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| :''For the dimension of the Cartan subgroup, see [[Rank of a Lie group]]''
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| In the [[Mathematics|mathematical]] subject of [[group theory]], the '''rank of a group''' ''G'', denoted rank(''G''), can refer to the smallest [[cardinality]] of a [[Generating set of a group|generating]] set for ''G'', that is
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| :<math> \operatorname{rank}(G)=\min\{ |X|: X\subseteq G, \langle X\rangle =G\}.</math>
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| If ''G'' is a [[finitely generated group]], then the rank of ''G'' is a nonnegative integer. The notion of rank of a group is a group-theoretic analog of the notion of [[dimension of a vector space]]. Indeed, for [[p-group|''p''-groups]], the rank of the group ''P'' is the dimension of the vector space ''P''/Φ(''P''), where Φ(''P'') is the [[Frattini subgroup]].
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| The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such as [[affine group]]s. To distinguish these different definitions, one sometimes calls this rank the '''subgroup rank'''. Explicitly, the subgroup rank of a group ''G'' is the maximum of the ranks of its subgroups:
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| :<math> \operatorname{sr}(G)=\max_{H \leq G} \min\{ |X|: X \subseteq H, \langle X \rangle = H \}.</math>
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| Sometimes the subgroup rank is restricted to abelian subgroups.
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| ==Known facts and examples==
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| *For a nontrivial group ''G'', we have rank(''G'')=1 if and only if ''G'' is a [[cyclic group]].
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| *For a [[free abelian group]] <math>\mathbb Z^n</math> we have <math> {\rm rank}(\mathbb Z^n)=n.</math>
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| *If ''X'' is a set and ''G'' = ''F''(''X'') is the [[free group]] with free basis ''X'' then rank(''G'') = |''X''|.
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| *If a group ''H'' is a [[group homomorphism|homomorphic image]] (or a [[quotient group]]) of a group ''G'' then rank(''H'') ≤ rank(''G'').
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| *If ''G'' is a finite non-abelian [[simple group]] (e.g. ''G = A<sub>n</sub>'', the [[alternating group]], for ''n'' > 4) then rank(''G'') = 2. This fact is a consequence of the [[Classification of finite simple groups]].
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| *If ''G'' is a finitely generated group and Φ(''G'') ≤ ''G'' is the [[Frattini subgroup]] of ''G'' (which is always normal in ''G'' so that the quotient group ''G''/Φ(''G'') is defined) then rank(''G'') = rank(''G''/Φ(''G'')).<ref name="Robinson">D. J. S. Robinson. ''A course in the theory of groups'', 2nd edn, Graduate Texts in Mathematics 80 (Springer-Verlag, 1996). ISBN 0-387-94461-3</ref>
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| *If ''G'' is the [[fundamental group]] of a closed (that is [[compact space|compact]] and without boundary) connected [[3-manifold]] ''M'' then rank(''G'')≤''g''(''M''), where ''g''(''M'') is the [[Heegaard genus]] of ''M''.<ref>Friedhelm Waldhausen. ''Some problems on 3-manifolds.'' Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 313–322, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978; ISBN 0-8218-1433-8</ref>
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| *If ''H'',''K'' ≤ ''F''(''X'') are [[finitely generated group|finitely generated]] subgroups of a [[free group]] ''F''(''X'') such that the intersection <math>L=H\cap K</math> is nontrivial, then ''L'' is finitely generated and
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| :rank(''L'') − 1 ≤ 2(rank(''K'') − 1)(rank(''H'') − 1).
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| :This result is due to [[Hanna Neumann]].<ref>Hanna Neumann. ''On the intersection of finitely generated free groups.''
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| [[Publicationes Mathematicae Debrecen]], vol. 4 (1956), 186–189.</ref><ref>Hanna Neumann. ''On the intersection of finitely generated free groups. Addendum.''
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| Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128</ref> The [[Hanna Neumann conjecture]] states that in fact one always has rank(''L'') − 1 ≤ (rank(''K'') − 1)(rank(''H'') − 1). The [[Hanna Neumann conjecture]] has recently been solved by Igor Mineyev<ref name="proof">Igor Minevev,
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| [http://annals.math.princeton.edu/2012/175-1/p11/ "Submultiplicativity and the Hanna Neumann Conjecture."] Ann. of Math., 175 (2012), no. 1, 393-414.</ref> and announced independently by Joel Friedman.<ref name="JF">{{cite web|url=http://www.math.ubc.ca/~jf/pubs/web_stuff/mehanna.html |title=Sheaves on Graphs and a Proof of the Hanna Neumann Conjecture |publisher=Math.ubc.ca |date= |accessdate=2012-06-12}}</ref>
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| *According to the classic [[Grushko theorem]], rank behaves additively with respect to taking [[free product]]s, that is, for any groups ''A'' and ''B'' we have
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| :rank(''A''<math>\ast</math>''B'') = rank(''A'') + rank(''B'').
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| *If <math>G=\langle x_1,\dots, x_n| r=1\rangle</math> is a [[one-relator group]] such that ''r'' is not a [[primitive element]] in the free group ''F''(''x''<sub>1</sub>,..., ''x''<sub>''n''</sub>), that is, ''r'' does not belong to a free basis of ''F''(''x''<sub>1</sub>,..., ''x''<sub>''n''</sub>), then rank(''G'') = ''n''.<ref>[[Wilhelm Magnus]], ''Uber freie Faktorgruppen und freie Untergruppen Gegebener Gruppen'', Monatshefte für Mathematik, vol. 47(1939), pp. 307–313. </ref><ref>[[Roger Lyndon|Roger C. Lyndon]] and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1; Proposition 5.11, p. 107</ref>
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| ==The rank problem==
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| There is an algorithmic problem studied in [[group theory]], known as the '''rank problem'''. The problem asks, for a particular class of [[finitely presented group]]s if there exists an algorithm that, given a finite presentation of a group from the class, computes the rank of that group. The rank problem is one of the harder algorithmic problems studied in group theory and relatively little is known about it. Known results include:
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| *The rank problem is algorithmically undecidable for the class of all [[finitely presented group]]s. Indeed, by a classical result of Adian-Rabin, there is no algorithm to decide if a finitely presented group is trivial, so even the question of whether rank(''G'')=0 is undecidable for finitely presented groups.<ref>W. W. Boone.
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| ''Decision problems about algebraic and logical systems as a whole and recursively enumerable degrees of unsolvability.'' 1968 Contributions to Math. Logic (Colloquium, Hannover, 1966) pp. 13 33 North-Holland, Amsterdam </ref><ref>Charles F. Miller, III. ''Decision problems for groups — survey and reflections.'' Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), pp. 1–59, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992; ISBN 0-387-97685-X</ref>
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| *The rank problem is decidable for finite groups and for finitely generated [[abelian group]]s.
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| *The rank problem is decidable for finitely generated [[nilpotent group]]s. The reason is that for such a group ''G'', the [[Frattini subgroup]] of ''G'' contains the [[commutator subgroup]] of ''G'' and hence the rank of ''G'' is equal to the rank of the [[abelianization]] of ''G''. <ref>John Lennox, and Derek J. S. Robinson. ''The theory of infinite soluble groups.'' Oxford Mathematical Monographs. The Clarendon Press, [[Oxford University Press]], Oxford, 2004. ISBN 0-19-850728-3 </ref>
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| *The rank problem is undecidable for [[word hyperbolic group]]s.<ref>G. Baumslag, C. F. Miller and H. Short. ''Unsolvable problems about small cancellation and word hyperbolic groups.'' Bulletin of the London Mathematical Society, vol. 26 (1994), pp. 97–101 </ref>
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| *The rank problem is decidable for torsion-free [[Kleinian group]]s.<ref> Ilya Kapovich, and Richard Weidmann. [http://msp.warwick.ac.uk/gt/2005/09/p012.xhtml ''Kleinian groups and the rank problem'']. [[Geometry & Topology|Geometry and Topology]], vol. 9 (2005), pp. 375–402 </ref>
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| *The rank problem is open for finitely generated virtually abelian groups (that is containing an abelian subgroup of finite [[index of a subgroup|index]]), for virtually free groups, and for [[3-manifold]] groups.
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| ==Generalizations and related notions==
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| The rank of a [[finitely generated group]] ''G'' can be equivalently defined as the smallest cardinality of a set ''X'' such that there exists an onto [[group homomorphism|homomorphism]] ''F''(''X'') → ''G'', where ''F''(''X'') is the [[free group]] with free basis ''X''. There is a dual notion of '''co-rank''' of a [[finitely generated group]] ''G'' defined as the ''largest'' [[cardinality]] of ''X'' such that there exists an onto [[group homomorphism|homomorphism]] ''G'' → ''F''(''X''). Unlike rank, co-rank is always algorithmically computable for [[finitely presented group]]s,<ref>John R. Stallings.
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| ''Problems about free quotients of groups.'' Geometric group theory (Columbus, OH, 1992), pp. 165–182, Ohio State Univ. Math. Res. Inst. Publ., 3, de Gruyter, Berlin, 1995. ISBN 3-11-014743-2</ref> using the algorithm of Makanin and [[Alexander Razborov|Razborov]] for solving systems of equations in free groups.<ref>A. A. Razborov.
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| ''Systems of equations in a free group.'' (in Russian) Izvestia Akademii Nauk SSSR, Seriya Matematischeskaya, vol. 48 (1984), no. 4, pp. 779–832. </ref><ref>G. S.Makanin
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| ''Equations in a free group.'' (Russian), Izvestia Akademii Nauk SSSR, Seriya Matematischeskaya, vol. 46 (1982), no. 6, pp. 1199–1273 </ref>
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| The notion of co-rank is related to the notion of a ''cut number'' for [[3-manifolds]].<ref>Shelly L. Harvey. [http://www.msp.warwick.ac.uk/gt/2002/06/p015.xhtml ''On the cut number of a 3-manifold.''] [[Geometry & Topology]], vol. 6 (2002), pp. 409–424 </ref>
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| If ''p'' is a [[prime number]], then the ''p''-'''rank''' of ''G'' is the largest rank of an [[elementary abelian group|elementary abelian]] ''p''-subgroup.<ref>{{Citation|last=Aschbacher|first=M.|title=Finite Group Theory|publisher=Cambridge University Press|year=2002|isbn=978-0-521-78675-1|page=5}}</ref> The '''sectional''' ''p''-'''rank''' is the largest rank of an elementary abelian ''p''-section (quotient of a subgroup).
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| ==Notes==
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| {{reflist}}
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| ==See also==
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| *[[Rank of an abelian group]]
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| *[[Prüfer rank]]
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| *[[Grushko theorem]]
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| *[[Free group]]
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| *[[Nielsen equivalence]]
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| [[Category:Group theory]]
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