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| [[Image:F2 Cayley Graph.png|right|thumb|Diagram showing what the [[Cayley graph]] for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''.]]
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| In [[mathematics]], the '''free group''' ''F''<sub>''S''</sub> over a given set ''S'' consists of all expressions (a.k.a. [[Word (group theory)|words]], or [[term (logic)#Formal definition|terms]]) that can be built from members of ''S'', considering two expressions different unless their equality follows from the [[group axioms]] (e.g. ''st'' = ''suu''<sup>−1</sup>''t'', but ''s'' ≠ ''t'' for ''s'',''t'',''u''∈''S''). The members of ''S'' are called '''generators''' of ''F''<sub>''S''</sub>.
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| An arbitrary [[group (mathematics)|group]] ''G'' is called '''free''' if it is [[group isomorphism|isomorphic]] to ''F''<sub>''S''</sub> for some [[subset]] ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in one and only one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''<sup>−1</sup>''t'').
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| A related but different notion is a [[free abelian group]], both notions are particular instances of a [[free object]] from [[universal algebra]].
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| == History ==
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| Free groups first arose in the study of [[hyperbolic geometry]], as examples of [[Fuchsian group]]s (discrete groups acting by [[isometry|isometries]] on the [[Hyperbolic geometry|hyperbolic plane]]). In an 1882 paper, [[Walther von Dyck]] pointed out that these groups have the simplest possible [[group presentation|presentations]].<ref>{{cite journal | last = von Dyck | first = Walther | authorlink = Walther von Dyck | title = Gruppentheoretische Studien | journal = Mathematische Annalen | volume = 20 | issue = 1 | pages = 1–44 | year = 1882 | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002246724&L=1 | doi = 10.1007/BF01443322 | ref = harv}}</ref> The algebraic study of free groups was initiated by [[Jakob Nielsen (mathematician)|Jakob Nielsen]] in 1924, who gave them their name and established many of their basic properties.<ref>{{cite journal | last = Nielsen | first = Jakob | authorlink = Jakob Nielsen (mathematician) | title = Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden | journal = [[Mathematische Annalen]] | volume = 78 | issue = 1 | pages = 385–397 | year = 1917 | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002266873&L=1 | doi = 10.1007/BF01457113 | ref = harv | jfm = 46.0175.01 | mr = 1511907 }}<!-- note the journal volume was published in 1964, but its JFM review and the text of the article use the date 1917.--></ref><ref>{{cite journal | last = Nielsen | first = Jakob | authorlink = Jakob Nielsen (mathematician) | title = On calculation with noncommutative factors and its application to group theory. (Translated from Danish) | journal = The Mathematical Scientist | volume = 6 (1981) | issue = 2 | pages = 73–85 | year = 1921 | ref = harv}}</ref><ref>{{cite journal | last = Nielsen | first = Jakob | authorlink = Jakob Nielsen (mathematician) | title = Die Isomorphismengruppe der freien Gruppen | journal = Mathematische Annalen | volume = 91 | issue = 3 | pages = 169–209 | year = 1924 | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002269813&L=1 | doi = 10.1007/BF01556078 | ref = harv}}</ref> [[Max Dehn]] realized the connection with topology, and obtained the first proof of the full [[Nielsen–Schreier theorem]].<ref>See {{cite journal | last = Magnus | first = Wilhelm | authorlink = Wilhelm Magnus | coauthors = [[Ruth Moufang|Moufang, Ruth]] | title = Max Dehn zum Gedächtnis | journal = Mathematische Annalen | volume = 127 | issue = 1 | pages = 215–227 | year = 1954 | url = http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002283808&L=1 | doi = 10.1007/BF01361121 | ref = harv}}.</ref> [[Otto Schreier]] published an algebraic proof of this result in 1927,<ref>{{cite journal | last = Schreier | first = Otto | authorlink = Otto Schreier | title = Die Untergruppen der freien Gruppen | journal = Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume = 5 | year = 1928 | pages = 161–183 | doi = 10.1007/BF02952517 | ref = harv}}</ref> and [[Kurt Reidemeister]] included a comprehensive treatment of free groups in his 1932 book on [[combinatorial topology]].<ref>{{cite book | last = Reidemeister | first = Kurt | authorlink = Kurt Reidemeister | title = Einführung in die kombinatorische Topologie | publisher = Wissenschaftliche Buchgesellschaft | date = 1972 (1932 original) | location = Darmstadt}}</ref> Later on in the 1930s, [[Wilhelm Magnus]] discovered the connection between the [[lower central series]] of free groups and [[free Lie algebra]]s.
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| == Examples ==
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| The group ('''Z''',+) of [[integer]]s is free; we can take ''S'' = {1}. A free group on a two-element set ''S'' occurs in the proof of the [[Banach–Tarski paradox]] and is described there.
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| On the other hand, any nontrivial finite group cannot be free, since the elements of a free generating set of a free group have infinite order.
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| In [[algebraic topology]], the [[fundamental group]] of a [[bouquet of circles|bouquet of ''k'' circles]] (a set of ''k'' loops having only one point in common) is the free group on a set of ''k'' elements.
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| == Construction ==
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| The '''free group''' ''F<sub>S</sub>'' with '''free generating set''' ''S'' can be constructed as follows. ''S'' is a set of symbols and we suppose for every ''s'' in ''S'' there is a corresponding "inverse" symbol, ''s''<sup>−1</sup>, in a set ''S''<sup>−1</sup>. Let ''T'' = ''S'' ∪ ''S''<sup>−1</sup>, and define a '''[[word (group theory)|word]]''' in ''S'' to be any written product of elements of ''T''. That is, a word in ''S'' is an element of the [[monoid]] generated by ''T''. The empty word is the word with no symbols at all. For example, if ''S'' = {''a'', ''b'', ''c''}, then ''T'' = {''a'', ''a''<sup>−1</sup>, ''b'', ''b''<sup>−1</sup>, ''c'', ''c''<sup>−1</sup>}, and
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| :<math>a b^3 c^{-1} c a^{-1} c\,</math> | |
| is a word in ''S''. If an element of ''S'' lies immediately next to its inverse, the word may be simplified by omitting the ''s'', ''s''<sup>−1</sup> pair:
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| :<math>a b^3 c^{-1} c a^{-1} c\;\;\longrightarrow\;\;a b^3 \, a^{-1} c.</math>
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| A word that cannot be simplified further is called '''reduced'''. The free group ''F<sub>S</sub>'' is defined to be the group of all reduced words in ''S''. The group operation in ''F<sub>S</sub>'' is [[concatenation]] of words (followed by reduction if necessary). The identity is the empty word. A word is called '''cyclically reduced''', if its first and last letter are not inverse to each other. Every word is [[Inner automorphism|conjugate]] to a cyclically reduced word, and a cyclically reduced conjugate of a cyclically reduced word is a cyclic permutation of the letters in the word. For instance ''b''<sup>−1</sup>''abcb'' is not cyclically reduced, but is conjugate to ''abc'', which is cyclically reduced. The only cyclically reduced conjugates of ''abc'' are ''abc'', ''bca'', and ''cab''.
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| == Universal property ==
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| The free group ''F<sub>S</sub>'' is the [[Universal (mathematics)|universal]] group generated by the set ''S''. This can be formalized by the following [[universal property]]: given any function ƒ from ''S'' to a group ''G'', there exists a unique [[group homomorphism|homomorphism]] ''φ'': ''F<sub>S</sub>'' → ''G'' making the following [[commutative diagram|diagram]] commute (where the unnamed mapping denotes the inclusion from ''S'' into ''F<sub>S</sub>''):
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| [[Image:Free Group Universal.svg|center|100px]]
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| That is, homomorphisms ''F<sub>S</sub>'' → ''G'' are in one-to-one correspondence with functions ''S'' → ''G''. For a non-free group, the presence of [[group presentation|relations]] would restrict the possible images of the generators under a homomorphism.
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| To see how this relates to the constructive definition, think of the mapping from ''S'' to ''F<sub>S</sub>'' as sending each symbol to a word consisting of that symbol. To construct ''φ'' for given ƒ, first note that ''φ'' sends the empty word to identity of ''G'' and it has to agree with ƒ on the elements of ''S''. For the remaining words (consisting of more than one symbol) ''φ'' can be uniquely extended since it is a homomorphism, i.e., ''φ''(''ab'') = ''φ''(''a'') ''φ''(''b'').
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| The above property characterizes free groups up to [[isomorphism]], and is sometimes used as an alternative definition. It is known as the [[universal property]] of free groups, and the generating set ''S'' is called a '''basis''' for ''F<sub>S</sub>''. The basis for a free group is not uniquely determined.
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| Being characterized by a universal property is the standard feature of [[free object]]s in [[universal algebra]]. In the language of [[category theory]], the construction of the free group (similar to most constructions of free objects) is a [[functor]] from the [[category of sets]] to the [[category of groups]]. This functor is [[left adjoint]] to the [[forgetful functor]] from groups to sets.
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| ==Facts and theorems==
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| Some properties of free groups follow readily from the definition:
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| #Any group ''G'' is the homomorphic image of some free group F(''S''). Let ''S'' be a set of ''[[Generating set of a group|generators]]'' of ''G''. The natural map ''f'': F(''S'') → ''G'' is an [[epimorphism]], which proves the claim. Equivalently, ''G'' is isomorphic to a [[quotient group]] of some free group F(''S''). The kernel of ''f'' is a set of ''relations'' in the [[Presentation of a group|presentation]] of ''G''. If ''S'' can be chosen to be finite here, then ''G'' is called '''finitely generated'''.
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| #If ''S'' has more than one element, then F(''S'') is not [[abelian group|abelian]], and in fact the [[center of a group|center]] of F(''S'') is trivial (that is, consists only of the identity element).
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| #Two free groups F(''S'') and F(''T'') are isomorphic if and only if ''S'' and ''T'' have the same [[cardinality]]. This cardinality is called the '''rank''' of the free group ''F''. Thus for every cardinal number ''k'', there is, [[up to]] isomorphism, exactly one free group of rank ''k''.
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| #A free group of finite rank ''n'' > 1 has an [[exponential growth|exponential]] [[growth rate (group theory)|growth rate]] of order 2''n'' − 1.
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| A few other related results are:
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| #The [[Nielsen–Schreier theorem]]: Every [[subgroup]] of a free group is free.
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| #A free group of rank ''k'' clearly has subgroups of every rank less than ''k''. Less obviously, a (''nonabelian!'') free group of rank at least 2 has subgroups of all [[countable set|countable]] ranks.
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| #The [[commutator subgroup]] of a free group of rank ''k'' > 1 has infinite rank; for example for F(''a'',''b''), it is freely generated by the [[commutator]]s [''a''<sup>''m''</sup>, ''b''<sup>''n''</sup>] for non-zero ''m'' and ''n''.
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| #The free group in two elements is [[SQ universal]]; the above follows as any SQ universal group has subgroups of all countable ranks.
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| #Any group that [[group action|acts]] on a tree, [[free action|freely]] and preserving the [[oriented graph|orientation]], is a free group of countable rank (given by 1 plus the [[Euler characteristic]] of the [[group action|quotient]] [[graph theory|graph]]).
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| #The [[Cayley graph]] of a free group of finite rank, with respect to a free generating set, is a [[tree (mathematics)|tree]] on which the group acts freely, preserving the orientation.
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| #The [[groupoid]] approach to these results, given in the work by P.J. Higgins below, is kind of extracted from an approach using [[covering space]]s. It allows more powerful results, for example on [[Grushko's theorem]], and a normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph.
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| # [[Grushko's theorem]] has the consequence that if a subset ''B'' of a free group ''F'' on ''n'' elements generates ''F'' and has ''n'' elements, then ''B'' generates ''F'' freely.
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| == Free abelian group ==
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| {{further2|[[free abelian group]]}}
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| The free abelian group on a set ''S'' is defined via its universal property in the analogous way, with obvious modifications:
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| Consider a pair (''F'', ''φ''), where ''F'' is an abelian group and ''φ'': ''S'' → ''F'' is a function. ''F'' is said to be the '''free abelian group on ''S'' with respect to ''φ'' ''' if for any abelian group ''G'' and any function ''ψ'': ''S'' → ''G'', there exists a unique homomorphism ''f'': ''F'' → ''G'' such that
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| :''f''(''φ''(''s'')) = ''ψ''(''s''), for all ''s'' in ''S''.
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| The free abelian group on ''S'' can be explicitly identified as the free group F(''S'') modulo the subgroup generated by its commutators, [F(''S''), F(''S'')], i.e.
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| its [[abelianisation]]. In other words, the free abelian group on ''S'' is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as the rank of its abelianisation as a free abelian group.
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| ==Tarski's problems==
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| Around 1945, [[Alfred Tarski]] asked whether the free groups on two or more generators have the same [[model theory|first order theory]], and whether this theory is [[decidability (logic)|decidable]]. {{harvtxt|Sela|2006}} answered the first question by showing that any two nonabelian free groups have the same first order theory, and {{harvtxt|Kharlampovich|Myasnikov|2006}} answered both questions, showing that this theory is decidable.
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| A similar unsolved (in 2011) question in [[free probability theory]] asks whether the [[von Neumann group algebra]]s of any two non-abelian finitely generated free groups are isomorphic.
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| ==See also==
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| * [[Generating set of a group]]
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| * [[Presentation of a group]]
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| * [[Nielsen transformation]], a factorization of elements of the [[automorphism group of a free group]]
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| * [[Free product]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Cite journal
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| |last=Kharlampovich|first= Olga|last2= Myasnikov|first2= Alexei
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| |title=Elementary theory of free non-abelian groups
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| |journal=J. Algebra |volume=302 |year=2006|issue= 2|pages= 451–552
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| |doi=10.1016/j.jalgebra.2006.03.033|ref=harv|postscript=<!--None-->
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| |mr=2293770 }}
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| *W. Magnus, A. Karrass and D. Solitar, "Combinatorial Group Theory", Dover (1976).
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| * P.J. Higgins, 1971, "Categories and Groupoids", van Nostrand, {New York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1–195.
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| *{{Cite journal
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| |last=Sela|first= Z.
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| |title=Diophantine geometry over groups. VI. The elementary theory of a free group.
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| |journal=Geom. Funct. Anal. 16 |year=2006|issue= 3|pages= 707–730|ref=harv|postscript=<!--None-->
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| |mr=2238945}}
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| *[[J.-P. Serre]], ''Trees'', Springer (2003) (English translation of "arbres, amalgames, SL<sub>2</sub>", 3rd edition, ''astérisque'' '''46''' (1983))
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| * P.J. Higgins, "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) {13}, (1976) 145–149.
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| * {{Cite book
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| | last=Aluffi
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| | first=Paolo
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| | title=Algebra: Chapter 0
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| | publisher=AMS Bookstore
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| | year=2009
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| | isbn=978-0-8218-4781-7
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| | url=http://books.google.com/books?id=deWkZWYbyHQC&pg=PA70
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| | page=70
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| | ref=harv
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| | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->
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| }}.
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| * {{Cite book
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| | last=Grillet
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| | first=Pierre Antoine
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| | title=Abstract algebra
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| | publisher=Springer
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| | year=2007
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| | isbn=978-0-387-71567-4
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| | url=http://books.google.com/books?id=LJtyhu8-xYwC&pg=PA27
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| | page=27
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| | ref=harv
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| | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->
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| }}.
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| {{DEFAULTSORT:Free Group}}
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| [[Category:Articles with inconsistent citation formats]]
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| [[Category:Group theory]]
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| [[Category:Geometric group theory]]
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| [[Category:Combinatorial group theory]]
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| [[Category:Free algebraic structures]]
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| [[Category:Properties of groups]]
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