Free abelian group (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Free abelian group" in English language version.

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  • Sims, Charles C. (1994), "Section 8.1: Free abelian groups", Computation with Finitely Presented Groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, p. 320, doi:10.1017/CBO9780511574702, ISBN 0-521-43213-8, MR 1267733
  • Fuchs, László (2015), "Section 3.1: Freeness and projectivity", Abelian Groups, Springer Monographs in Mathematics, Cham: Springer, pp. 75–80, doi:10.1007/978-3-319-19422-6, ISBN 978-3-319-19421-9, MR 3467030
  • As stated explicitly, for instance, in Hartley, Brian; Turull, Alexandre (1994), "On characters of coprime operator groups and the Glauberman character correspondence", Journal für die Reine und Angewandte Mathematik, 1994 (451): 175–219, doi:10.1515/crll.1994.451.175, MR 1277300, S2CID 118116330, proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups"
  • Baer, Reinhold (1937), "Abelian groups without elements of finite order", Duke Mathematical Journal, 3 (1): 68–122, doi:10.1215/S0012-7094-37-00308-9, hdl:10338.dmlcz/100591, MR 1545974
  • Specker, Ernst (1950), "Additive Gruppen von Folgen ganzer Zahlen", Portugaliae Math., 9: 131–140, MR 0039719
  • van Glabbeek, Rob; Goltz, Ursula; Schicke-Uffmann, Jens-Wolfhard (2013), "On characterising distributability", Logical Methods in Computer Science, 9 (3): 3:17, 58, arXiv:1309.3883, doi:10.2168/LMCS-9(3:17)2013, MR 3109601, S2CID 17046529
  • Corner, A. L. S. (2008), "Groups of units of orders in Q-algebras", Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 9–61, doi:10.1515/9783110203035.9, MR 2513226. See in particular the proof of Lemma H.4, p. 36, which uses this fact.
  • Blass, Andreas (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society, 255: 31–59, doi:10.1090/S0002-9947-1979-0542870-6, JSTOR 1998165, MR 0542870. For the connection to free objects, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group in this model that is a subgroup of a free abelian group , where is a set of atoms and is a finite integer. Blass writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free.
  • Machì, Antonio (2012), "Theorem 4.10", Groups: An introduction to ideas and methods of the theory of groups, Unitext, vol. 58, Milan: Springer, p. 172, doi:10.1007/978-88-470-2421-2, ISBN 978-88-470-2420-5, MR 2987234
  • Johnson, D. L. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, vol. 42, Cambridge University Press, p. 9, ISBN 978-0-521-23108-4, MR 0695161
  • Appendix 2 §2, page 880 of Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001
  • Bridson, Martin R.; Vogtmann, Karen (2006), "Automorphism groups of free groups, surface groups and free abelian groups", in Farb, Benson (ed.), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74, Providence, Rhode Island: American Mathematical Society, pp. 301–316, arXiv:math/0507612, doi:10.1090/pspum/074/2264548, MR 2264548, S2CID 17710182
  • Tolstykh, Vladimir (2005), "What does the automorphism group of a free abelian group A know about A?", in Blass, Andreas; Zhang, Yi (eds.), Logic and its Applications, Contemporary Mathematics, vol. 380, Providence, Rhode Island: American Mathematical Society, pp. 283–296, arXiv:math/0701752, doi:10.1090/conm/380/07117, MR 2167584, S2CID 18107280
  • Stein, Sherman K.; Szabó, Sándor (1994), Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs, vol. 25, Washington, DC: Mathematical Association of America, p. 198, ISBN 0-88385-028-1, MR 1311249
  • Higman, Graham (1940), "The units of group-rings", Proceedings of the London Mathematical Society, Second Series, 46: 231–248, doi:10.1112/plms/s2-46.1.231, MR 0002137
  • Ayoub, Raymond G.; Ayoub, Christine (1969), "On the group ring of a finite abelian group", Bulletin of the Australian Mathematical Society, 1 (2): 245–261, doi:10.1017/S0004972700041496, MR 0252526

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  • Sims, Charles C. (1994), "Section 8.1: Free abelian groups", Computation with Finitely Presented Groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, p. 320, doi:10.1017/CBO9780511574702, ISBN 0-521-43213-8, MR 1267733
  • Fuchs, László (2015), "Section 3.1: Freeness and projectivity", Abelian Groups, Springer Monographs in Mathematics, Cham: Springer, pp. 75–80, doi:10.1007/978-3-319-19422-6, ISBN 978-3-319-19421-9, MR 3467030
  • As stated explicitly, for instance, in Hartley, Brian; Turull, Alexandre (1994), "On characters of coprime operator groups and the Glauberman character correspondence", Journal für die Reine und Angewandte Mathematik, 1994 (451): 175–219, doi:10.1515/crll.1994.451.175, MR 1277300, S2CID 118116330, proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups"
  • Baer, Reinhold (1937), "Abelian groups without elements of finite order", Duke Mathematical Journal, 3 (1): 68–122, doi:10.1215/S0012-7094-37-00308-9, hdl:10338.dmlcz/100591, MR 1545974
  • van Glabbeek, Rob; Goltz, Ursula; Schicke-Uffmann, Jens-Wolfhard (2013), "On characterising distributability", Logical Methods in Computer Science, 9 (3): 3:17, 58, arXiv:1309.3883, doi:10.2168/LMCS-9(3:17)2013, MR 3109601, S2CID 17046529
  • Corner, A. L. S. (2008), "Groups of units of orders in Q-algebras", Models, modules and abelian groups, Walter de Gruyter, Berlin, pp. 9–61, doi:10.1515/9783110203035.9, MR 2513226. See in particular the proof of Lemma H.4, p. 36, which uses this fact.
  • Blass, Andreas (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society, 255: 31–59, doi:10.1090/S0002-9947-1979-0542870-6, JSTOR 1998165, MR 0542870. For the connection to free objects, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group in this model that is a subgroup of a free abelian group , where is a set of atoms and is a finite integer. Blass writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free.
  • Machì, Antonio (2012), "Theorem 4.10", Groups: An introduction to ideas and methods of the theory of groups, Unitext, vol. 58, Milan: Springer, p. 172, doi:10.1007/978-88-470-2421-2, ISBN 978-88-470-2420-5, MR 2987234
  • Bridson, Martin R.; Vogtmann, Karen (2006), "Automorphism groups of free groups, surface groups and free abelian groups", in Farb, Benson (ed.), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics, vol. 74, Providence, Rhode Island: American Mathematical Society, pp. 301–316, arXiv:math/0507612, doi:10.1090/pspum/074/2264548, MR 2264548, S2CID 17710182
  • Tolstykh, Vladimir (2005), "What does the automorphism group of a free abelian group A know about A?", in Blass, Andreas; Zhang, Yi (eds.), Logic and its Applications, Contemporary Mathematics, vol. 380, Providence, Rhode Island: American Mathematical Society, pp. 283–296, arXiv:math/0701752, doi:10.1090/conm/380/07117, MR 2167584, S2CID 18107280
  • Higman, Graham (1940), "The units of group-rings", Proceedings of the London Mathematical Society, Second Series, 46: 231–248, doi:10.1112/plms/s2-46.1.231, MR 0002137
  • Ayoub, Raymond G.; Ayoub, Christine (1969), "On the group ring of a finite abelian group", Bulletin of the Australian Mathematical Society, 1 (2): 245–261, doi:10.1017/S0004972700041496, MR 0252526

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  • Blass, Andreas (1979), "Injectivity, projectivity, and the axiom of choice", Transactions of the American Mathematical Society, 255: 31–59, doi:10.1090/S0002-9947-1979-0542870-6, JSTOR 1998165, MR 0542870. For the connection to free objects, see Corollary 1.2. Example 7.1 provides a model of set theory without choice, and a non-free projective abelian group in this model that is a subgroup of a free abelian group , where is a set of atoms and is a finite integer. Blass writes that this model makes the use of choice essential in proving that every projective group is free; by the same reasoning it also shows that choice is essential in proving that subgroups of free groups are free.

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