|
:''For the dimension of the Cartan subgroup, see Rank of a Lie group'' In the mathematical subject of group theory, the rank of a group ''G'', denoted rank(''G''), can refer to the smallest cardinality of a generating set for ''G'', that is : 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''-groups, the rank of the group ''P'' is the dimension of the vector space ''P''/Φ(''P''), where Φ(''P'') is the Frattini subgroup. 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 groups. 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: : Sometimes the subgroup rank is restricted to abelian subgroups. ==Known facts and examples== *For a nontrivial group ''G'', we have rank(''G'')=1 if and only if ''G'' is a cyclic group. *For a free abelian group we have *If ''X'' is a set and ''G'' = ''F''(''X'') is the free group with free basis ''X'' then rank(''G'') = |''X''|. *If a group ''H'' is a homomorphic image (or a quotient group) of a group ''G'' then rank(''H'') ≤ rank(''G''). *If ''G'' is a finite non-abelian simple group (e.g. ''G = An'', the alternating group, for ''n'' > 4) then rank(''G'') = 2. This fact is a consequence of the Classification of finite simple groups. *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'')).〔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〕 *If ''G'' is the fundamental group of a closed (that is compact and without boundary) connected 3-manifold ''M'' then rank(''G'')≤''g''(''M''), where ''g''(''M'') is the Heegaard genus of ''M''.〔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〕 *If ''H'',''K'' ≤ ''F''(''X'') are finitely generated subgroups of a free group ''F''(''X'') such that the intersection is nontrivial, then ''L'' is finitely generated and :rank(''L'') − 1 ≤ 2(rank(''K'') − 1)(rank(''H'') − 1). :This result is due to Hanna Neumann.〔Hanna Neumann. ''On the intersection of finitely generated free groups.'' Publicationes Mathematicae Debrecen, vol. 4 (1956), 186–189.〕〔Hanna Neumann. ''On the intersection of finitely generated free groups. Addendum.'' Publicationes Mathematicae Debrecen, vol. 5 (1957), p. 128〕 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〔Igor Minevev, ("Submultiplicativity and the Hanna Neumann Conjecture." ) Ann. of Math., 175 (2012), no. 1, 393-414.〕 and announced independently by Joel Friedman. *According to the classic Grushko theorem, rank behaves additively with respect to taking free products, that is, for any groups ''A'' and ''B'' we have :rank(''A''''B'') = rank(''A'') + rank(''B''). *If is a one-relator group such that ''r'' is not a primitive element in the free group ''F''(''x''1,..., ''x''''n''), that is, ''r'' does not belong to a free basis of ''F''(''x''1,..., ''x''''n''), then rank(''G'') = ''n''.〔Wilhelm Magnus, ''Uber freie Faktorgruppen und freie Untergruppen Gegebener Gruppen'', Monatshefte für Mathematik, vol. 47(1939), pp. 307–313. 〕〔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〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「rank of a group」の詳細全文を読む スポンサード リンク
|