Group (mathematics) (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Group (mathematics)" in English language version.

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Some authors include an additional axiom referred to as the closure under the operation " $\cdot$ ", which means that $a \cdot b$ is an element of $G$ for every $a$ and $b$ in $G$ . This condition is subsumed by requiring " $\cdot$ " to be a binary operation on $G$ . See Lang 2002. 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
The MathSciNet database of mathematics publications lists 1,779 research papers on group theory and its generalizations written in 2020 alone. See MathSciNet 2021. MathSciNet (2021), List of papers reviewed on MathSciNet on "Group theory and its generalizations" (MSC code 20), published in 2020, retrieved 14 May 2021
See, for example, Lang 2002, Lang 2005, Herstein 1996 and Herstein 1975. 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 Lang, Serge (2005), Undergraduate Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3. Herstein, Israel Nathan (1996), Abstract Algebra (3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc., ISBN 978-0-13-374562-7, MR 1375019. Herstein, Israel Nathan (1975), Topics in Algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR 0356988.
An example is group cohomology of a group which equals the singular cohomology of its classifying space, see Weibel 1994, §8.2. Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4, MR 1269324, OCLC 36131259
Elements which do have multiplicative inverses are called units, see Lang 2002, p. 84, §II.1. 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
For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See Lang 2002, Theorem IV.1.9. The notions of torsion of a module and simple algebras are other instances of this principle. 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
More precisely, the monodromy action on the vector space of solutions of the differential equations is considered. See Kuga 1993, pp. 105–113. Kuga, Michio (1993), Galois' Dream: Group Theory and Differential Equations, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3688-3, MR 1199112.
Up to isomorphism, there are about 49 billion groups of order up to 2000. See Besche, Eick & O'Brien 2001. Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001), "The groups of order at most 2000", Electronic Research Announcements of the American Mathematical Society, 7: 1–4, doi:10.1090/S1079-6762-01-00087-7, MR 1826989.
Herstein 1975, p. 26, §2. Herstein, Israel Nathan (1975), Topics in Algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR 0356988.
Hall 1967, p. 1, §1.1: "The idea of a group is one which pervades the whole of mathematics both pure and applied." Hall, G. G. (1967), Applied Group Theory, American Elsevier Publishing Co., Inc., New York, MR 0219593, an elementary introduction.
Lang 2002, p. 3, I.§1 and p. 7, I.§2. 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
Herstein 1975, p. 54, §2.6. Herstein, Israel Nathan (1975), Topics in Algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR 0356988.
Kleiner 1986. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
Kleiner 1986, p. 202. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
Lie 1973. Lie, Sophus (1973), Gesammelte Abhandlungen. Band 1 [Collected papers. Volume 1] (in German), New York: Johnson Reprint Corp., MR 0392459.
Kleiner 1986, p. 204. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
Mackey 1976. Mackey, George Whitelaw (1976), The Theory of Unitary Group Representations, University of Chicago Press, MR 0396826
Ledermann 1953, pp. 4–5, §1.2. Ledermann, Walter (1953), Introduction to the Theory of Finite Groups, Oliver and Boyd, Edinburgh and London, MR 0054593.
Lang 2002, p. 7, §I.2. 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
Lang 2002, p. 12, §I.2. 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
Lang 2002, p. 9, §I.2. 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
Coornaert, Delzant & Papadopoulos 1990. Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes [Geometry and Group Theory], Lecture Notes in Mathematics (in French), vol. 1441, Berlin, New York: Springer-Verlag, ISBN 978-3-540-52977-4, MR 1075994.
For example, class groups and Picard groups; see Neukirch 1999, in particular §§I.12 and I.13 Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021
Seress 1997. Seress, Ákos (1997), "An Introduction to Computational Group Theory" (PDF), Notices of the American Mathematical Society, 44 (6): 671–679, MR 1452069.
Rosen 2000, p. 54, (Theorem 2.1). Rosen, Kenneth H. (2000), Elementary Number Theory and its Applications (4th ed.), Addison-Wesley, ISBN 978-0-201-87073-2, MR 1739433.
Lang 2002, pp. 26, 29, §I.5. 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
Ellis 2019. Ellis, Graham (2019), "6.4 Triangle groups", An Invitation to Computational Homotopy, Oxford University Press, pp. 441–444, doi:10.1093/oso/9780198832973.001.0001, ISBN 978-0-19-883298-0, MR 3971587.
Conway et al. 2001. See also Bishop 1993 Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie, 42 (2): 475–507, arXiv:math.MG/9911185, MR 1865535. Bishop, David H. L. (1993), Group Theory and Chemistry, New York: Dover Publications, ISBN 978-0-486-67355-4.
Mumford, Fogarty & Kirwan 1994. Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric Invariant Theory, vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56963-3, MR 1304906.
Kuipers 1999. Kuipers, Jack B. (1999), Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality, Princeton University Press, Bibcode:1999qrsp.book.....K, ISBN 978-0-691-05872-6, MR 1670862.
Fulton & Harris 1991. Fulton, William; Harris, Joe (1991), Representation Theory: A First Course, Graduate Texts in Mathematics, Readings in Mathematics, vol. 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249
Serre 1977. Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90190-9, MR 0450380.
Lang 2002, Chapter VI (see in particular p. 273 for concrete examples). 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
Lang 2002, p. 292, (Theorem VI.7.2). 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
Kurzweil & Stellmacher 2004, p. 3. Kurzweil, Hans; Stellmacher, Bernd (2004), The Theory of Finite Groups, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-40510-0, MR 2014408.
Artin 2018, Proposition 6.4.3. See also Lang 2002, p. 77 for similar results. 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
Neukirch 1999. Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021
Shatz 1972. Shatz, Stephen S. (1972), Profinite Groups, Arithmetic, and Geometry, Princeton University Press, ISBN 978-0-691-08017-8, MR 0347778
Borel 1991. Borel, Armand (1991), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012.
Naber 2003. Naber, Gregory L. (2003), The Geometry of Minkowski Spacetime, New York: Dover Publications, ISBN 978-0-486-43235-9, MR 2044239.
Dudek 2001. Dudek, Wiesław A. (2001), "On some old and new problems in $n$ -ary groups" (PDF), Quasigroups and Related Systems, 8: 15–36, MR 1876783.

ams.org

This was crucial to the classification of finite simple groups, for example. See Aschbacher 2004. Aschbacher, Michael (2004), "The status of the classification of the finite simple groups" (PDF), Notices of the American Mathematical Society, 51 (7): 736–740.
Up to isomorphism, there are about 49 billion groups of order up to 2000. See Besche, Eick & O'Brien 2001. Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001), "The groups of order at most 2000", Electronic Research Announcements of the American Mathematical Society, 7: 1–4, doi:10.1090/S1079-6762-01-00087-7, MR 1826989.
Seress 1997. Seress, Ákos (1997), "An Introduction to Computational Group Theory" (PDF), Notices of the American Mathematical Society, 44 (6): 671–679, MR 1452069.
Aschbacher 2004, p. 737. Aschbacher, Michael (2004), "The status of the classification of the finite simple groups" (PDF), Notices of the American Mathematical Society, 51 (7): 736–740.

archive.org

More precisely, the monodromy action on the vector space of solutions of the differential equations is considered. See Kuga 1993, pp. 105–113. Kuga, Michio (1993), Galois' Dream: Group Theory and Differential Equations, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3688-3, MR 1199112.
Jordan 1870. Jordan, Camille (1870), Traité des substitutions et des équations algébriques [Study of Substitutions and Algebraic Equations] (in French), Paris: Gauthier-Villars.
Serre 1977. Serre, Jean-Pierre (1977), Linear Representations of Finite Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90190-9, MR 0450380.
Milne 1980. Milne, James S. (1980), Étale Cohomology, Princeton University Press, ISBN 978-0-691-08238-7
Weinberg 1972. Weinberg, Steven (1972), Gravitation and Cosmology, New York: John Wiley & Sons, ISBN 0-471-92567-5.

arxiv.org

Conway et al. 2001. See also Bishop 1993 Conway, John Horton; Delgado Friedrichs, Olaf; Huson, Daniel H.; Thurston, William P. (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie, 42 (2): 475–507, arXiv:math.MG/9911185, MR 1865535. Bishop, David H. L. (1993), Group Theory and Chemistry, New York: Dover Publications, ISBN 978-0-486-67355-4.

books.google.com

Cook 2009, p. 24. Cook, Mariana R. (2009), Mathematicians: An Outer View of the Inner World, Princeton, N.J.: Princeton University Press, ISBN 978-0-691-13951-7
Magnus, Karrass & Solitar 2004, pp. 56–67, §1.6. Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (2004) [1966], Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations, Courier, ISBN 978-0-486-43830-6

cornell.edu

pi.math.cornell.edu

Hatcher 2002, p. 30, Chapter I. Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1.

doi.org

However, a group is not determined by its lattice of subgroups. See Suzuki 1951. Suzuki, Michio (1951), "On the lattice of subgroups of finite groups", Transactions of the American Mathematical Society, 70 (2): 345–371, doi:10.2307/1990375, JSTOR 1990375.
Up to isomorphism, there are about 49 billion groups of order up to 2000. See Besche, Eick & O'Brien 2001. Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001), "The groups of order at most 2000", Electronic Research Announcements of the American Mathematical Society, 7: 1–4, doi:10.1090/S1079-6762-01-00087-7, MR 1826989.
Kleiner 1986. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
Kleiner 1986, p. 202. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
Kleiner 1986, p. 204. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
von Dyck 1882. von Dyck, Walther (1882), "Gruppentheoretische Studien (Group-theoretical studies)", Mathematische Annalen (in German), 20 (1): 1–44, doi:10.1007/BF01443322, S2CID 179178038, archived from the original on 2014-02-22.
Solomon 2018. Solomon, Ronald (2018), "The classification of finite simple groups: A progress report", Notices of the AMS, 65 (6): 1, doi:10.1090/noti1689
Ellis 2019. Ellis, Graham (2019), "6.4 Triangle groups", An Invitation to Computational Homotopy, Oxford University Press, pp. 441–444, doi:10.1093/oso/9780198832973.001.0001, ISBN 978-0-19-883298-0, MR 3971587.

gutenberg.org

Smith 1906. Smith, David Eugene (1906), History of Modern Mathematics, Mathematical Monographs, No. 1.

harvard.edu

ui.adsabs.harvard.edu

Kuipers 1999. Kuipers, Jack B. (1999), Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality, Princeton University Press, Bibcode:1999qrsp.book.....K, ISBN 978-0-691-05872-6, MR 1670862.

idsi.md

ibn.idsi.md

Dudek 2001. Dudek, Wiesław A. (2001), "On some old and new problems in $n$ -ary groups" (PDF), Quasigroups and Related Systems, 8: 15–36, MR 1876783.

jstor.org

However, a group is not determined by its lattice of subgroups. See Suzuki 1951. Suzuki, Michio (1951), "On the lattice of subgroups of finite groups", Transactions of the American Mathematical Society, 70 (2): 345–371, doi:10.2307/1990375, JSTOR 1990375.
Kleiner 1986. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
Kleiner 1986, p. 202. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.
Kleiner 1986, p. 204. Kleiner, Israel (1986), "The evolution of group theory: A brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, JSTOR 2690312, MR 0863090.

numdam.org

More rigorously, every group is the symmetry group of some graph; see Frucht's theorem, Frucht 1939. Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe [Construction of graphs with prescribed group]", Compositio Mathematica (in German), 6: 239–50, archived from the original on 2008-12-01.

semanticscholar.org

api.semanticscholar.org

von Dyck 1882. von Dyck, Walther (1882), "Gruppentheoretische Studien (Group-theoretical studies)", Mathematische Annalen (in German), 20 (1): 1–44, doi:10.1007/BF01443322, S2CID 179178038, archived from the original on 2014-02-22.

umich.edu

quod.lib.umich.edu

Galois 1908. Galois, Évariste (1908), Tannery, Jules (ed.), Manuscrits de Évariste Galois [Évariste Galois' Manuscripts] (in French), Paris: Gauthier-Villars (Galois work was first published by Joseph Liouville in 1843).

hti.umich.edu

Cayley 1889. Cayley, Arthur (1889), The Collected Mathematical Papers of Arthur Cayley, vol. II (1851–1860), Cambridge University Press.

uni-goettingen.de

gdz.sub.uni-goettingen.de

von Dyck 1882. von Dyck, Walther (1882), "Gruppentheoretische Studien (Group-theoretical studies)", Mathematische Annalen (in German), 20 (1): 1–44, doi:10.1007/BF01443322, S2CID 179178038, archived from the original on 2014-02-22.

web.archive.org

More rigorously, every group is the symmetry group of some graph; see Frucht's theorem, Frucht 1939. Frucht, R. (1939), "Herstellung von Graphen mit vorgegebener abstrakter Gruppe [Construction of graphs with prescribed group]", Compositio Mathematica (in German), 6: 239–50, archived from the original on 2008-12-01.
von Dyck 1882. von Dyck, Walther (1882), "Gruppentheoretische Studien (Group-theoretical studies)", Mathematische Annalen (in German), 20 (1): 1–44, doi:10.1007/BF01443322, S2CID 179178038, archived from the original on 2014-02-22.

wiktionary.org

en.wiktionary.org

The word homomorphism derives from Greek ὁμός—the same and μορφή—structure. See Schwartzman 1994, p. 108. Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Mathematical Association of America, ISBN 978-0-88385-511-9.

worldcat.org

search.worldcat.org

An example is group cohomology of a group which equals the singular cohomology of its classifying space, see Weibel 1994, §8.2. Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, ISBN 978-0-521-55987-4, MR 1269324, OCLC 36131259
Ledermann 1973, p. 3, §I.1. Ledermann, Walter (1973), Introduction to Group Theory, New York: Barnes and Noble, OCLC 795613.
Ledermann 1973, p. 39, §II.12. Ledermann, Walter (1973), Introduction to Group Theory, New York: Barnes and Noble, OCLC 795613.
Zee 2010, p. 228. Zee, A. (2010), Quantum Field Theory in a Nutshell (second ed.), Princeton, N.J.: Princeton University Press, ISBN 978-0-691-14034-6, OCLC 768477138
Zee 2010. Zee, A. (2010), Quantum Field Theory in a Nutshell (second ed.), Princeton, N.J.: Princeton University Press, ISBN 978-0-691-14034-6, OCLC 768477138

zbmath.org

For example, class groups and Picard groups; see Neukirch 1999, in particular §§I.12 and I.13 Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021
Neukirch 1999. Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859, Zbl 0956.11021