
## is the category for finitely presented groups
## or subgroups of a finitely presented group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareCategory( "IsSubgroupFpGroup", IsSubgroupFgGroup );
# implications for the full family
InstallTrueMethod(CanEasilyTestMembership, IsSubgroupFgGroup and IsWholeFamily);
#############################################################################
##
#F IsFpGroup(<G>)
##
## <#GAPDoc Label="IsFpGroup">
## <ManSection>
## <Filt Name="IsFpGroup" Arg='G'/>
##
## <Description>
## is a synonym for
## <C>IsSubgroupFpGroup(<A>G</A>) and IsGroupOfFamily(<A>G</A>)</C>.
## <P/>
## Free groups are a special case of finitely presented groups,
## namely finitely presented groups with no relators.
##
## <P/>
## Note that <C>FreeGroup(infinity)</C> (which exists e.g. for purposes of
## rewriting presentations with further generators) satisfies this filter,
## though of course it is not finitely generated (and thus not finitely
## presented). <C>IsFpGroup</C> thus is not a proper property test and
## slightly misnamed for the sake of its most prominent uses.
## <P/>
## Another special case are groups given by polycyclic presentations.
## &GAP; uses a special representation for these groups which is created
## in a different way.
## See chapter <Ref Chap="Pc Groups"/> for details.
## <Example><![CDATA[
## gap> g:=FreeGroup(2);
## <free group on the generators [ f1, f2 ]>
## gap> IsFpGroup(g);
## true
## gap> h:=CyclicGroup(2);
## <pc group of size 2 with 1 generator>
## gap> IsFpGroup(h);
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsFpGroup", IsSubgroupFpGroup and IsGroupOfFamily );
#############################################################################
##
#C IsElementOfFpGroup
##
## <ManSection>
## <Filt Name="IsElementOfFpGroup" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategory( "IsElementOfFpGroup",
IsMultiplicativeElementWithInverse and IsAssociativeElement );
#############################################################################
##
#C IsElementOfFpGroupCollection
##
## <ManSection>
## <Filt Name="IsElementOfFpGroupCollection" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryCollections( "IsElementOfFpGroup" );
#############################################################################
##
#m IsSubgroupFpGroup
##
InstallTrueMethod(IsSubgroupFpGroup,IsGroup and IsElementOfFpGroupCollection);
## free groups also are to be fp
InstallTrueMethod(IsSubgroupFpGroup,IsGroup and IsAssocWordCollection);
#############################################################################
##
#M IsGeneratorsOfMagmaWithInverses( <fpelmscoll> )
##
InstallTrueMethod( IsGeneratorsOfMagmaWithInverses,
IsElementOfFpGroupCollection );
#############################################################################
##
#C IsElementOfFpGroupFamily
##
## <ManSection>
## <Filt Name="IsElementOfFpGroupFamily" Arg='obj' Type='Category'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareCategoryFamily( "IsElementOfFpGroup" );
#############################################################################
##
#A FpElmEqualityMethod(<fam>)
##
## <ManSection>
## <Attr Name="FpElmEqualityMethod" Arg='fam'/>
##
## <Description>
## If <A>fam</A> is the elements family of a finitely presented group this
## attribute returns a function <C>equal(<A>left</A>, <A>right</A>)</C> that will be
## used to compare elements in <A>fam</A>.
## </Description>
## </ManSection>
##
DeclareAttribute( "FpElmEqualityMethod",IsElementOfFpGroupFamily);
#############################################################################
##
#A FpElmComparisonMethod(<fam>)
##
## <#GAPDoc Label="FpElmComparisonMethod">
## <ManSection>
## <Attr Name="FpElmComparisonMethod" Arg='fam'/>
##
## <Description>
## If <A>fam</A> is the elements family of a finitely presented group this
## attribute returns a function <C>smaller(<A>left</A>, <A>right</A>)</C>
## that will be used to compare elements in <A>fam</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "FpElmComparisonMethod",IsElementOfFpGroupFamily);
#############################################################################
##
#F SetReducedMultiplication(<f>)
#F SetReducedMultiplication(<e>)
#F SetReducedMultiplication(<fam>)
##
## <#GAPDoc Label="SetReducedMultiplication">
## <ManSection>
## <Func Name="SetReducedMultiplication" Arg='obj'/>
##
## <Description>
## For an FpGroup <A>obj</A>, an element <A>obj</A> of it or the family
## <A>obj</A> of its elements,
## this function will force immediate reduction when multiplying, keeping
## words short at extra cost per multiplication.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("SetReducedMultiplication");
#############################################################################
##
#A FpElementNFFunction(<fam>)
##
## <ManSection>
## <Attr Name="FpElementNFFunction" Arg='fam'/>
##
## <Description>
## If <A>fam</A> is the elements family of a finitely presented group this
## attribute returns a function <A>f</A>, which, when applied to the
## <b>underlying element</b> of an element of <A>fam</A> returns a <b>normal
## form</b> (whose format is not defined and will differ on the method used).
## This normal form can be used (and is used by
## <Ref Func="SetReducedMultiplication"/>) to
## compare elements or to reduce long products.
## </Description>
## </ManSection>
##
DeclareAttribute( "FpElementNFFunction",IsElementOfFpGroupFamily);
# #############################################################################
# ##
# #A FpElmKBRWS(<fam>)
# ##
# ## <ManSection>
# ## <Attr Name="FpElmKBRWS" Arg='fam'/>
# ##
# ## <Description>
# ## If <A>fam</A> is the elements family of a finitely presented group this
# ## attribute returns a list [<A>iso</A>,<A>k</A>,<A>id</A>] where <A>iso</A> is a isomorphism to an
# ## fp monoid, <A>k</A> a confluent rewriting system for the image of <A>iso</A> and
# ## <A>id</A> the element in the free monoid corresponding to the image of the
# ## identity element under <A>iso</A>.
# ## </Description>
# ## </ManSection>
# ##
#DeclareAttribute( "FpElmKBRWS",IsElementOfFpGroupFamily);
#############################################################################
##
#O ElementOfFpGroup( <fam>, <word> )
##
## <#GAPDoc Label="ElementOfFpGroup">
## <ManSection>
## <Oper Name="ElementOfFpGroup" Arg='fam, word'/>
##
## <Description>
## If <A>fam</A> is the elements family of a finitely presented group
## and <A>word</A> is a word in the free generators underlying this
## finitely presented group, this operation creates the element with the
## representative <A>word</A> in the free group.
## <Example><![CDATA[
## gap> ge := ElementOfFpGroup( FamilyObj( g.1 ), f.1*f.2 );
## a*b
## gap> ge in f;
## false
## gap> ge in g;
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ElementOfFpGroup",
[ IsElementOfFpGroupFamily, IsAssocWordWithInverse ] );
#############################################################################
##
#V TCENUM
#V GAPTCENUM
##
## <ManSection>
## <Var Name="TCENUM"/>
## <Var Name="GAPTCENUM"/>
##
## <Description>
## TCENUM is a global record variable whose components contain functions
## used for coset enumeration. By default <C>TCENUM</C> is assigned to
## <C>GAPTCENUM</C>, which contains the coset enumeration functions provided by
## the GAP library.
## </Description>
## </ManSection>
##
BindGlobal("GAPTCENUM",rec(name:="GAP Felsch-type enumerator"));
TCENUM:=GAPTCENUM;
#############################################################################
##
#F CosetTableFromGensAndRels( <fgens>, <grels>, <fsgens> )
##
## <#GAPDoc Label="CosetTableFromGensAndRels">
## <ManSection>
## <Func Name="CosetTableFromGensAndRels" Arg='fgens, grels, fsgens'/>
##
## <Description>
## <Index Key="TCENUM"><C>TCENUM</C></Index>
## <Index Key="GAPTCENUM"><C>GAPTCENUM</C></Index>
## is an internal function which is called by the functions
## <Ref Oper="CosetTable"/>, <Ref Attr="CosetTableInWholeGroup"/>
## and others.
## It is, in fact, the workhorse that performs a Todd-Coxeter
## coset enumeration.
## <A>fgens</A> must be a set of free generators and <A>grels</A> a set
## of relators in these generators. <A>fsgens</A> are subgroup generators
## expressed as words in these generators. The function returns a coset
## table with respect to <A>fgens</A>.
## <P/>
## <Ref Func="CosetTableFromGensAndRels"/> will call
## <C>TCENUM.CosetTableFromGensAndRels</C>.
## This makes it possible to replace the built-in coset enumerator with
## another one by assigning <C>TCENUM</C> to another record.
## <P/>
## The library version which is used by default performs a standard Felsch
## strategy coset enumeration. You can call this function explicitly as
## <C>GAPTCENUM.CosetTableFromGensAndRels</C> even if other coset enumerators
## are installed.
## <P/>
## The expected parameters are
## <List>
## <Mark><A>fgens</A></Mark>
## <Item>
## generators of the free group <A>F</A>
## </Item>
## <Mark><A>grels</A></Mark>
## <Item>
## relators as words in <A>F</A>
## </Item>
## <Mark><A>fsgens</A></Mark>
## <Item>
## subgroup generators as words in <A>F</A>.
## </Item>
## </List>
## <P/>
## <Ref Func="CosetTableFromGensAndRels"/> processes two options (see
## chapter <Ref Chap="Options Stack"/>):
## <List>
## <Mark><C>max</C></Mark>
## <Item>
## The limit of the number of cosets to be defined. If the
## enumeration does not finish with this number of cosets, an error is
## raised and the user is asked whether she wants to continue. The
## default value is the value given in the variable
## <C>CosetTableDefaultMaxLimit</C>. (Due to the algorithm the actual
## limit used can be a bit higher than the number given.)
## </Item>
## <Mark><C>silent</C></Mark>
## <Item>
## If set to <K>true</K> the algorithm will not raise the error
## mentioned under option <C>max</C> but silently return <K>fail</K>.
## This can be useful if an enumeration is only wanted unless it becomes
## too big.
## </Item>
## </List>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("CosetTableFromGensAndRels");
#############################################################################
##
#F IndexCosetTab( <table> )
##
## <ManSection>
## <Func Name="IndexCosetTab" Arg='table'/>
##
## <Description>
## this function returns <C>Length(table[1])</C>, but the table might be empty
## for a no-generator group, in which case 1 is returned.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("IndexCosetTab");
#############################################################################
##
#F StandardizeTable( <table>, <standard> )
##
## <#GAPDoc Label="StandardizeTable">
## <ManSection>
## <Func Name="StandardizeTable" Arg='table, standard'/>
##
## <Description>
## standardizes the given coset table <A>table</A>. The second argument is
## optional. It defines the standard to be used, its values may be
## <C>"lenlex"</C> or <C>"semilenlex"</C> specifying the new or the old
## convention, respectively.
## If no value for the parameter <A>standard</A> is provided the
## function will use the global variable <Ref Var="CosetTableStandard"/>
## instead.
## Note that the function alters the given table, it does not create a copy.
## <Example><![CDATA[
## gap> StandardizeTable( tab, "semilenlex" );
## gap> PrintArray( TransposedMat( tab ) );
## [ [ 1, 1, 2, 4 ],
## [ 3, 3, 4, 1 ],
## [ 2, 2, 3, 3 ],
## [ 5, 5, 1, 2 ],
## [ 4, 4, 5, 5 ] ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("StandardizeTable");
#############################################################################
##
#F StandardizeTable2( <table>, <table2>, <standard> )
##
## <ManSection>
## <Func Name="StandardizeTable2" Arg='table, table2, standard'/>
##
## <Description>
## standardizes the augmented coset table given by <A>table</A> and <A>table2</A>.
## The third argument is optional. It defines the standard to be used, its
## values may be <C>"lenlex"</C> or <C>"semilenlex"</C> specifying the new or the old
## convention, respectively. If no value for the parameter <A>standard</A> is
## provided the function will use the global variable <C>CosetTableStandard</C>
## instead. Note that the function alters the given table, it does not
## create a copy.
## <P/>
## Warning: The function alters just the two tables. Any further lists
## involved in the object <E>augmented coset table</E> which refer to these two
## tables will not be updated.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("StandardizeTable2");
#############################################################################
##
#A CosetTableInWholeGroup(< H >)
#O TryCosetTableInWholeGroup(< H >)
##
## <#GAPDoc Label="CosetTableInWholeGroup">
## <ManSection>
## <Attr Name="CosetTableInWholeGroup" Arg='H'/>
## <Oper Name="TryCosetTableInWholeGroup" Arg='H'/>
##
## <Description>
## is equivalent to <C>CosetTable(<A>G</A>,<A>H</A>)</C> where <A>G</A> is
## the (unique) finitely presented group such that <A>H</A> is a subgroup
## of <A>G</A>.
## It overrides a <C>silent</C> option
## (see <Ref Func="CosetTableFromGensAndRels"/>) with <K>false</K>.
## <P/>
## The variant <Ref Oper="TryCosetTableInWholeGroup"/> does not override the
## <C>silent</C> option with <K>false</K> in case a coset table is only
## wanted if not too expensive.
## It will store a result that is not <K>fail</K> in the attribute
## <Ref Attr="CosetTableInWholeGroup"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "CosetTableInWholeGroup", IsGroup );
DeclareOperation( "TryCosetTableInWholeGroup", [IsGroup] );
InstallTrueMethod(CanEasilyTestMembership,
IsSubgroupFpGroup and HasCosetTableInWholeGroup);
#############################################################################
##
#A CosetTableNormalClosureInWholeGroup(< H >)
##
## <ManSection>
## <Attr Name="CosetTableNormalClosureInWholeGroup" Arg='H'/>
##
## <Description>
## is equivalent to <C>CosetTableNormalClosure(<A>G</A>,<A>H</A>)</C> where <A>G</A> is the
## (unique) finitely presented group such that <A>H</A> is a subgroup of <A>G</A>.
## It overrides a <C>silent</C> option (see <Ref Func="CosetTableFromGensAndRels"/>) with
## <K>false</K>.
## </Description>
## </ManSection>
##
DeclareAttribute( "CosetTableNormalClosureInWholeGroup", IsGroup );
#############################################################################
##
#F TracedCosetFpGroup( <tab>, <word>, <pt> )
##
## <#GAPDoc Label="TracedCosetFpGroup">
## <ManSection>
## <Func Name="TracedCosetFpGroup" Arg='tab, word, pt'/>
##
## <Description>
## Traces the coset number <A>pt</A> under the word <A>word</A> through the
## coset table <A>tab</A>.
## (Note: <A>word</A> must be in the free group, use
## <Ref Oper="UnderlyingElement" Label="fp group elements"/> if in doubt.)
## <Example><![CDATA[
## gap> TracedCosetFpGroup(tab,UnderlyingElement(g.1),2);
## 4
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("TracedCosetFpGroup");
#############################################################################
##
#F SubgroupOfWholeGroupByCosetTable( <fpfam>, <tab> )
##
## <#GAPDoc Label="SubgroupOfWholeGroupByCosetTable">
## <ManSection>
## <Func Name="SubgroupOfWholeGroupByCosetTable" Arg='fpfam, tab'/>
##
## <Description>
## takes a family <A>fpfam</A> of an FpGroup and a standardized coset
## table <A>tab</A>
## and returns the subgroup of <A>fpfam</A><C>!.wholeGroup</C> defined by
## this coset table. The function will not check whether the coset table is
## standardized.
## See also <Ref Oper="CosetTableBySubgroup"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("SubgroupOfWholeGroupByCosetTable");
#############################################################################
##
#F SubgroupOfWholeGroupByQuotientSubgroup( <fpfam>, <Q>, <U> )
##
## <#GAPDoc Label="SubgroupOfWholeGroupByQuotientSubgroup">
## <ManSection>
## <Func Name="SubgroupOfWholeGroupByQuotientSubgroup" Arg='fpfam, Q, U'/>
##
## <Description>
## takes a FpGroup family <A>fpfam</A>, a finitely generated group <A>Q</A>
## such that the fp generators of <A>fpfam</A> can be mapped by an
## epimorphism <M>phi</M> onto the <Ref Attr="GeneratorsOfGroup"/> value
## of <A>Q</A>, and a subgroup <A>U</A> of <A>Q</A>.
## It returns the subgroup of <A>fpfam</A><C>!.wholeGroup</C> which is
## the full preimage of <A>U</A> under <M>phi</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("SubgroupOfWholeGroupByQuotientSubgroup");
#############################################################################
##
#R IsSubgroupOfWholeGroupByQuotientRep(<G>)
##
## <#GAPDoc Label="IsSubgroupOfWholeGroupByQuotientRep">
## <ManSection>
## <Filt Name="IsSubgroupOfWholeGroupByQuotientRep" Arg='G'
## Type='Representation'/>
##
## <Description>
## is the representation for subgroups of an FpGroup, given by a quotient
## subgroup. The components <A>G</A><C>!.quot</C> and <A>G</A><C>!.sub</C>
## hold quotient, respectively subgroup.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation("IsSubgroupOfWholeGroupByQuotientRep",
IsSubgroupFpGroup and IsComponentObjectRep,["quot","sub"]);
#############################################################################
##
#F DefiningQuotientHomomorphism(<U>)
##
## <#GAPDoc Label="DefiningQuotientHomomorphism">
## <ManSection>
## <Func Name="DefiningQuotientHomomorphism" Arg='U'/>
##
## <Description>
## if <A>U</A> is a subgroup in quotient representation
## (<Ref Filt="IsSubgroupOfWholeGroupByQuotientRep"/>),
## this function returns the
## defining homomorphism from the whole group to <A>U</A><C>!.quot</C>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("DefiningQuotientHomomorphism");
#############################################################################
##
#A AsSubgroupOfWholeGroupByQuotient(<U>)
##
## <#GAPDoc Label="AsSubgroupOfWholeGroupByQuotient">
## <ManSection>
## <Attr Name="AsSubgroupOfWholeGroupByQuotient" Arg='U'/>
##
## <Description>
## returns the same subgroup in the representation
## <Ref Attr="AsSubgroupOfWholeGroupByQuotient"/>.
## <P/>
## See also <Ref Func="SubgroupOfWholeGroupByCosetTable"/>
## and <Ref Oper="CosetTableBySubgroup"/>.
## <P/>
## This technique is used by &GAP; for example to represent the derived
## subgroup, which is obtained from the quotient <M>G/G'</M>.
## <Example><![CDATA[
## gap> f:=FreeGroup(2);;g:=f/[f.1^6,f.2^6,(f.1*f.2)^6];;
## gap> d:=DerivedSubgroup(g);
## Group(<fp, no generators known>)
## gap> Index(g,d);
## 36
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("AsSubgroupOfWholeGroupByQuotient", IsSubgroupFpGroup);
############################################################################
##
#O LowIndexSubgroupsFpGroupIterator( <G>[, <H>], <index>[, <excluded>] )
#O LowIndexSubgroupsFpGroup( <G>[, <H>], <index>[, <excluded>] )
##
## <#GAPDoc Label="LowIndexSubgroupsFpGroupIterator">
## <ManSection>
## <Oper Name="LowIndexSubgroupsFpGroupIterator"
## Arg='G[, H], index[, excluded]'/>
## <Oper Name="LowIndexSubgroupsFpGroup" Arg='G[, H], index[, excluded]'/>
##
## <Description>
## <Index Subkey="for low index subgroups">iterator</Index>
## These functions compute representatives of the conjugacy classes of
## subgroups of the finitely presented group <A>G</A> that contain the
## subgroup <A>H</A> of <A>G</A> and that have index less than or equal to
## <A>index</A>.
## <P/>
## <Ref Oper="LowIndexSubgroupsFpGroupIterator"/> returns an iterator
## (see <Ref Sect="Iterators"/>)
## that can be used to run over these subgroups,
## and <Ref Oper="LowIndexSubgroupsFpGroup"/> returns the list of these
## subgroups.
## If one is interested only in one or a few subgroups up to a given index
## then preferably the iterator should be used.
## <P/>
## If the optional argument <A>excluded</A> has been specified, then it is
## expected to be a list of words in the free generators of the underlying
## free group of <A>G</A>, and <Ref Oper="LowIndexSubgroupsFpGroup"/>
## returns only those subgroups of index at most <A>index</A> that contain
## <A>H</A>, but do not contain any conjugate of any of the group elements
## defined by these words.
## <P/>
## If not given, <A>H</A> defaults to the trivial subgroup.
## <P/>
## The algorithm used finds the requested subgroups
## by systematically running through a tree of all potential coset tables
## of <A>G</A> of length at most <A>index</A> (where it skips all branches
## of that tree for which it knows in advance that they cannot provide new
## classes of such subgroups).
## The time required to do this depends, of course, on the presentation of
## <A>G</A>, but in general it will grow exponentially with
## the value of <A>index</A>. So you should be careful with the choice of
## <A>index</A>.
## <Example><![CDATA[
## gap> li:=LowIndexSubgroupsFpGroup( g, TrivialSubgroup( g ), 10 );
## [ Group(<fp, no generators known>), Group(<fp, no generators known>),
## Group(<fp, no generators known>), Group(<fp, no generators known>) ]
## ]]></Example>
## <P/>
## By default, the algorithm computes no generating sets for the subgroups.
## This can be enforced with <Ref Attr="GeneratorsOfGroup"/>:
## <Example><![CDATA[
## gap> GeneratorsOfGroup(li[2]);
## [ a, b*a*b^-1 ]
## ]]></Example>
## <P/>
## If we are interested just in one (proper) subgroup of index at most
## <M>10</M>, we can use the function that returns an iterator.
## The first subgroup found is the group itself,
## except if a list of excluded elements is entered (see below),
## so we look at the second subgroup.
## <P/>
## <Example><![CDATA[
## gap> iter:= LowIndexSubgroupsFpGroupIterator( g, 10 );;
## gap> s1:= NextIterator( iter );; Index( g, s1 );
## 1
## gap> IsDoneIterator( iter );
## false
## gap> s2:= NextIterator( iter );; s2 = li[2];
## true
## ]]></Example>
## <P/>
## As an example for an application of the optional parameter
## <A>excluded</A>, we
## compute all conjugacy classes of torsion free subgroups of index at most
## <M>24</M> in the group <M>G =
## \langle x,y,z \mid x^2, y^4, z^3, (xy)^3, (yz)^2, (xz)^3 \rangle</M>.
## It is know from theory that each torsion element of this
## group is conjugate to a power of <M>x</M>, <M>y</M>, <M>z</M>, <M>xy</M>,
## <M>xz</M>, or <M>yz</M>.
## (Note that this includes conjugates of <M>y^2</M>.)
## <P/>
## <Example><![CDATA[
## gap> F := FreeGroup( "x", "y", "z" );;
## gap> x := F.1;; y := F.2;; z := F.3;;
## gap> G := F / [ x^2, y^4, z^3, (x*y)^3, (y*z)^2, (x*z)^3 ];;
## gap> torsion := [ x, y, y^2, z, x*y, x*z, y*z ];;
## gap> SetInfoLevel( InfoFpGroup, 2 );
## gap> lis := LowIndexSubgroupsFpGroup(G, TrivialSubgroup(G), 24, torsion);;
## #I LowIndexSubgroupsFpGroup called
## #I class 1 of index 24 and length 8
## #I class 2 of index 24 and length 24
## #I class 3 of index 24 and length 24
## #I class 4 of index 24 and length 24
## #I class 5 of index 24 and length 24
## #I LowIndexSubgroupsFpGroup done. Found 5 classes
## gap> SetInfoLevel( InfoFpGroup, 0 );
## ]]></Example>
## <P/>
## If a particular image group is desired, the operation
## <Ref Oper="GQuotients"/>
## (see <Ref Sect="Quotient Methods"/>) can be useful as well.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "LowIndexSubgroupsFpGroupIterator",
[ IsSubgroupFpGroup, IsPosInt ] );
DeclareOperation( "LowIndexSubgroupsFpGroupIterator",
[ IsSubgroupFpGroup, IsSubgroupFpGroup, IsPosInt ] );
DeclareOperation( "LowIndexSubgroupsFpGroupIterator",
[ IsSubgroupFpGroup and IsWholeFamily, IsPosInt, IsList ] );
DeclareOperation( "LowIndexSubgroupsFpGroupIterator",
[ IsSubgroupFpGroup and IsWholeFamily, IsSubgroupFpGroup, IsPosInt,
IsList ] );
DeclareOperation("LowIndexSubgroupsFpGroup",
[IsSubgroupFpGroup,IsSubgroupFpGroup,IsPosInt]);
############################################################################
##
#F MostFrequentGeneratorFpGroup( <G> )
##
## <#GAPDoc Label="MostFrequentGeneratorFpGroup">
## <ManSection>
## <Func Name="MostFrequentGeneratorFpGroup" Arg='G'/>
##
## <Description>
## is an internal function which is used in some applications of coset
## table methods. It returns the first of those generators of the given
## finitely presented group <A>G</A> which occur most frequently in the
## relators.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("MostFrequentGeneratorFpGroup");
#############################################################################
##
#A FreeGeneratorsOfFpGroup( <G> )
#O FreeGeneratorsOfWholeGroup( <U> )
##
## <#GAPDoc Label="FreeGeneratorsOfFpGroup">
## <ManSection>
## <Attr Name="FreeGeneratorsOfFpGroup" Arg='G'/>
## <Oper Name="FreeGeneratorsOfWholeGroup" Arg='U'/>
##
## <Description>
## <Ref Attr="FreeGeneratorsOfFpGroup"/> returns the underlying free
## generators corresponding to the generators of the finitely presented
## group <A>G</A> which must be a full FpGroup.
## <P/>
## <Ref Oper="FreeGeneratorsOfWholeGroup"/> also works for subgroups of an
## FpGroup and returns the free generators of the full group that defines
## the family.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "FreeGeneratorsOfFpGroup",
IsSubgroupFpGroup and IsGroupOfFamily );
DeclareOperation( "FreeGeneratorsOfWholeGroup",
[IsSubgroupFpGroup] );
############################################################################
##
#A RelatorsOfFpGroup(<G>)
##
## <#GAPDoc Label="RelatorsOfFpGroup">
## <ManSection>
## <Attr Name="RelatorsOfFpGroup" Arg='G'/>
##
## <Description>
## returns the relators of the finitely presented group <A>G</A> as words
## in the free generators provided by the
## <Ref Attr="FreeGeneratorsOfFpGroup"/> value of <A>G</A>.
## <Example><![CDATA[
## gap> f := FreeGroup( "a", "b" );;
## gap> g := f / [ f.1^5, f.2^2, f.1^f.2*f.1 ];
## <fp group on the generators [ a, b ]>
## gap> Size( g );
## 10
## gap> FreeGroupOfFpGroup( g ) = f;
## true
## gap> FreeGeneratorsOfFpGroup( g );
## [ a, b ]
## gap> RelatorsOfFpGroup( g );
## [ a^5, b^2, b^-1*a*b*a ]
## ]]></Example>
## <P/>
## Note that these attributes are only available for the <E>full</E>
## finitely presented group.
## It is possible (for example by using <Ref Func="Subgroup"/>) to
## construct a subgroup of index <M>1</M> which is not identical to the
## whole group.
## The latter one can be obtained in this situation via
## <Ref Func="Parent"/>.
## <P/>
## Elements of a finitely presented group are not words, but are represented
## using a word from the free group as representative. The following two
## commands obtain this representative, respectively create an element in the
## finitely presented group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("RelatorsOfFpGroup",IsSubgroupFpGroup and IsGroupOfFamily);
#############################################################################
##
#A FreeGroupOfFpGroup(<G>)
##
## <#GAPDoc Label="FreeGroupOfFpGroup">
## <ManSection>
## <Attr Name="FreeGroupOfFpGroup" Arg='G'/>
##
## <Description>
## returns the underlying free group for the finitely presented group
## <A>G</A>.
## This is the group generated by the free generators provided by the
## <Ref Attr="FreeGeneratorsOfFpGroup"/> value of <A>G</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("FreeGroupOfFpGroup",IsSubgroupFpGroup and IsGroupOfFamily);
#############################################################################
##
#A IndicesInvolutaryGenerators( <G> )
##
## <#GAPDoc Label="IndicesInvolutaryGenerators">
## <ManSection>
## <Attr Name="IndicesInvolutaryGenerators" Arg='G'/>
##
## <Description>
## returns the indices of those generators of the finitely presented group
## <A>G</A> which are known to be involutions. This knowledge is used by
## internal functions to improve the performance of coset enumerations.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("IndicesInvolutaryGenerators",
IsSubgroupFpGroup and IsGroupOfFamily);
############################################################################
##
#F RelatorRepresentatives(<rels>)
##
## <ManSection>
## <Func Name="RelatorRepresentatives" Arg='rels'/>
##
## <Description>
## returns a set of relators, that contains for each relator in the list
## <A>rels</A> its minimal cyclical permutation (which is automatically
## cyclically reduced).
## </Description>
## </ManSection>
##
DeclareGlobalFunction("RelatorRepresentatives");
#############################################################################
##
#F RelsSortedByStartGen( <gens>, <rels>, <table> )
##
## <ManSection>
## <Func Name="RelsSortedByStartGen" Arg='gens, rels, table'/>
##
## <Description>
## is a subroutine of the Felsch Todd-Coxeter and the Reduced
## Reidemeister-Schreier routines. It returns a list which for each
## generator or inverse generator in <A>gens</A> contains a list of all
## cyclically reduced relators, starting with that element, which can be
## obtained by conjugating or inverting the given relators <A>rels</A>. The
## relators are represented as lists of the coset table columns from the
## table <A>table</A> corresponding to the generators and, in addition, as lists
## of the respective column numbers.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("RelsSortedByStartGen");
#############################################################################
##
#A IsomorphismPermGroupOrFailFpGroup( <G> [,<max>] )
##
## <ManSection>
## <Attr Name="IsomorphismPermGroupOrFailFpGroup" Arg='G [,max]'/>
##
## <Description>
## returns an isomorphism <M>\varphi</M> from the fp group <A>G</A> onto
## a permutation group <A>P</A> which is isomorphic to <A>G</A>, if one can be found
## with reasonable effort and of reasonable degree. The function
## returns <K>fail</K> otherwise.
## <P/>
## The optional argument <C>max</C> can be used to override the default maximal
## size of a coset table used (and thus the maximal degree of the resulting
## permutation).
## </Description>
## </ManSection>
##
DeclareGlobalFunction("IsomorphismPermGroupOrFailFpGroup");
#############################################################################
##
#F SubgroupGeneratorsCosetTable(<freegens>,<fprels>,<table>)
##
## <ManSection>
## <Func Name="SubgroupGeneratorsCosetTable" Arg='freegens,fprels,table'/>
##
## <Description>
## determinates subgroup generators for the subgroup given by the coset
## table <A>table</A> from the free generators <A>freegens</A>,
## the relators <A>fprels</A> (as words in <A>freegens</A>).
## It returns words in <A>freegens</A>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SubgroupGeneratorsCosetTable" );
#############################################################################
##
#F LiftFactorFpHom(<hom>,<G>,<N>,<dec>)
##
## <ManSection>
## <Func Name="LiftFactorFpHom" Arg='hom,G,N,dec'/>
##
## <Description>
## Let <A>hom</A> be an epimorphism from a group <A>G</A> to a finitely presented
## group <A>F</A> with kernel <A>M</A> and <M>M/N</M> a chief factor.
## If <M>M/N</M> is abelian, then <A>dec</A> is a modulo pcgs. Otherwise <A>dec</A> is a
## homomorphism from <A>M</A> onto a finitely presented group, with kernel <A>N</A>.
## This function
## constructs a new fp group <A>F2</A> isomorphic to <M>G/N</M> and returns an
## epimorphism from <A>G</A> onto <A>F2</A>.
## <P/>
## No test of the arguments is performed.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "LiftFactorFpHom" );
DeclareGlobalFunction( "IsomorphismFpGroupByChiefSeriesFactor" );
#############################################################################
##
#F ComplementFactorFpHom(<hom>,<M>,<N>,<C>,<Ggens>,<Cgens>)
##
## <ManSection>
## <Func Name="ComplementFactorFpHom" Arg='hom,M,N,C,Ggens,Cgens'/>
##
## <Description>
## Let <A>hom</A> be an epimorphism from a group <C>G</C> to a finitely presented
## group <A>F</A> with kernel <A>M</A> and <M>M/N</M> be elementary abelian and <M>C/N</M> a
## complement to <A>M</A> in <M>G/N</M>. The set <A>Cgens</A> is a set of generators of
## <A>C</A> modulo <A>N</A>, <A>Ggens</A> are corresponding representatives in <C>G</C>.
## This function constructs a new epimorphism from <A>C</A> onto <A>F</A>.
## <P/>
## No test of the arguments is performed.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "ComplementFactorFpHom" );
#############################################################################
##
#F FactorGroupFpGroupByRels( <G>, <elts> )
##
## <#GAPDoc Label="FactorGroupFpGroupByRels">
## <ManSection>
## <Func Name="FactorGroupFpGroupByRels" Arg='G, elts'/>
##
## <Description>
## returns the factor group <A>G</A>/<M>N</M> of <A>G</A> by
## the normal closure <M>N</M> of <A>elts</A>
## where <A>elts</A> is expected to be a list of elements of <A>G</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "FactorGroupFpGroupByRels" );
#############################################################################
##
#F ExcludedOrders( <fpgrp>[,<ords>] )
#A StoredExcludedOrders( <fpgrp> )
##
## <ManSection>
## <Func Name="ExcludedOrders" Arg='fpgrp[,ords]'/>
## <Attr Name="StoredExcludedOrders" Arg='fpgrp'/>
##
## <Description>
## for a (full) finitely presented group <A>fpgrp</A> this attribute returns
## a list of orders, corresponding to <Ref Func="GeneratorsOfGroup"/>,
## for which the presentation collapses.
## (That is, the group becomes trivial when a relator <M>g_i^o</M> is
## added.) If given, the list <A>ords</A> contains a set of
## orders corresponding to the generators which are explicitly to be
## tested.
## (The mutable attribute <Ref Func="StoredExcludedOrders"/> is used to
## store results.)
## </Description>
## </ManSection>
##
DeclareGlobalFunction("ExcludedOrders");
DeclareAttribute( "StoredExcludedOrders",IsSubgroupFpGroup,"mutable");
#############################################################################
##
#F NewmanInfinityCriterion(<G>,<p>)
##
## <#GAPDoc Label="NewmanInfinityCriterion">
## <ManSection>
## <Func Name="NewmanInfinityCriterion" Arg='G, p'/>
##
## <Description>
## Let <A>G</A> be a finitely presented group and <A>p</A> a prime that
## divides the order of the commutator factor group of <A>G</A>.
## This function applies an infinity criterion due to M. F. Newman
## <Cite Key="New90"/> to <A>G</A>.
## (See <Cite Key="Joh97" Where="chapter 16"/> for a more explicit
## description.)
## It returns <K>true</K>
## if the criterion succeeds in proving that <A>G</A> is infinite and
## <K>fail</K> otherwise.
## <P/>
## Note that the criterion uses the number of generators and
## relations in the presentation of <A>G</A>.
## Reduction of the presentation via Tietze transformations
## (<Ref Attr="IsomorphismSimplifiedFpGroup"/>) therefore might
## produce an isomorphic group, for which the criterion will work better.
## <Example><![CDATA[
## gap> g:=FibonacciGroup(2,9);
## <fp group on the generators [ f1, f2, f3, f4, f5, f6, f7, f8, f9 ]>
## gap> hom:=EpimorphismNilpotentQuotient(g,2);;
## gap> k:=Kernel(hom);;
## gap> Index(g,k);
## 152
## gap> AbelianInvariants(k);
## [ 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5 ]
## gap> NewmanInfinityCriterion(Kernel(hom),5);
## true
## ]]></Example>
## <P/>
## This proves that the subgroup <C>k</C>
## (and thus the whole group <C>g</C>) is infinite.
## (This is the original example from <Cite Key="New90"/>.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("NewmanInfinityCriterion");
#############################################################################
##
#F FibonacciGroup(<r>,<n>)
#F FibonacciGroup(<n>)
##
## <ManSection>
## <Func Name="FibonacciGroup" Arg='r,n'/>
## <Func Name="FibonacciGroup" Arg='n'/>
##
## <Description>
## This function returns the <E>Fibonacci group</E> with parameters <A>r</A>, <A>n</A>.
## This is a finitely presented group with <A>n</A> generators <M>x_i</M> and <A>n</A>
## relators <M>x_i\cdot\cdots\cdot x_{r+i-1}/x_{r+i}</M> (with indices reduced
## modulo <A>n</A>).
## <P/>
## If <A>r</A> is omitted, it defaults to 2.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("FibonacciGroup");
#############################################################################
##
#A FPFaithHom(<fam>)
##
## <ManSection>
## <Attr Name="FPFaithHom" Arg='fam'/>
##
## <Description>
## For the elements family <A>fam</A> of a finite fp group <A>G</A> this returns an
## isomorphism to a permutation
## or a pc group isomorphic to <A>G</A>.
## </Description>
## </ManSection>
##
DeclareAttribute("FPFaithHom",IsFamily);
#############################################################################
##
#F ParseRelators(<gens>,<rels>)
##
## <#GAPDoc Label="ParseRelators">
## <ManSection>
## <Func Name="ParseRelators" Arg='gens, rels'/>
##
## <Description>
## Will translate a list of relations as given in print, e.g.
## <M>x y^2 = (x y^3 x)^2 xy = yzx</M> into relators.
## <A>gens</A> must be a list of generators of a free group,
## each being displayed by a single letter.
## <A>rels</A> is a string that lists a sequence of equalities.
## These must be written in the letters which are the names of
## the generators in <A>gens</A>.
## Change of upper/lower case is interpreted to indicate inverses.
## <P/>
## <Example><![CDATA[
## gap> f:=FreeGroup("x","y","z");;
## gap> AssignGeneratorVariables(f);
## #I Assigned the global variables [ x, y, z ]
## gap> r:=ParseRelators([x,y,z],
## > "x^2 = y^5 = z^3 = (xyxyxy^4)^2 = (xz)^2 = (y^2z)^2 = 1");
## [ x^2, y^5, z^3, (x*z)^2, (y^2*z)^2, ((x*y)^3*y^3)^2 ]
## gap> g:=f/r;
## <fp group on the generators [ x, y, z ]>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("ParseRelators");
#############################################################################
##
#F StringFactorizationWord(<w>)
##
## <#GAPDoc Label="StringFactorizationWord">
## <ManSection>
## <Func Name="StringFactorizationWord" Arg='w'/>
##
## <Description>
## returns a string that expresses a given word <A>w</A> in compact form
## written as a string. Inverses are expressed by changing the upper/lower
## case of the generators, recurring expressions are written as products.
## <Example><![CDATA[
## gap> StringFactorizationWord(z^-1*x*y*y*y*x*x*y*y*y*x*y^-1*x);
## "Z(xy3x)2Yx"
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("StringFactorizationWord");
# used to test whether abeliniazation can be mapped in GQuotients
DeclareGlobalFunction("CanMapFiniteAbelianInvariants");
# map fpgrp->fpmon creator
DeclareGlobalFunction("MakeFpGroupToMonoidHomType1");
# used in homomorphisms
DeclareGlobalName("TRIVIAL_FP_GROUP");
DeclareAttribute("CyclicSubgroupFpGroup", IsFpGroup);
