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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, Frank Celler.
##
## 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 methods for free groups.
##
## Free groups are treated as special cases of finitely presented groups.
## In addition, elements of a free group are
## (associative) words, that is they have a normal form that allows an easy
## equalitity test.
##
#############################################################################
##
#M Iterator( <G> )
##
## The implementation of iterator and enumerator for free groups is more
## complicated than for free semigroups and monoids, since one has to be
## careful to avoid cancellation of generators and their inverses when
## building words.
## So the iterator for a free group of rank $n$ uses the following ordering.
## Enumerate signless words (that is, forget about the signs of exponents)
## as given by the enumerator of free monoids, and for each such word
## consisting of $k$ pairs of generators/exponents, enumerate all
## $2^k$ possibilities of signs for the exponents.
##
## The enumerator for a free group uses a different succession, in order to
## make the bijection of words and positive integers easy to calculate.
##
## There are exactly $2n (2n-1)^{l-1}$ words of length $l$, for $l > 0$.
##
## So the word corresponding to the integer
## $m = 1 + \sum_{i=1}^{l-1} 2n (2n-1)^{i-1} + m^{\prime}$,
## with $1 \leq m^{\prime} \leq 2n (2n-1)^l$,
## is the $m^{\prime}$-th word of length $l$.
##
## Write $m^{\prime} - 1 = c_1 - 1 + \sum_{i=2}^l (c_i - 1) 2n (2n-1)^{i-2}$
## where $1 \leq c_1 \leq 2n$ and $1 \leq c_i \leq 2n-1$ for
## $2 \leq i \leq l$.
##
## Let $(s_1, s_2, \ldots, s_{2n}) = (g_1, g_1^{-1}, g_2, \ldots, g_n^{-1})$
## and translate the coefficient vector $(c_1, c_2, \ldots, c_l)$ to
## $s(c_1) s(c_2) \cdots s(c_l)$, defined by $s(c_1) = s_{c_1}$, and
## \[ s(c_{i+1}) = \left\{ \begin{array}{lcl}
## s_{c_{i+1}} & ; & c_i \equiv 1 \bmod 2, c_{i+1} \leq c_i \\
## s_{c_{i+1}} & ; & c_i \equiv 0 \bmod 2, c_{i+1} \leq c_{i-2} \\
## s_{c_{i+1}+1} & ; & \mbox{\rm otherwise}
## \end{array} \right. \]
##
BindGlobal( "NextIterator_FreeGroup", function( iter )
local word, oldword, exp, len, pos, i;
# Increase the counter.
# Get the next sign distribution of same length if possible.
word:= iter!.word;
oldword:= ShallowCopy( word );
exp:= iter!.exp;
len:= Length( word );
pos:= 2;
while pos <= len and word[ pos ] < 0 do
pos:= pos + 2;
od;
if pos <= len then
for i in [ 2, 4 .. pos ] do
word[i]:= - word[i];
od;
else
# We have enumerated all sign vectors,
# so we must take the next tuple.
FreeSemigroup_NextWordExp( iter );
fi;
return ObjByExtRep( iter!.family, 1, exp, oldword );
end );
BindGlobal( "ShallowCopy_FreeGroup", iter -> rec(
family := iter!.family,
nrgenerators := iter!.nrgenerators,
exp := iter!.exp,
word := ShallowCopy( iter!.word ),
length := iter!.length,
counter := ShallowCopy( iter!.counter ) ) );
InstallMethod( Iterator,
"for a free group",
[ IsAssocWordWithInverseCollection and IsWholeFamily and IsGroup ],
G -> IteratorByFunctions( rec(
IsDoneIterator := ReturnFalse,
NextIterator := NextIterator_FreeGroup,
ShallowCopy := ShallowCopy_FreeGroup,
family := ElementsFamily( FamilyObj( G ) ),
nrgenerators := Length( GeneratorsOfGroup( G ) ),
exp := 0,
word := [],
length := 0,
counter := [ 0, 0 ] ) ) );
#############################################################################
##
#M Enumerator( <G> )
##
BindGlobal( "ElementNumber_FreeGroup",
function( enum, nr )
local n, 2n, nn, l, power, word, exp, maxexp, cc, sign, i, c;
if nr = 1 then
return One( enum!.family );
fi;
n:= enum!.nrgenerators;
2n:= 2 * n;
nn:= 2n - 1;
# Compute the length of the word corresponding to `nr'.
l:= 0;
power:= 2n;
nr:= nr - 1;
while 0 < nr do
nr:= nr - power;
l:= l+1;
power:= power * nn;
od;
nr:= nr + power / nn - 1;
# Compute the vector of the `(nr + 1)'-th element of length `l'.
exp:= 0;
maxexp:= 1;
c:= nr mod 2n;
nr:= ( nr - c ) / 2n;
cc:= c;
if c mod 2 = 0 then
sign:= 1;
else
sign:= -1;
c:= c-1;
fi;
word:= [ c/2 + 1 ];
for i in [ 1 .. l ] do
# translate `c'
if cc < c or ( cc mod 2 = 1 and cc-2 < c ) then
c:= c+1;
fi;
if c = cc then
exp:= exp + 1;
else
Add( word, sign * exp );
if maxexp < exp then
maxexp:= exp;
fi;
exp:= 1;
cc:= c;
if c mod 2 = 0 then
sign:= 1;
else
sign:= -1;
c:= c-1;
fi;
Add( word, c/2 + 1 );
fi;
c:= nr mod nn;
nr:= ( nr - c ) / nn;
od;
Add( word, sign * exp );
# Return the element.
return ObjByExtRep( enum!.family, 1, maxexp, word );
end );
BindGlobal( "NumberElement_FreeGroup",
function( enum, elm )
local l, len, i, n, 2n, nn, nr, j, power, c, cc, exp;
if not IsCollsElms( FamilyObj( enum ), FamilyObj( elm ) ) then
return fail;
fi;
elm:= ExtRepOfObj( elm );
l:= Length( elm );
if l = 0 then
return 1;
fi;
# Calculate the length of the word.
len:= 0;
for i in [ 2, 4 .. l ] do
exp:= elm[i];
if 0 < exp then
len:= len + elm[i];
else
len:= len - elm[i];
fi;
od;
# Calculate the number of words of smaller length, plus 1.
n:= enum!.nrgenerators;
2n:= 2 * n;
nn:= 2n - 1;
nr:= 2;
power:= 2n;
for i in [ 1 .. len-1 ] do
nr:= nr + power;
power:= power * nn;
od;
# Add the position in the words of length 'len'.
c:= 2 * elm[1] - 1;
exp:= elm[2];
if 0 < exp then
c:= c-1;
else
exp:= -exp;
fi;
nr:= nr + c;
power:= 2n;
cc:= c;
c:= c - ( c mod 2 );
for j in [ 2 .. exp ] do
nr:= nr + c * power;
power:= power * nn;
od;
for i in [ 4, 6 .. l ] do
c:= 2 * elm[ i-1 ] - 1;
exp:= elm[i];
if 0 < exp then
c:= c-1;
else
exp:= -exp;
fi;
if cc < c or ( cc mod 2 = 1 and cc - 2 < c ) then
cc:= c;
c:= c - 1;
else
cc:= c;
fi;
nr:= nr + c * power;
power:= power * nn;
c:= cc - ( cc mod 2 );
for j in [ 2 .. exp ] do
nr:= nr + c * power;
power:= power * nn;
od;
od;
return nr;
end );
InstallMethod( Enumerator,
"for enumerator of a free group",
[ IsAssocWordWithInverseCollection and IsWholeFamily and IsGroup ],
G -> EnumeratorByFunctions( G, rec(
NumberElement := NumberElement_FreeGroup,
ElementNumber := ElementNumber_FreeGroup,
family := ElementsFamily( FamilyObj( G ) ),
nrgenerators := Length( ElementsFamily(
FamilyObj( G ) )!.names ) ) ) );
#############################################################################
##
#M IsWholeFamily( <G> )
##
## If all magma generators of the family are among the group generators
## of <G> then <G> contains the whole family of its elements.
##
InstallMethod( IsWholeFamily,
"for a free group",
[ IsAssocWordWithInverseCollection and IsGroup ],
function( M )
if IsSubset( MagmaGeneratorsOfFamily( ElementsFamily( FamilyObj( M ) ) ),
GeneratorsOfGroup( M ) ) then
return true;
else
TryNextMethod();
fi;
end );
#############################################################################
##
#M Random( <M> )
##
#T isn't this a generic group method? (without guarantee about distribution)
##
InstallMethodWithRandomSource( Random,
"for a random source and a free group",
[ IsRandomSource, IsAssocWordWithInverseCollection and IsGroup ],
function( rs, M )
local len, result, gens, i;
# Get a random length for the word.
len:= Random( rs, Integers );
if 0 < len then
len:= 2 * len;
elif len < 0 then
len:= -2 * len - 1;
else
return One( M );
fi;
# Multiply 'len' random generator powers.
gens:= GeneratorsOfGroup( M );
result:= Random( rs, gens ) ^ Random( rs, Integers );
for i in [ 2 .. len ] do
result:= result * Random( rs, gens ) ^ Random( rs, Integers );
od;
# Return the result.
return result;
end );
#############################################################################
##
#M Size( <G> ) . . . . . . . . . . . . . . . . . . . . . . for a free group
##
InstallMethod( Size,
"for a free group",
[ IsAssocWordWithInverseCollection and IsGroup ],
function( G )
if IsTrivial( G ) then
return 1;
else
return infinity;
fi;
end );
#############################################################################
##
#M IsCommutative( <G> ) . . . . . . . . . . . . . . . . . . for a free group
##
InstallMethod( IsCommutative,
"for a free group",
[ IsFreeGroup and HasIsFinitelyGeneratedGroup ],
function( G )
if not IsFinitelyGeneratedGroup( G ) then
return false;
fi;
TryNextMethod();
end );
#############################################################################
##
#M IsSolvableGroup( <G> ) . . . . . . . . . . . . . . . . . for a free group
##
InstallMethod( IsSolvableGroup,
"for a free group",
[ IsFreeGroup ],
10, # rank it higher than the method in the fr package
G -> IsAbelian( G ) );
#############################################################################
##
#M MagmaGeneratorsOfFamily( <F> )
##
InstallMethod( MagmaGeneratorsOfFamily,
"for a family of assoc. words",
[ IsAssocWordWithInverseFamily ],
function( F )
local gens;
# Make the generators.
gens:= List( [ 1 .. Length( F!.names ) ],
i -> ObjByExtRep( F, 1, 1, [ i, 1 ] ) );
Append( gens, List( [ 1 .. Length( F!.names ) ],
i -> ObjByExtRep( F, 1, 1, [ i, -1 ] ) ) );
Add( gens, One( F ) );
# Return the magma generators.
return gens;
end );
#############################################################################
##
#M Order <elm> )
##
InstallMethod( Order,
"free group element",
[ IsElementOfFreeGroup ],
0,
function(elt)
if IsOne(elt) then
return 1;
else
return infinity;
fi;
end);
# the following method returns a lex-minimal generating set for a free group
# it relies on the (unguaranteed) ordering of free group elements, that
# inverses of generators come before the generators, and generators of low
# number come before those of higher number.
InstallMethod( GeneratorsSmallest,
"for a free group",
[ IsFreeGroup ],
x->List(GeneratorsOfGroup(x),Inverse));
#############################################################################
##
#F FreeGroup( [<wfilt>,]<rank>[, <name>] ) . . . . free group of given rank
#F FreeGroup( [<wfilt>,][<name1>[, <name2>[, ...]]] )
#F FreeGroup( [<wfilt>,]<names> )
#F FreeGroup( [<wfilt>,]infinity[, <name>][, <init>] )
##
InstallGlobalFunction( FreeGroup, function ( arg )
local rank, # number of generators
F, # family of free group element objects
G, # free group, result
processed;
processed := FreeXArgumentProcessor( "FreeGroup", "f", arg, true, true );
rank := Length( processed.names );
# Construct the family of element objects of our group.
F := NewFamily( "FreeGroupElementsFamily",
IsAssocWordWithInverse and IsElementOfFreeGroup,
CanEasilySortElements,
CanEasilySortElements and processed.lesy );
# Install the data (names, no. of bits available for exponents, types).
StoreInfoFreeMagma( F, processed.names, IsAssocWordWithInverse and
IsElementOfFreeGroup );
# Make the group.
if rank = 0 then
G:= GroupByGenerators( [], One( F ) );
elif rank < infinity then
G:= GroupByGenerators( List( [ 1 .. rank ],
i -> ObjByExtRep( F, 1, 1, [ i, 1 ] ) ) );
else
G:= GroupByGenerators( InfiniteListOfGenerators( F ) );
fi;
SetIsWholeFamily( G, true );
# Store whether group is finitely generated / trivial / abelian / solvable.
SetIsFinitelyGeneratedGroup( G, rank < infinity );
SetIsTrivial( G, rank = 0 );
SetIsAbelian( G, rank <= 1 );
SetIsSolvableGroup( G, rank <= 1 );
# Store the whole group in the family.
FamilyObj(G)!.wholeGroup := G;
F!.freeGroup:=G;
SetFilterObj(G,IsGroupOfFamily);
return G;
end );
#############################################################################
##
#M FreeGeneratorsOfFpGroup( <F> )
##
InstallMethod( FreeGeneratorsOfFpGroup,
"for a free group",
[ IsSubgroupFpGroup and IsGroupOfFamily and IsFreeGroup ],
GeneratorsOfGroup );
#############################################################################
##
#M RelatorsOfFpGroup( <F> )
##
InstallMethod( RelatorsOfFpGroup,
"for a free group",
[ IsSubgroupFpGroup and IsGroupOfFamily and IsFreeGroup ],
F -> [] );
#############################################################################
##
#M FreeGroupOfFpGroup( <F> )
##
InstallMethod( FreeGroupOfFpGroup,
"for a free group",
[ IsSubgroupFpGroup and IsGroupOfFamily and IsFreeGroup ],
IdFunc );
#############################################################################
##
#M UnderlyingElement( w )
##
InstallMethod( UnderlyingElement,
"for an element of a free group",
[ IsElementOfFreeGroup ],
IdFunc );
#############################################################################
##
#M ElementOfFpGroup( w )
##
InstallOtherMethod( ElementOfFpGroup,
"for a family of free group elements, and an assoc. word",
[ IsElementOfFreeGroupFamily and IsAssocWordWithInverseFamily,
IsAssocWordWithInverse ],
function( fam, w ) return w; end );
#############################################################################
##
#M ViewObj(<G>)
##
InstallMethod( ViewObj,
"subgroup of free group",
[ IsFreeGroup ],
function(G)
if IsGroupOfFamily(G) then
if IsEmpty(GeneratorsOfGroup(G)) then
Print("<free group of rank zero>");
elif Length(GeneratorsOfGroup(G)) > GAPInfo.ViewLength * 10 then
Print("<free group with ",Length(GeneratorsOfGroup(G))," generators>");
else
Print("<free group on the generators ",GeneratorsOfGroup(G),">");
fi;
else
Print("Group(");
if HasGeneratorsOfGroup(G) then
if not IsBound(G!.gensWordLengthSum) then
G!.gensWordLengthSum:=Sum(List(GeneratorsOfGroup(G),Length));
fi;
if G!.gensWordLengthSum <= GAPInfo.ViewLength * 30 then
Print(GeneratorsOfGroup(G));
else
Print("<",Pluralize(Length(GeneratorsOfGroup(G)),"generator"),">");
fi;
else
Print("<free, no generators known>");
fi;
Print(")");
fi;
end);
#############################################################################
##
#M \.( <F>, <n> ) . . . . . . . . . . access to generators of a free group
##
InstallAccessToGenerators( IsSubgroupFpGroup and IsGroupOfFamily
and IsFreeGroup,
"free group containing the whole family",
GeneratorsOfMagmaWithInverses );
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