自由阿贝尔群 (Chinese Wikipedia)

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ams.org

Sims, Charles C., Section 8.1: Free abelian groups, Computation with Finitely Presented Groups, Encyclopedia of Mathematics and its Applications 48, Cambridge University Press: 320, 1994 [2023-09-01], ISBN 0-521-43213-8, MR 1267733, doi:10.1017/CBO9780511574702, （原始内容存档于2023-09-01）
Fuchs, László, Section 3.1: Freeness and projectivity, Abelian Groups, Springer Monographs in Mathematics, Cham: Springer: 75–80, 2015 [2023-09-01], ISBN 978-3-319-19421-9, MR 3467030, doi:10.1007/978-3-319-19422-6, （原始内容存档于2023-09-01）
As stated explicitly, for instance, in Hartley, Brian; Turull, Alexandre, On characters of coprime operator groups and the Glauberman character correspondence, Journal für die Reine und Angewandte Mathematik, 1994, 1994 (451): 175–219, MR 1277300, S2CID 118116330, doi:10.1515/crll.1994.451.175 , proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups"
Baer, Reinhold, Abelian groups without elements of finite order, Duke Mathematical Journal, 1937, 3 (1): 68–122, MR 1545974, doi:10.1215/S0012-7094-37-00308-9, hdl:10338.dmlcz/100591 
Specker, Ernst, Additive Gruppen von Folgen ganzer Zahlen, Portugaliae Math., 1950, 9: 131–140, MR 0039719
van Glabbeek, Rob; Goltz, Ursula; Schicke-Uffmann, Jens-Wolfhard, On characterising distributability, Logical Methods in Computer Science, 2013, 9 (3): 3:17, 58, MR 3109601, S2CID 17046529, arXiv:1309.3883 , doi:10.2168/LMCS-9(3:17)2013
Corner, A. L. S., Groups of units of orders in Q-algebras, Models, modules and abelian groups, Walter de Gruyter, Berlin: 9–61, 2008, MR 2513226, doi:10.1515/9783110203035.9 . See in particular the proof of Lemma H.4, p. 36, which uses this fact.
Blass, Andreas, Injectivity, projectivity, and the axiom of choice, Transactions of the American Mathematical Society, 1979, 255: 31–59, JSTOR 1998165, MR 0542870, doi:10.1090/S0002-9947-1979-0542870-6  . 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 $P$ in this model that is a subgroup of a free abelian group ${\textstyle {\bigl (}\mathbb {Z} ^{(A)}{\bigr )}^{n}}$ , where $A$ is a set of atoms and $n$ 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, Theorem 4.10, Groups: An introduction to ideas and methods of the theory of groups, Unitext 58, Milan: Springer: 172, 2012, ISBN 978-88-470-2420-5, MR 2987234, doi:10.1007/978-88-470-2421-2
Johnson, D. L., Topics in the Theory of Group Presentations, London Mathematical Society lecture note series 42, Cambridge University Press: 9, 1980, ISBN 978-0-521-23108-4, MR 0695161
Appendix 2 §2, page 880 of Lang, Serge, Algebra, Graduate Texts in Mathematics 211 Revised third, New York: Springer-Verlag, 2002, ISBN 978-0-387-95385-4, MR 1878556
Bridson, Martin R.; Vogtmann, Karen, Automorphism groups of free groups, surface groups and free abelian groups, Farb, Benson (编), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics 74, Providence, Rhode Island: American Mathematical Society: 301–316, 2006, MR 2264548, S2CID 17710182, arXiv:math/0507612 , doi:10.1090/pspum/074/2264548
Tolstykh, Vladimir, What does the automorphism group of a free abelian group $A$ know about $A$ ?, Blass, Andreas; Zhang, Yi (编), Logic and its Applications, Contemporary Mathematics 380, Providence, Rhode Island: American Mathematical Society: 283–296, 2005, MR 2167584, S2CID 18107280, arXiv:math/0701752 , doi:10.1090/conm/380/07117
Stein, Sherman K.; Szabó, Sándor, Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs 25, Washington, DC: Mathematical Association of America: 198, 1994, ISBN 0-88385-028-1, MR 1311249
Higman, Graham, The units of group-rings, Proceedings of the London Mathematical Society, Second Series, 1940, 46: 231–248, MR 0002137, doi:10.1112/plms/s2-46.1.231
Ayoub, Raymond G.; Ayoub, Christine, On the group ring of a finite abelian group, Bulletin of the Australian Mathematical Society, 1969, 1 (2): 245–261, MR 0252526, doi:10.1017/S0004972700041496

arxiv.org

Bradley, David M., Counting the positive rationals: A brief survey, 2005, Bibcode:2005math......9025B, arXiv:math/0509025 
van Glabbeek, Rob; Goltz, Ursula; Schicke-Uffmann, Jens-Wolfhard, On characterising distributability, Logical Methods in Computer Science, 2013, 9 (3): 3:17, 58, MR 3109601, S2CID 17046529, arXiv:1309.3883 , doi:10.2168/LMCS-9(3:17)2013
Bridson, Martin R.; Vogtmann, Karen, Automorphism groups of free groups, surface groups and free abelian groups, Farb, Benson (编), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics 74, Providence, Rhode Island: American Mathematical Society: 301–316, 2006, MR 2264548, S2CID 17710182, arXiv:math/0507612 , doi:10.1090/pspum/074/2264548
Tolstykh, Vladimir, What does the automorphism group of a free abelian group $A$ know about $A$ ?, Blass, Andreas; Zhang, Yi (编), Logic and its Applications, Contemporary Mathematics 380, Providence, Rhode Island: American Mathematical Society: 283–296, 2005, MR 2167584, S2CID 18107280, arXiv:math/0701752 , doi:10.1090/conm/380/07117

books.google.com

Sims, Charles C., Section 8.1: Free abelian groups, Computation with Finitely Presented Groups, Encyclopedia of Mathematics and its Applications 48, Cambridge University Press: 320, 1994 [2023-09-01], ISBN 0-521-43213-8, MR 1267733, doi:10.1017/CBO9780511574702, （原始内容存档于2023-09-01）
Vick, James W., Homology Theory: An Introduction to Algebraic Topology, Graduate Texts in Mathematics 145, Springer: 4, 70, 1994 [2023-09-01], ISBN 9780387941264, （原始内容存档于2023-08-09）
Fuchs, László, Section 3.1: Freeness and projectivity, Abelian Groups, Springer Monographs in Mathematics, Cham: Springer: 75–80, 2015 [2023-09-01], ISBN 978-3-319-19421-9, MR 3467030, doi:10.1007/978-3-319-19422-6, （原始内容存档于2023-09-01）
Mollin, Richard A., Advanced Number Theory with Applications, CRC Press: 182, 2011, ISBN 9781420083293
Bremner, Murray R., Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications, CRC Press: 6, 2011, ISBN 9781439807026
Lee, John M., Free Abelian Groups, Introduction to Topological Manifolds, Graduate Texts in Mathematics 202 2nd, Springer: 244–248, 2010, ISBN 9781441979407
Mac Lane, Saunders, Homology, Classics in Mathematics, Springer: 93, 1995, ISBN 9783540586623
Kaplansky, Irving, Set Theory and Metric Spaces, AMS Chelsea Publishing Series 298, American Mathematical Society: 124–125, 2001, ISBN 9780821826942
Hungerford, Thomas W., II.1 Free abelian groups, Algebra, Graduate Texts in Mathematics 73, Springer: 70–75, 1974, ISBN 9780387905181 . See in particular Theorem 1.1, pp. 72–73, and the remarks following it.
Joshi, K. D., Applied Discrete Structures, New Age International: 45–46, 1997, ISBN 9788122408263
Johnson, D. L., Symmetries, Springer undergraduate mathematics series, Springer: 71, 2001 [2023-09-01], ISBN 9781852332709, （原始内容存档于2023-09-02）
Sahai, Vivek; Bist, Vikas, Algebra, Alpha Science International Ltd.: 152, 2003, ISBN 9781842651575
Rotman, Joseph J., Advanced Modern Algebra, American Mathematical Society: 450, 2015, ISBN 9780821884201
Corner, A. L. S., Groups of units of orders in Q-algebras, Models, modules and abelian groups, Walter de Gruyter, Berlin: 9–61, 2008, MR 2513226, doi:10.1515/9783110203035.9 . See in particular the proof of Lemma H.4, p. 36, which uses this fact.
Hatcher, Allen, Algebraic Topology, Cambridge University Press: 196, 2002, ISBN 9780521795401
Vermani, L. R., An Elementary Approach to Homological Algebra, Monographs and Surveys in Pure and Applied Mathematics, CRC Press: 80, 2004, ISBN 9780203484081
Hofmann, Karl H.; Morris, Sidney A., The Structure of Compact Groups: A Primer for Students - A Handbook for the Expert, De Gruyter Studies in Mathematics 25 2nd, Walter de Gruyter: 640, 2006, ISBN 9783110199772
Rotman, Joseph J., An Introduction to Algebraic Topology, Graduate Texts in Mathematics 119, Springer: 61–62, 1988, ISBN 9780387966786
The theorem that free abelian groups are projective is equivalent to the axiom of choice; see Moore, Gregory H., Zermelo's Axiom of Choice: Its Origins, Development, and Influence, Courier Dover Publications: xii, 2012, ISBN 9780486488417
Cavagnaro, Catherine; Haight, William T. II, Dictionary of Classical and Theoretical Mathematics, Comprehensive Dictionary of Mathematics 3, CRC Press: 15, 2001, ISBN 9781584880509
Edelsbrunner, Herbert; Harer, John, Computational Topology: An Introduction, Providence, Rhode Island: American Mathematical Society: 79–81, 2010, ISBN 9780821849255
Dedekind, Richard; Weber, Heinrich, Theory of Algebraic Functions of One Variable, History of mathematics 39, Translated by John Stillwell, American Mathematical Society: 13–15, 2012, ISBN 9780821890349
Miranda, Rick, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics 5, American Mathematical Society: 129, 1995, ISBN 9780821802687

doi.org

Sims, Charles C., Section 8.1: Free abelian groups, Computation with Finitely Presented Groups, Encyclopedia of Mathematics and its Applications 48, Cambridge University Press: 320, 1994 [2023-09-01], ISBN 0-521-43213-8, MR 1267733, doi:10.1017/CBO9780511574702, （原始内容存档于2023-09-01）
Fuchs, László, Section 3.1: Freeness and projectivity, Abelian Groups, Springer Monographs in Mathematics, Cham: Springer: 75–80, 2015 [2023-09-01], ISBN 978-3-319-19421-9, MR 3467030, doi:10.1007/978-3-319-19422-6, （原始内容存档于2023-09-01）
As stated explicitly, for instance, in Hartley, Brian; Turull, Alexandre, On characters of coprime operator groups and the Glauberman character correspondence, Journal für die Reine und Angewandte Mathematik, 1994, 1994 (451): 175–219, MR 1277300, S2CID 118116330, doi:10.1515/crll.1994.451.175 , proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups"
Baer, Reinhold, Abelian groups without elements of finite order, Duke Mathematical Journal, 1937, 3 (1): 68–122, MR 1545974, doi:10.1215/S0012-7094-37-00308-9, hdl:10338.dmlcz/100591 
van Glabbeek, Rob; Goltz, Ursula; Schicke-Uffmann, Jens-Wolfhard, On characterising distributability, Logical Methods in Computer Science, 2013, 9 (3): 3:17, 58, MR 3109601, S2CID 17046529, arXiv:1309.3883 , doi:10.2168/LMCS-9(3:17)2013
Corner, A. L. S., Groups of units of orders in Q-algebras, Models, modules and abelian groups, Walter de Gruyter, Berlin: 9–61, 2008, MR 2513226, doi:10.1515/9783110203035.9 . See in particular the proof of Lemma H.4, p. 36, which uses this fact.
Blass, Andreas, Injectivity, projectivity, and the axiom of choice, Transactions of the American Mathematical Society, 1979, 255: 31–59, JSTOR 1998165, MR 0542870, doi:10.1090/S0002-9947-1979-0542870-6  . 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 $P$ in this model that is a subgroup of a free abelian group ${\textstyle {\bigl (}\mathbb {Z} ^{(A)}{\bigr )}^{n}}$ , where $A$ is a set of atoms and $n$ 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, Theorem 4.10, Groups: An introduction to ideas and methods of the theory of groups, Unitext 58, Milan: Springer: 172, 2012, ISBN 978-88-470-2420-5, MR 2987234, doi:10.1007/978-88-470-2421-2
Bridson, Martin R.; Vogtmann, Karen, Automorphism groups of free groups, surface groups and free abelian groups, Farb, Benson (编), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics 74, Providence, Rhode Island: American Mathematical Society: 301–316, 2006, MR 2264548, S2CID 17710182, arXiv:math/0507612 , doi:10.1090/pspum/074/2264548
Tolstykh, Vladimir, What does the automorphism group of a free abelian group $A$ know about $A$ ?, Blass, Andreas; Zhang, Yi (编), Logic and its Applications, Contemporary Mathematics 380, Providence, Rhode Island: American Mathematical Society: 283–296, 2005, MR 2167584, S2CID 18107280, arXiv:math/0701752 , doi:10.1090/conm/380/07117
Higman, Graham, The units of group-rings, Proceedings of the London Mathematical Society, Second Series, 1940, 46: 231–248, MR 0002137, doi:10.1112/plms/s2-46.1.231
Ayoub, Raymond G.; Ayoub, Christine, On the group ring of a finite abelian group, Bulletin of the Australian Mathematical Society, 1969, 1 (2): 245–261, MR 0252526, doi:10.1017/S0004972700041496

handle.net

hdl.handle.net

Baer, Reinhold, Abelian groups without elements of finite order, Duke Mathematical Journal, 1937, 3 (1): 68–122, MR 1545974, doi:10.1215/S0012-7094-37-00308-9, hdl:10338.dmlcz/100591

harvard.edu

ui.adsabs.harvard.edu

Bradley, David M., Counting the positive rationals: A brief survey, 2005, Bibcode:2005math......9025B, arXiv:math/0509025

jstor.org

Blass, Andreas, Injectivity, projectivity, and the axiom of choice, Transactions of the American Mathematical Society, 1979, 255: 31–59, JSTOR 1998165, MR 0542870, doi:10.1090/S0002-9947-1979-0542870-6  . 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 $P$ in this model that is a subgroup of a free abelian group ${\textstyle {\bigl (}\mathbb {Z} ^{(A)}{\bigr )}^{n}}$ , where $A$ is a set of atoms and $n$ 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.

semanticscholar.org

api.semanticscholar.org

As stated explicitly, for instance, in Hartley, Brian; Turull, Alexandre, On characters of coprime operator groups and the Glauberman character correspondence, Journal für die Reine und Angewandte Mathematik, 1994, 1994 (451): 175–219, MR 1277300, S2CID 118116330, doi:10.1515/crll.1994.451.175 , proof of Lemma 2.3: "the trivial group is the direct product of the empty family of groups"
van Glabbeek, Rob; Goltz, Ursula; Schicke-Uffmann, Jens-Wolfhard, On characterising distributability, Logical Methods in Computer Science, 2013, 9 (3): 3:17, 58, MR 3109601, S2CID 17046529, arXiv:1309.3883 , doi:10.2168/LMCS-9(3:17)2013
Bridson, Martin R.; Vogtmann, Karen, Automorphism groups of free groups, surface groups and free abelian groups, Farb, Benson (编), Problems on mapping class groups and related topics, Proceedings of Symposia in Pure Mathematics 74, Providence, Rhode Island: American Mathematical Society: 301–316, 2006, MR 2264548, S2CID 17710182, arXiv:math/0507612 , doi:10.1090/pspum/074/2264548
Tolstykh, Vladimir, What does the automorphism group of a free abelian group $A$ know about $A$ ?, Blass, Andreas; Zhang, Yi (编), Logic and its Applications, Contemporary Mathematics 380, Providence, Rhode Island: American Mathematical Society: 283–296, 2005, MR 2167584, S2CID 18107280, arXiv:math/0701752 , doi:10.1090/conm/380/07117

web.archive.org

Sims, Charles C., Section 8.1: Free abelian groups, Computation with Finitely Presented Groups, Encyclopedia of Mathematics and its Applications 48, Cambridge University Press: 320, 1994 [2023-09-01], ISBN 0-521-43213-8, MR 1267733, doi:10.1017/CBO9780511574702, （原始内容存档于2023-09-01）
Vick, James W., Homology Theory: An Introduction to Algebraic Topology, Graduate Texts in Mathematics 145, Springer: 4, 70, 1994 [2023-09-01], ISBN 9780387941264, （原始内容存档于2023-08-09）
Fuchs, László, Section 3.1: Freeness and projectivity, Abelian Groups, Springer Monographs in Mathematics, Cham: Springer: 75–80, 2015 [2023-09-01], ISBN 978-3-319-19421-9, MR 3467030, doi:10.1007/978-3-319-19422-6, （原始内容存档于2023-09-01）
Johnson, D. L., Symmetries, Springer undergraduate mathematics series, Springer: 71, 2001 [2023-09-01], ISBN 9781852332709, （原始内容存档于2023-09-02）