Teorema di Nielsen-Schreier

Nielsen–Schreier theorem In group theory, una branca della matematica, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free.[1][2][3] It is named after Jakob Nielsen and Otto Schreier.

Contenuti 1 Enunciato del teorema 2 Esempio 3 Prova 4 Axiomatic foundations 5 Storia 6 Guarda anche 7 Appunti 8 References Statement of the theorem A free group may be defined from a group presentation consisting of a set of generators with no relations. Questo è, every element is a product of some sequence of generators and their inverses, but these elements do not obey any equations except those trivially following from gg−1 = 1. The elements of a free group may be described as all possible reduced words, those strings of generators and their inverses in which no generator is adjacent to its own inverse. Two reduced words may be multiplied by concatenating them and then removing any generator-inverse pairs that result from the concatenation.

The Nielsen–Schreier theorem states that if H is a subgroup of a free group G, then H is itself isomorphic to a free group. Questo è, there exists a set S of elements which generate H, with no nontrivial relations among the elements of S.

The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the subgroup has finite index: if G is a free group of rank n (free on n generators), and H is a subgroup of finite index [G : H] = e, then H is free of rank {displaystyle 1+e(n{-}1)} .[4] Example Let G be the free group with two generators {stile di visualizzazione a,b} , and let H be the subgroup consisting of all reduced words of even length (products of an even number of letters {stile di visualizzazione a,b,un^{-1},b^{-1}} ). Then H is generated by its six elements {displaystyle p=aa, q=ab, r=ba, s=bb, t=ab^{-1}, u=a^{-1}b.} A factorization of any reduced word in H into these generators and their inverses may be constructed simply by taking consecutive pairs of letters in the reduced word. Tuttavia, this is not a free presentation of H because the last three generators can be written in terms of the first three as {displaystyle s=rp^{-1}q, t=pr^{-1}, u=p^{-1}q} . Rather, H is generated as a free group by the three elements {displaystyle p=aa, q=ab, r=ba,} which have no relations among them; or instead by several other triples of the six generators.[5] Ulteriore, G is free on n = 2 generators, H has index e = [G : H] = 2 in G, and H is free on 1 + e(n–1) = 3 generators. The Nielsen–Schreier theorem states that like H, every subgroup of a free group can be generated as a free group, and if the index of H is finite, its rank is given by the index formula.

Proof The free group G = π1(X) has n = 2 generators corresponding to loops a,b from the base point P in X. The subgroup H of even-length words, with index e = [G : H] = 2, corresponds to the covering graph Y with two vertices corresponding to the cosets H and H' = aH = bH = a−1H = b−1H, and two lifted edges for each of the original loop-edges a,b. Contracting one of the edges of Y gives a homotopy equivalence to a bouquet of three circles, so that H = π1(Y) is a free group on three generators, for example aa, ab, ba.

A short proof of the Nielsen–Schreier theorem uses the algebraic topology of fundamental groups and covering spaces.[1] A free group G on a set of generators is the fundamental group of a bouquet of circles, a topological graph X with a single vertex and with a loop-edge for each generator.[6] Any subgroup H of the fundamental group is itself the fundamental group of a connected covering space Y → X. The space Y is a (possibly infinite) topological graph, the Schreier coset graph having one vertex for each coset in G/H.[7] In any connected topological graph, it is possible to shrink the edges of a spanning tree of the graph, producing a bouquet of circles that has the same fundamental group H. Since H is the fundamental group of a bouquet of circles, it is itself free.[6] Simplicial homology allows the computation of the rank of H, which is equal to h1(Y), the first Betti number of the covering space, the number of independent cycles. For G free of rank n, the graph X has n edges and 1 vertex; assuming H has finite index [G : H] = e, the covering graph Y has en edges and e vertices. The first Betti number of a graph is equal to the number of edges, minus the number of vertices, plus the number of connected components; hence the rank of H is: {stile di visualizzazione h_{1}(Y),=,en-e+1,=,1+e(n{-}1).} This proof is due to Reinhold Baer and Friedrich Levi (1936); the original proof by Schreier forms the Schreier graph in a different way as a quotient of the Cayley graph of G modulo the action of H.[8] According to Schreier's subgroup lemma, a set of generators for a free presentation of H may be constructed from cycles in the covering graph formed by concatenating a spanning tree path from a base point (the coset of the identity) to one of the cosets, a single non-tree edge, and an inverse spanning tree path from the other endpoint of the edge back to the base point.[9][8] Axiomatic foundations Although several different proofs of the Nielsen–Schreier theorem are known, they all depend on the axiom of choice. In the proof based on fundamental groups of bouquets, per esempio, the axiom of choice appears in the guise of the statement that every connected graph has a spanning tree. The use of this axiom is necessary, as there exist models of Zermelo–Fraenkel set theory in which the axiom of choice and the Nielsen–Schreier theorem are both false. The Nielsen–Schreier theorem in turn implies a weaker version of the axiom of choice, for finite sets.[10][11] History The Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian.[3] Jakob Nielsen (1921) originally proved a restricted form of the theorem, stating that any finitely-generated subgroup of a free group is free. His proof involves performing a sequence of Nielsen transformations on the subgroup's generating set that reduce their length (as reduced words in the free group from which they are drawn).[1][12] Otto Schreier proved the Nielsen–Schreier theorem in its full generality in his 1926 habilitation thesis, Die Untergruppen der freien Gruppe, also published in 1927 in Abh. math. Sem. Hamburg. Univ.[13][14] The topological proof based on fundamental groups of bouquets of circles is due to Reinhold Baer and Friedrich Levi (1936). Another topological proof, based on the Bass–Serre theory of group actions on trees, was published by Jean-Pierre Serre (1970).[15] See also Fundamental theorem of cyclic groups, a similar result for cyclic groups that in the infinite case may be seen as a special case of the Nielsen–Schreier theorem Kurosh subgroup theorem Notes ^ Jump up to: a b c Stillwell (1993), Sezione 2.2.4, The Nielsen–Schreier Theorem, pp. 103–104. ^ Magnus, Karass & Solitar 1976, Corollario 2.9, p. 95. ^ Salta su: a b Johnson (1980), Sezione 2, The Nielsen–Schreier Theorem, pp. 9–23. ^ Fried & Jarden (2008), p. 355 ^ Johnson (1997), ex. 15, p. 12. ^ Salta su: a b Stillwell (1993), Sezione 2.1.8, Freeness of the Generators, p. 97. ^ Stillwell (1993), Sezione 2.2.2, The Subgroup Property, pp. 100–101. ^ Salta su: a b Bollobas, Bela (1998). "Chapter VIII.1". Modern Graph Theory. Casa editrice Springer. p. 262. ISBN 978-0-387-98488-9. ^ Stillwell (1993), Sezione 2.2.6, Schreier Transversals, pp. 105–106. ^ Leuchli (1962) ^ Howard (1985). ^ Magnus, Karass & Solitar 1976, Sezione 3.2, A Reduction Process, pp. 121–140. ^ O'Connor, John J.; Robertson, Edmund F., "Otto Schreier", Archivio di storia della matematica di MacTutor, University of St Andrews ^ Hansen, Vagn Lundsgaard (1986), Jakob Nielsen, Collected Mathematical Papers: 1913-1932, Birkhauser, p. 117, ISBN 978-0-8176-3140-6. ^ Rotman (1995), The Nielsen–Schreier Theorem, pp. 383–387. References Baer, Reinhold; Levi, Friedrich (1936), "Freie Produkte und ihre Untergruppen", Composizione matematica, 3: 391–398. Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Risultati della matematica e loro aree di confine. 3. Folge, vol. 11 (33a ed.), Springer-Verlag, p. 70, ISBN 978-3-540-77269-9, Zbl 1145.12001. Howard, Paul E. (1985), "Subgroups of a free group and the axiom of choice", Il diario della logica simbolica, 50 (2): 458–467, doi:10.2307/2274234, JSTOR 2274234, SIG 0793126. Johnson, D. l. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, vol. 42, Cambridge University Press, ISBN 978-0-521-23108-4. Johnson, D. l. (1997), Presentations of Groups, London Mathematical Society student texts, vol. 15 (2nd ed.), Cambridge University Press, ISBN 978-0-521-58542-2. Läuchli, Hans (1962), "Auswahlaxiom in der Algebra", Commentarii Mathematici Helvetici, 37: 1–18, doi:10.1007/bf02566957, hdl:20.500.11850/131689, SIG 0143705, S2CID 186223589. Magnus, Guglielmo; Karrass, Abramo; Solitario, Paperino (1976), Combinatorial Group Theory (2nd revised ed.), Pubblicazioni di Dover. Nielsen, Jakob (1921), "Om regning med ikke-kommutative faktorer og dens anvendelse i gruppeteorien", Matematica. Tidsskrift B (in Danish), 1921: 78–94, JFM 48.0123.03. Rotman, Giuseppe J. (1995), Introduzione alla teoria dei gruppi, Testi di laurea in Matematica, vol. 148 (4th ed.), Springer-Verlag, ISBN 978-0-387-94285-8. Stretto, J.-P. (1970), Groupes Discretes, Extrait de I'Annuaire du College de France, Parigi. Stretto, J.-P. (1980), Trees, Springer-Verlag, ISBN 3-540-10103-9. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Testi di laurea in Matematica, vol. 72 (2nd ed.), Springer-Verlag. Categorie: Properties of groupsAxiom of choiceTheorems in algebra

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