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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Werner Nickel.
##
## 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
##
## 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.
##
#############################################################################
##
#F IsElementOfFreeGroup . . . . . . . . . . . . . elements in a free group
##
## <ManSection>
## <Func Name="IsElementOfFreeGroup" Arg='obj'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareSynonym( "IsElementOfFreeGroup", IsAssocWordWithInverse );
DeclareSynonym( "IsElementOfFreeGroupFamily",IsAssocWordWithInverseFamily );
#############################################################################
##
#F FreeGroup( [<wfilt>,]<rank>[, <name>] )
#F FreeGroup( [<wfilt>,][<name1>[, <name2>[, ...]]] )
#F FreeGroup( [<wfilt>,]<names> )
#F FreeGroup( [<wfilt>,]infinity[, <name>][, <init>] )
##
## <#GAPDoc Label="FreeGroup">
## <ManSection>
## <Heading>FreeGroup</Heading>
## <Func Name="FreeGroup" Arg='[wfilt, ]rank[, name]'
## Label="for given rank"/>
## <Func Name="FreeGroup" Arg='[wfilt, ][name1[, name2[, ...]]]'
## Label="for various names"/>
## <Func Name="FreeGroup" Arg='[wfilt, ]names'
## Label="for a list of names"/>
## <Func Name="FreeGroup" Arg='[wfilt, ]infinity[, name][, init]'
## Label="for infinitely many generators"/>
##
## <Description>
## <C>FreeGroup</C> returns a free group. The number of
## generators, and the labels given to the generators, can be specified in
## several different ways.
## Warning: the labels of generators are only an aid for printing,
## and do not necessarily distinguish generators;
## see the examples at the end of
## <Ref Func="FreeSemigroup" Label="for given rank"/>
## for more information.
## <List>
## <Mark>
## 1: For a given rank, and an optional generator name prefix
## </Mark>
## <Item>
## Called with a nonnegative integer <A>rank</A>,
## <Ref Func="FreeGroup" Label="for given rank"/> returns
## a free group on <A>rank</A> generators.
## The optional argument <A>name</A> must be a string;
## its default value is <C>"f"</C>. <P/>
##
## If <A>name</A> is not given but the <C>generatorNames</C> option is,
## then this option is respected as described in
## Section <Ref Sect="Generator Names"/>. <P/>
##
## Otherwise, the generators of the returned free group are labelled
## <A>name</A><C>1</C>, ..., <A>name</A><C>k</C>,
## where <C>k</C> is the value of <A>rank</A>. <P/>
## </Item>
## <Mark>2: For given generator names</Mark>
## <Item>
## Called with various nonempty strings,
## <Ref Func="FreeGroup" Label="for various names"/> returns
## a free group on as many generators as arguments, which are labelled
## <A>name1</A>, <A>name2</A>, etc.
## </Item>
## <Mark>3: For a given list of generator names</Mark>
## <Item>
## Called with a finite list <A>names</A> of
## nonempty strings,
## <Ref Func="FreeGroup" Label="for a list of names"/> returns
## a free group on <C>Length(<A>names</A>)</C> generators, whose
## <C>i</C>-th generator is labelled <A>names</A><C>[i]</C>.
## </Item>
## <Mark>
## 4: For the rank <K>infinity</K>,
## an optional default generator name prefix,
## and an optional finite list of generator names
## </Mark>
## <Item>
## Called in the fourth form,
## <Ref Func="FreeGroup" Label="for infinitely many generators"/>
## returns a free group on infinitely many generators.
## The optional argument <A>name</A> must be a string; its default value is
## <C>"f"</C>,
## and the optional argument <A>init</A> must be a finite list of
## nonempty strings; its default value is an empty list.
## The generators are initially labelled according to the list <A>init</A>,
## followed by
## <A>name</A><C>i</C> for each <C>i</C> in the range from
## <C>Length(<A>init</A>)+1</C> to <K>infinity</K>.
## </Item>
## </List>
## If the optional first argument <A>wfilt</A> is given, then it must be either
## <C>IsSyllableWordsFamily</C>, <C>IsLetterWordsFamily</C>,
## <C>IsWLetterWordsFamily</C>, or <C>IsBLetterWordsFamily</C>.
## This filter specifies the representation used for the elements of
## the free group
## (see <Ref Sect="Representations for Associative Words"/>).
## If no such filter is given, a letter representation is used.
## <P/>
## (For interfacing to old code that omits the representation flag, use of
## the syllable representation is also triggered by setting the option
## <C>FreeGroupFamilyType</C> to the string <C>"syllable"</C>; this is
## overwritten by the optional first argument if it is given.)
##
## <Example><![CDATA[
## gap> FreeGroup(5);
## <free group on the generators [ f1, f2, f3, f4, f5 ]>
## gap> FreeGroup(4, "gen");
## <free group on the generators [ gen1, gen2, gen3, gen4 ]>
## gap> FreeGroup(3 : generatorNames := "ack");
## <free group on the generators [ ack1, ack2, ack3 ]>
## gap> FreeGroup(2 : generatorNames := ["u", "v", "w"]);
## <free group on the generators [ u, v ]>
## gap> FreeGroup();
## <free group of rank zero>
## gap> FreeGroup("a", "b", "c");
## <free group on the generators [ a, b, c ]>
## gap> FreeGroup(["x", "y"]);
## <free group on the generators [ x, y ]>
## gap> FreeGroup(infinity);
## <free group with infinity generators>
## gap> F := FreeGroup(infinity, "g", ["a", "b"]);
## <free group with infinity generators>
## gap> GeneratorsOfGroup(F){[1..4]};
## [ a, b, g3, g4 ]
## gap> GeneratorsOfGroup(FreeGroup(infinity, "gen")){[1..3]};
## [ gen1, gen2, gen3 ]
## gap> FreeGroup(IsSyllableWordsFamily, 50);
## <free group with 50 generators>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "FreeGroup" );
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