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Journal articles on the topic 'Mathematics ; Group theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 July 2024

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1

Wang, Mingshen. "Group Theory in Number Theory." Theoretical and Natural Science 5, no. 1 (2023): 9–13. http://dx.doi.org/10.54254/2753-8818/5/20230254.

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The theory of groups exists in many fields of mathematics and has made a great impact on many fields of mathematics. In this article, this paper first introduces the history of group theory and elementary number theory, and then lists the definitions of group, ring, field and the most basic prime and integer and divisor in number theory that need to be used in this article. Then from the definitions, step by step, Euler's theorem, Bzout's lemma, Wilson's theorem and Fermat's Little theorem in elementary number theory are proved by means of definitions of group theory, cyclic groups, and even p

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2

Stonehewer, Stewart. "MATHEMATICAL WORKS I: GROUP THEORY." Bulletin of the London Mathematical Society 28, no. 2 (1996): 219–20. http://dx.doi.org/10.1112/blms/28.2.219.

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3

Pride, Stephen J. "CONTRIBUTIONS TO GROUP THEORY (Contemporary Mathematics, 33)." Bulletin of the London Mathematical Society 17, no. 6 (1985): 610–12. http://dx.doi.org/10.1112/blms/17.6.610.

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4

Qiu, Shengqi. "Group theory behind Rubiks Cube." Theoretical and Natural Science 9, no. 1 (2023): 151–56. http://dx.doi.org/10.54254/2753-8818/9/20240732.

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The Rubiks Cube is a widely recognized puzzle. The mathematics behind the Rubiks Cube is group theory. Group theory studies algebraic structures in mathematics such as groups, rings, and fields. The operation of the Rubiks Cube is rotation, which can be considered an operation of a group. The combination of two rotations of the Rubiks Cube can be considered the association of two operations of a group. The rotations and the combination operation of two rotations form a group called the Rubiks Cube group, and this paper presents the order of this group which is also the quantity of possible val

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5

Gordon, Gary. "USING WALLPAPER GROUPS TO MOTIVATE GROUP THEORY." PRIMUS 6, no. 4 (1996): 355–65. http://dx.doi.org/10.1080/10511979608965838.

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6

Clay, Matt. "Geometric Group Theory." Notices of the American Mathematical Society 69, no. 10 (2022): 1. http://dx.doi.org/10.1090/noti2572.

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7

Knapp, A. W., Andrew Baker, and Wulf Rossmann. "Matrix Groups: An Introduction to Lie Group Theory." American Mathematical Monthly 110, no. 5 (2003): 446. http://dx.doi.org/10.2307/3647845.

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8

Alperin, J. L. "Book Review: Group theory." Bulletin of the American Mathematical Society 17, no. 2 (1987): 339–41. http://dx.doi.org/10.1090/s0273-0979-1987-15583-2.

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9

Huetinck, Linda. "Group Theory: It's a SNAP." Mathematics Teacher 89, no. 4 (1996): 342–46. http://dx.doi.org/10.5951/mt.89.4.0342.

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Martin Gardner has compared the concept of a mathematical group to the grin of the Cheshue Cat. “The body of the cat (algebra as traditionally taught) vanishes, leaving only an abstract grin. A grin implies something amusing. Perhaps we can make group theory less mysterious if we do not take it too seriously” (Gardner 1966). A game like “It's a SNAP” can be used to introduce mathematical groups and make them appear less mysterious. This mathematics manipulative allows students to play with the concepts of group theory and develop an understanding of modern algebra.

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10

SAPIR, MARK V. "SOME GROUP THEORY PROBLEMS." International Journal of Algebra and Computation 17, no. 05n06 (2007): 1189–214. http://dx.doi.org/10.1142/s0218196707003925.

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This is a survey of some problems in geometric group theory that I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For each problem, I try to mention the author. If the author is not given, the problem, to the best of my knowledge, was formulated by me first.

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11

Becker, Oren, Alexander Lubotzky, and Jonathan Mosheiff. "Testability in group theory." Israel Journal of Mathematics 256, no. 1 (2023): 61–90. http://dx.doi.org/10.1007/s11856-023-2503-y.

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AbstractThis paper is a journal counterpart to [5], in which we initiate the study of property testing problems concerning a finite system of relations E between permutations, generalizing the study of stability in permutations. To every such system E, a group Γ = ΓE is associated and the testability of E depends only on Γ (just like in Galois theory, where the solvability of a polynomial is determined by the solvability of the associated group). This leads to the notion of testable groups, and, more generally, Benjamini–Schramm rigid groups. The paper presents an ensemble of tools to check if

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12

Steingart, Alma. "A group theory of group theory: Collaborative mathematics and the ‘uninvention’ of a 1000-page proof." Social Studies of Science 42, no. 2 (2012): 185–213. http://dx.doi.org/10.1177/0306312712436547.

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13

Mrope, Fadhili Mustafa. "Exploring group concepts in abstract algebra through ChatGPT." Union: Jurnal Ilmiah Pendidikan Matematika 12, no. 2 (2024): 258–73. http://dx.doi.org/10.30738/union.v12i2.17156.

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Traditional methods often struggle to engage students and effectively communicate the intricacies of abstract algebra. This study aims to explore the future of mathematics teaching, with a focus on integrating group theory concepts into Abstract Algebra using ChatGPT, an advanced language processing AI. Chat transcripts edited with ChatGPT were analyzed to gain insights into the evolving landscape of mathematics education, specifically concerning group theory. Group theory is a fundamental branch of Abstract Algebra poised to shape the future of mathematics instruction. ChatGPT facilitates int

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14

Streater, R. F. "APPLICATIONS OF GROUP THEORY IN PHYSICS AND MATHEMATICAL PHYSICS (Lectures in Applied Mathematics 21)." Bulletin of the London Mathematical Society 19, no. 5 (1987): 500. http://dx.doi.org/10.1112/blms/19.5.500a.

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15

Senashov, V. I. "Applications of group theory in crystallography." IOP Conference Series: Materials Science and Engineering 1230, no. 1 (2022): 012018. http://dx.doi.org/10.1088/1757-899x/1230/1/012018.

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Abstract Group theory is a powerful tool for studying symmetric physical systems. Such systems include, in particular, molecules and crystals with symmetry. Group theory serves to explain the most important characteristics of atomic spectra. Group theory is also applied to the problems of atomic and nuclear physics. This paper gives examples of the use of the apparatus of group theory in research on crystallography, quantum mechanics, elementary particle physics. In particular, in these studies matrix groups and representations of unitary groups are actively used. For such groups we give an ov

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16

Makhnev, A. A. "School-conferences on group theory." Proceedings of the Steklov Institute of Mathematics 267, S1 (2009): 244–47. http://dx.doi.org/10.1134/s0081543809070220.

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17

Beardon, Alan F. "Complex Exponents and Group Theory." Mathematics Magazine 93, no. 3 (2020): 186–92. http://dx.doi.org/10.1080/0025570x.2020.1736876.

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18

Elliott, George A., Toshikazu Natsume, and Ryszard Nest. "The Heisenberg group andK-theory." K-Theory 7, no. 5 (1993): 409–28. http://dx.doi.org/10.1007/bf00961535.

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19

Salce, Luigi. "Abelian Group Theory in Italy." Rocky Mountain Journal of Mathematics 32, no. 4 (2002): 1229–43. http://dx.doi.org/10.1216/rmjm/1181070019.

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20

Allhebi, Bayan Khaled. "PROBABILISTIC STRATEGIES IN GROUP THEORY." Far East Journal of Mathematical Sciences (FJMS) 123, no. 2 (2020): 181–208. http://dx.doi.org/10.17654/ms123020181.

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21

Moore, Vardeman G. "Recursive functions in group theory." Illinois Journal of Mathematics 30, no. 2 (1986): 284–94. http://dx.doi.org/10.1215/ijm/1256044637.

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22

Humphreys, J. E. "Book Review: Group theory and physics." Bulletin of the American Mathematical Society 32, no. 4 (1995): 455–58. http://dx.doi.org/10.1090/s0273-0979-1995-00612-9.

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23

Murphy, G. J., and L. Tuset. "Aspects of compact quantum group theory." Proceedings of the American Mathematical Society 132, no. 10 (2004): 3055–67. http://dx.doi.org/10.1090/s0002-9939-04-07400-3.

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24

Artamonov, V. A., and A. A. Bovdi. "Integral group rings: Groups of units and classical K-theory." Journal of Soviet Mathematics 57, no. 2 (1991): 2931–58. http://dx.doi.org/10.1007/bf01099283.

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25

Wang, Xiao Gang. "Significance of Mathematization of Philosophical Problems from the Angle of Broadspectrum Philosophy." Advanced Materials Research 433-440 (January 2012): 6315–18. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.6315.

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Whether philosophy can realize mathematization has long been controversial. As the mathematics develops a nonquantative branch- structural mathematics, however, mathematization of philosophy has a turnaround. Broadspectrum philosophy which makes use of structural mathematics has established a generally applicable as well as precise mathematical model for many philosophical problems, giving a positive answer to whether the philosophy can be mathematized. Mathematizaiton of philosophy allows more accurate and clear distinction of people’s expression in meaning, gives ideas the visible characteri

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26

Eklof, Paul C. "The Affinity of Set Theory and Abelian Group Theory." Rocky Mountain Journal of Mathematics 32, no. 4 (2002): 1119–34. http://dx.doi.org/10.1216/rmjm/1181070012.

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27

Cui, Can, Chenqin Gan, Changwang Ren, and Zhangying Mo. "Lagrange’s Theorem in Group Theory." Journal of Physics: Conference Series 2381, no. 1 (2022): 012100. http://dx.doi.org/10.1088/1742-6596/2381/1/012100.

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Abstract The structure and behavior of molecules and crystals depend on their different symmetries. Thus, group theory is an essential technique in some fields of chemistry. Within mathematics itself, group theory is very closely linked to symmetry in geometry. Lagrange’s theorem is a statement in group theory that can be viewed as an extension of the number theoretical result of Euler’s theorem. It is seen as a significant lemma for proving more complicated results in group theory. The main intention of this dissertation is to prove Lagrange’s theorem which illustrates that every quadratic ir

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28

Zeng, Ziyu. "Some connections between galois group and free group." Journal of Physics: Conference Series 2634, no. 1 (2023): 012001. http://dx.doi.org/10.1088/1742-6596/2634/1/012001.

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Abstract Free Group and Galois Theory are both new contents of modern mathematics with being widely studied, while there are few studies about the connections between them. We can find that the automorphisms in the definition of Galois Group is like the symbols in free group. The structure of subgroups in free groups is a significant subject which is originated from the Group Theory. Thus, this paper will introduce some basic theorems of Galois Group, free group and give a proof of the set of every subfield of a field is isomorphic to a free Galois Group.

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29

Melhuish, Kathleen. "Three Conceptual Replication Studies in Group Theory." Journal for Research in Mathematics Education 49, no. 1 (2018): 9–38. http://dx.doi.org/10.5951/jresematheduc.49.1.0009.

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Many studies in mathematics education research occur with a nonrepresentative sample and are never replicated. To challenge this paradigm, I designed a large-scale study evaluating student conceptions in group theory that surveyed a national, representative sample of students. By replicating questions previously used to build theory around student understanding of subgroups, cyclic groups, and isomorphism with over 800 students, I establish the utility of replication studies to (a) validate previous results, (b) establish the prevalence of various student conceptions, and (c) reexamine theoret

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30

Alexanderson, Gerald L., and Brian Hayes. "Group Theory in the Bedroom." College Mathematics Journal 37, no. 4 (2006): 322. http://dx.doi.org/10.2307/27646368.

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31

Lynch, Mark A. M. "Generating quasigroups: a group theory investigation." International Journal of Mathematical Education in Science and Technology 42, no. 6 (2011): 806–12. http://dx.doi.org/10.1080/0020739x.2011.562318.

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32

Auslander, L., and R. Tolimieri. "Radar Ambiguity Functions and Group Theory." SIAM Journal on Mathematical Analysis 16, no. 3 (1985): 577–601. http://dx.doi.org/10.1137/0516043.

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33

Webb, David L. "G-theory of group rings for groups of square-free order." K-Theory 1, no. 4 (1987): 417–22. http://dx.doi.org/10.1007/bf00539626.

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34

Kuku, Aderemi O., and Guoping Tang. "Higher K -theory of group-rings of virtually infinite cyclic groups." Mathematische Annalen 325, no. 4 (2003): 711–26. http://dx.doi.org/10.1007/s00208-002-0397-2.

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35

Toledo, Domingo. "Book Review: Geometric group theory, Vol. 2: Asymptotic invariants of infinite groups." Bulletin of the American Mathematical Society 33, no. 03 (1996): 395–99. http://dx.doi.org/10.1090/s0273-0979-96-00669-6.

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36

Chen, Quan-Guo, Yang-Lei Pang, and Ding-Guo Wang. "Galois theory for comatrix group corings." Studia Scientiarum Mathematicarum Hungarica 55, no. 3 (2018): 281–92. http://dx.doi.org/10.1556/012.2018.55.3.1399.

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37

Garimella, Ramesh. "A Counter Example in Group Theory." Missouri Journal of Mathematical Sciences 3, no. 2 (1991): 77–78. http://dx.doi.org/10.35834/1991/0302077.

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38

Neumann, Peter M. "GROUP THEORY: essays for Philip Hall." Bulletin of the London Mathematical Society 18, no. 3 (1986): 311. http://dx.doi.org/10.1112/blms/18.3.311.

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39

Neumann, Peter M. "Two Combinatorial Problems in Group Theory." Bulletin of the London Mathematical Society 21, no. 5 (1989): 456–58. http://dx.doi.org/10.1112/blms/21.5.456.

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40

Dunwoody, M. J. "COMBINATORIAL GROUP THEORY: A TOPOLOGICAL APPROACH." Bulletin of the London Mathematical Society 24, no. 2 (1992): 191–92. http://dx.doi.org/10.1112/blms/24.2.191.

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41

Goppa, V. D. "Group representations and algebraic information theory." Izvestiya: Mathematics 59, no. 6 (1995): 1123–47. http://dx.doi.org/10.1070/im1995v059n06abeh000051.

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42

Ren, Yunxiang. "Universal skein theory for group actions." Advances in Mathematics 356 (November 2019): 106804. http://dx.doi.org/10.1016/j.aim.2019.106804.

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43

Lesh, Kathryn. "Infinite loop spaces from group theory." Mathematische Zeitschrift 225, no. 3 (1997): 467–83. http://dx.doi.org/10.1007/pl00004622.

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44

Léandre, Rémi. "Positivity theorem in semi-group theory." Mathematische Zeitschrift 258, no. 4 (2007): 893–914. http://dx.doi.org/10.1007/s00209-007-0204-6.

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45

Moretó, A. "Some problems in number theory that arise from group theory." Publicacions Matemàtiques EXTRA (June 1, 2007): 181–91. http://dx.doi.org/10.5565/publmat_pjtn05_09.

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46

Ralph, William J. "Category and group rings in homotopy theory." Transactions of the American Mathematical Society 299, no. 1 (1987): 205. http://dx.doi.org/10.1090/s0002-9947-1987-0869408-2.

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47

Brauer, R. "Book Review: The theory of group representations." Bulletin of the American Mathematical Society 37, no. 01 (1999): 100——100. http://dx.doi.org/10.1090/s0273-0979-99-00823-x.

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48

Satoh, Takao. "On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three." Proceedings of the Edinburgh Mathematical Society 64, no. 2 (2021): 338–63. http://dx.doi.org/10.1017/s0013091521000171.

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AbstractIn this paper, we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm {IA}_3$ of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of $\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm {IA}_

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49

Fomin, A. A. "To Quotient Divisible Group Theory. I." Journal of Mathematical Sciences 197, no. 5 (2014): 688–97. http://dx.doi.org/10.1007/s10958-014-1752-z.

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50

Martina, L. "Exotic Galileian group in field theory." Journal of Mathematical Sciences 136, no. 6 (2006): 4465–69. http://dx.doi.org/10.1007/s10958-006-0237-0.

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