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Zeitschriftenartikel zum Thema „Permutation groups“

Autor: Grafiati

Veröffentlicht am 4. Juni 2021

Zuletzt aktualisiert am 9. Juni 2025

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1

Niemenmaa, Markku. "Decomposition of Transformation Groups of Permutation Machines." Fundamenta Informaticae 10, no. 4 (October 1, 1987): 363–67. http://dx.doi.org/10.3233/fi-1987-10403.

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By a permutation machine we mean a triple (Q,S,F), where Q and S are finite sets and F is a function Q × S → Q which defines a permutation on Q for every element from S. These permutations generate a permutation group G and by considering the structure of G we can obtain efficient ways to decompose the transformation group (Q,G). In this paper we first consider the situation where G is half-transitive and after this we show how to use our result in the general non-transitive case.

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2

Burns, J. M., B. Goldsmith, B. Hartley, and R. Sandling. "On quasi-permutation representations of finite groups." Glasgow Mathematical Journal 36, no. 3 (September 1994): 301–8. http://dx.doi.org/10.1017/s0017089500030901.

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In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

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3

Bigelow, Stephen. "Supplements of bounded permutation groups." Journal of Symbolic Logic 63, no. 1 (March 1998): 89–102. http://dx.doi.org/10.2307/2586590.

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AbstractLet λ ≤ κ be infinite cardinals and let Ω be a set of cardinality κ. The bounded permutation group Bλ(Ω), or simply Bλ, is the group consisting of all permutations of Ω which move fewer than λ points in Ω. We say that a permutation group G acting on Ω is a supplement of Bλ if BλG is the full symmetric group on Ω.In [7], Macpherson and Neumann claimed to have classified all supplements of bounded permutation groups. Specifically, they claimed to have proved that a group G acting on the set Ω is a supplement of Bλ if and only if there exists Δ ⊂ Ω with ∣Δ∣ < λ such that the setwise stabiliser G{Δ} acts as the full symmetric group on Ω ∖ Δ. However I have found a mistake in their proof. The aim of this paper is to examine conditions under which Macpherson and Neumann's claim holds, as well as conditions under which a counterexample can be constructed. In the process we will discover surprising links with cardinal arithmetic and Shelah's recently developed pcf theory.

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4

Tovstyuk, K. D., C. C. Tovstyuk, and O. O. Danylevych. "The Permutation Group Theory and Electrons Interaction." International Journal of Modern Physics B 17, no. 21 (August 20, 2003): 3813–30. http://dx.doi.org/10.1142/s0217979203021812.

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The new mathematical formalism for the Green's functions of interacting electrons in crystals is constructed. It is based on the theory of Green's functions and permutation groups. We constructed a new object of permutation groups, which we call double permutation (DP). DP allows one to take into consideration the symmetry of the ground state as well as energy and momentum conservation in every virtual interaction. We developed the classification of double permutations and proved the theorem, which allows the selection of classes of associated double permutations (ADP). The Green's functions are constructed for series of ADP. We separate in the DP the convolving columns by replacing the initial interaction between the particles with the effective interaction. In convoluting the series for Green's functions, we use the methods developed for permutation groups schemes of Young–Yamanuti.

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5

Cohen, Stephen D. "Permutation polynomials and primitive permutation groups." Archiv der Mathematik 57, no. 5 (November 1991): 417–23. http://dx.doi.org/10.1007/bf01246737.

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6

Senashov, Vasily S., Konstantin A. Filippov, and Anatoly K. Shlepkin. "Regular permutations and their applications in crystallography." E3S Web of Conferences 525 (2024): 04002. http://dx.doi.org/10.1051/e3sconf/202452504002.

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The representation of a group G in the form of regular permutations is widely used for studying the structure of finite groups, in particular, parameters like the group density function. This is related to the increased potential of computer technologies for conducting calculations. The work addresses the problem of calculation regular permutations with restrictions on the structure of the degree and order of permutations. The considered regular permutations have the same nontrivial order, which divides the degree of the permutation. Examples of the application of permutation groups in crystallography and crystal chemistry are provided.

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7

Boy de la Tour, Thierry, and Mnacho Echenim. "On leaf permutative theories and occurrence permutation groups." Electronic Notes in Theoretical Computer Science 86, no. 1 (May 2003): 61–75. http://dx.doi.org/10.1016/s1571-0661(04)80653-4.

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8

Cameron, Peter J. "Cofinitary Permutation Groups." Bulletin of the London Mathematical Society 28, no. 2 (March 1996): 113–40. http://dx.doi.org/10.1112/blms/28.2.113.

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9

Lucchini, A., F. Menegazzo, and M. Morigi. "Generating Permutation Groups." Communications in Algebra 32, no. 5 (December 31, 2004): 1729–46. http://dx.doi.org/10.1081/agb-120029899.

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10

Kearnes, Keith A. "Collapsing permutation groups." Algebra Universalis 45, no. 1 (February 1, 2001): 35–51. http://dx.doi.org/10.1007/s000120050200.

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11

Gu, Yutong. "Introduction of Several Special Groups and Their Applications to Rubik’s Cube." Highlights in Science, Engineering and Technology 47 (May 11, 2023): 172–75. http://dx.doi.org/10.54097/hset.v47i.8186.

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Group theory is the subject that aims to study the symmetries and structures of groups in mathematics. This work provides an introduction to group theory and explores some potential applications of group theory on complex geometric objects like the Rubik's cube. To this end, the concepts of symmetric group, permutation group, and cyclic group are introduced, and the famous Lagrange’s theorem and Cayley’s theorem are mentioned briefly. The former theorem establishes that a subgroup’s order must be a divisor of the parent group’s order. Concerning the permutation group, it is a set of permutations that form a group under composition. Hence, the various groups that can be formed by the Rubik's cube are discussed, including the group of all possible permutations of the cube's stickers, and the subgroups that are generated through permutations of the six basic movements embedded in Rubik’s cube. Overall, this essay provides an accessible introduction to group theory and its applications to the popular Rubik's cube.

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12

Sousa, Renata Teófilo de, Francisco Régis Vieira Alves, and Ana Paula Florêncio Aires. "Rubik’s cube and GeoGebra: a visual exploration of permutation groups." Revista do Instituto GeoGebra Internacional de São Paulo 13, no. 3 (November 18, 2024): 027–44. http://dx.doi.org/10.23925/2237-9657.2024.v13i3p027-044.

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This article aims to use the construction of the Rubik’s Cube in GeoGebra as a primary tool for visualization and manipulation in the teaching of permutation groups in Abstract Algebra. We bring a brief discussion about the concept of group, aspects of the Rubik's cube, the Rubik’s group as a group of permutations and possibilities for its exploration in GeoGebra. Based on this study, we recognize the potential to delve into permutation groups in Abstract Algebra through a visual interface that associates their properties with a tangible and manipulable object. Additionally, there is the potential for simulating their movements using Dynamic Geometry software, such as GeoGebra.

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13

Tombotamunoa, W. J. Lawson, and ItoroUbom Udo-Akpan. "EFFICIENT COMPUTATION OF THE M-CLOSURE FOR SOLVABLE PERMUTATION GROUPS." GPH - International Journal of Mathematics 07, no. 03 (March 17, 2024): 11–18. https://doi.org/10.5281/zenodo.10827540.

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<strong>This research presents a novel approach to efficiently compute the m-closure of solvable permutation groups of degree n. The m-closure is an essential concept in group theory, particularly in understanding the structure and properties of permutation groups. We propose an algorithm that constructs the m-closure with a time complexity of n<sup>O</sup><sup>(m)</sup>, significantly improving the computational efficiency compared to existing methods. Through rigorous mathematical analysis and computational experiments, we demonstrate the effectiveness and scalability of our approach.</strong>

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14

Guralnick, Robert M., and David Perkinson. "Permutation polytopes and indecomposable elements in permutation groups." Journal of Combinatorial Theory, Series A 113, no. 7 (October 2006): 1243–56. http://dx.doi.org/10.1016/j.jcta.2005.11.004.

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15

Khashaev, Arthur A. "On the membership problem for finite automata over symmetric groups." Discrete Mathematics and Applications 32, no. 6 (December 1, 2022): 383–89. http://dx.doi.org/10.1515/dma-2022-0033.

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Abstract We consider automata in which transitions are labelled with arbitrary permutations. The language of such an automaton consists of compositions of permutations for all possible admissible computation paths. The membership problem for finite automata over symmetric groups is the following decision problem: does a given permutation belong to the language of a given automaton? We show that this problem is NP-complete. We also propose an efficient algorithm for the case of strongly connected automata.

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16

Liebeck, Martin W., and Aner Shalev. "Simple groups, permutation groups, and probability." Journal of the American Mathematical Society 12, no. 2 (1999): 497–520. http://dx.doi.org/10.1090/s0894-0347-99-00288-x.

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17

Chalapathi, T., and R. V. "Graphs of Permutation Groups." International Journal of Computer Applications 179, no. 3 (December 15, 2017): 14–19. http://dx.doi.org/10.5120/ijca2017915872.

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18

Burness, Timothy C., and Emily V. Hall. "Almost elusive permutation groups." Journal of Algebra 594 (March 2022): 519–43. http://dx.doi.org/10.1016/j.jalgebra.2021.11.037.

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19

Lucchini, Andrea, Marta Morigi, and Mariapia Moscatiello. "Primitive permutation IBIS groups." Journal of Combinatorial Theory, Series A 184 (November 2021): 105516. http://dx.doi.org/10.1016/j.jcta.2021.105516.

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20

Conway, John H., Alexander Hulpke, and John McKay. "On Transitive Permutation Groups." LMS Journal of Computation and Mathematics 1 (1998): 1–8. http://dx.doi.org/10.1112/s1461157000000115.

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21

Gerner, M. "Predicate-Induced Permutation Groups." Journal of Semantics 29, no. 1 (October 13, 2011): 109–44. http://dx.doi.org/10.1093/jos/ffr007.

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22

Adeleke, S. A., and Peter M. Neumann. "Infinite Bounded Permutation Groups." Journal of the London Mathematical Society 53, no. 2 (April 1996): 230–42. http://dx.doi.org/10.1112/jlms/53.2.230.

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23

KOVÁCS, L. G., and M. F. NEWMAN. "GENERATING TRANSITIVE PERMUTATION GROUPS." Quarterly Journal of Mathematics 39, no. 3 (1988): 361–72. http://dx.doi.org/10.1093/qmath/39.3.361.

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24

Neumann, Peter M. "Some Primitive Permutation Groups." Proceedings of the London Mathematical Society s3-50, no. 2 (March 1985): 265–81. http://dx.doi.org/10.1112/plms/s3-50.2.265.

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25

Atkinson, M. D. "Permutation Involvement and Groups." Quarterly Journal of Mathematics 52, no. 4 (December 1, 2001): 415–21. http://dx.doi.org/10.1093/qjmath/52.4.415.

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26

Cossey, John. "Quotients of permutation groups." Bulletin of the Australian Mathematical Society 57, no. 3 (June 1998): 493–95. http://dx.doi.org/10.1017/s0004972700031907.

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If G is a finite permutation group of degree d and N is a normal subgroup of G, Derek Holt has given conditions which show that in some important special cases the least degree of a faithful permutation representation of the quotient G/N will be no larger than d. His conditions do not apply in all cases of interest and he remarks that it would be interesting to know if G/F(G) has a faithful representation of degree no larger than d (where F(G) is the Fitting subgroup of G). We prove in this note that this is the case.

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27

Cigler, Grega. "Permutation-like matrix groups." Linear Algebra and its Applications 422, no. 2-3 (April 2007): 486–505. http://dx.doi.org/10.1016/j.laa.2006.11.007.

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28

Mazurov, V. D. "2-Transitive permutation groups." Siberian Mathematical Journal 31, no. 4 (1991): 615–17. http://dx.doi.org/10.1007/bf00970632.

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29

Galvin, Fred. "Almost disjoint permutation groups." Proceedings of the American Mathematical Society 124, no. 6 (1996): 1723–25. http://dx.doi.org/10.1090/s0002-9939-96-03264-9.

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30

Hulpke, Alexander. "Constructing transitive permutation groups." Journal of Symbolic Computation 39, no. 1 (January 2005): 1–30. http://dx.doi.org/10.1016/j.jsc.2004.08.002.

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31

Cameron, Peter J. "Cycle-closed permutation groups." Journal of Algebraic Combinatorics 5, no. 4 (October 1996): 315–22. http://dx.doi.org/10.1007/bf00193181.

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32

Grech, Mariusz. "Graphical cyclic permutation groups." Discrete Mathematics 337 (December 2014): 25–33. http://dx.doi.org/10.1016/j.disc.2014.08.006.

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33

Bichon, Julien. "ALGEBRAIC QUANTUM PERMUTATION GROUPS." Asian-European Journal of Mathematics 01, no. 01 (March 2008): 1–13. http://dx.doi.org/10.1142/s1793557108000023.

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We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If 𝕂 is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra 𝕂n: this is a refinement of Wang's universality theorem for the (compact) quantum permutation group. We also prove a structural result for Hopf algebras having a non-ergodic coaction on the diagonal algebra 𝕂n, on which we determine the possible group gradings when 𝕂 is algebraically closed and has characteristic zero.

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34

Neumann, Peter M., and Cheryl E. Praeger. "Three-star permutation groups." Illinois Journal of Mathematics 47, no. 1-2 (March 2003): 445–52. http://dx.doi.org/10.1215/ijm/1258488164.

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35

Kuzucuoğlu, M. "Barely transitive permutation groups." Archiv der Mathematik 55, no. 6 (December 1990): 521–32. http://dx.doi.org/10.1007/bf01191686.

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36

Li, Jiongsheng. "TheL-sharp permutation groups." Science in China Series A: Mathematics 43, no. 1 (January 2000): 22–27. http://dx.doi.org/10.1007/bf02903844.

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37

Anagnostopoulou-Merkouri, Marina, Peter J. Cameron, and Enoch Suleiman. "Pre-primitive permutation groups." Journal of Algebra 636 (December 2023): 695–715. http://dx.doi.org/10.1016/j.jalgebra.2023.09.012.

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38

Franchi, Clara. "Abelian sharp permutation groups." Journal of Algebra 283, no. 1 (January 2005): 1–5. http://dx.doi.org/10.1016/j.jalgebra.2004.06.031.

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39

Pearson, Mike, and Ian Short. "Magic letter groups." Mathematical Gazette 91, no. 522 (November 2007): 493–99. http://dx.doi.org/10.1017/s0025557200182130.

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Certain numeric puzzles, known as ‘magic letters’, each have a finite permutation group associated with them in a natural manner. We describe how the isomorphism type of these permutation groups relates to the structure of the magic letters.

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40

Vesanen, Ari. "Finite classical groups and multiplication groups of loops." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 3 (May 1995): 425–29. http://dx.doi.org/10.1017/s0305004100073278.

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Let Q be a loop; then the left and right translations La(x) = ax and Ra(x) = xa are permutations of Q. The permutation group M(Q) = 〈La, Ra | a ε Q〉 is called the multiplication group of Q; it is well known that the structure of M(Q) reflects strongly the structure of Q (cf. [1] and [8], for example). It is thus an interesting question, which groups can be represented as multiplication groups of loops. In particular, it seems important to classify the finite simple groups that are multiplication groups of loops. In [3] it was proved that the alternating groups An are multiplication groups of loops, whenever n ≥ 6; in this paper we consider the finite classical groups and prove the following theorems

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41

Khalil, Shuker, Manal Al-Labadi, Enoch Suleiman, and Namuma Yerima. "New structure of algebras using permutations in symmetric groups." Journal of Discrete Mathematical Sciences and Cryptography 27, no. 5 (2024): 1611–18. http://dx.doi.org/10.47974/jdmsc-2003.

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The permutation BG-algebras were first introduced as a novel kind of algebra. In this work, their basic qualities were investigated to better understand how they relate to one another and how they might be combined to construct various sorts of superstructures. We investigated permutation BG-algebras, permutation group-derived, permutation BGsubalgebra, and permutation BG-normal. Furthermore, we explored certain novel concepts in permutation theory for the initial time. We additionally looked into BG-algebra isomorphism theorems, BG-algebra homomorphism, quotient permutation BG-algebras, and equivalence relations.

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42

Grech, Mariusz, and Andrzej Kisielewicz. "Cyclic Permutation Groups that are Automorphism Groups of Graphs." Graphs and Combinatorics 35, no. 6 (September 13, 2019): 1405–32. http://dx.doi.org/10.1007/s00373-019-02096-1.

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Abstract In this paper we establish conditions for a permutation group generated by a single permutation to be an automorphism group of a graph. This solves the so called concrete version of König’s problem for the case of cyclic groups. We establish also similar conditions for the symmetry groups of other related structures: digraphs, supergraphs, and boolean functions.

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43

Burov, Dmitry A. "Subgroups of direct products of groups invariant under the action of permutations on factors." Discrete Mathematics and Applications 30, no. 4 (August 26, 2020): 243–55. http://dx.doi.org/10.1515/dma-2020-0021.

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AbstractWe study subgroups of the direct product of two groups invariant under the action of permutations on factors. An invariance criterion for the subdirect product of two groups under the action of permutations on factors is put forward. Under certain additional constraints on permutations, we describe the subgroups of the direct product of a finite number of groups that are invariant under the action of permutations on factors. We describe the subgroups of the additive group of vector space over a finite field of characteristic 2 which are invariant under the coordinatewise action of inversion permutation of nonzero elements of the field.

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44

Gayathri, K. S. and M. Rajeshwari∗ Mahalakshmi. "Group Actions of wreath product −→Sn on permutation groups." Scandinavian Journal of Information Systems 35, no. 1 (April 7, 2023): 733–48. https://doi.org/10.5281/zenodo.7807470.

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Wreath product −→Sn = Z2 o Sn of type Bn is a subgroup of the symmetric group S2n. In this paper we determine the group actions of wreath product −→Sn on a finite set. Stabilizer and orbit of a point, imprimitivity of −→Snarealsodiscussed. Also the same is illustrated with an application for n = 3

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45

Bowler, Nathan, and Thomas Forster. "Normal subgroups of infinite symmetric groups, with an application to stratified set theory." Journal of Symbolic Logic 74, no. 1 (March 2009): 17–26. http://dx.doi.org/10.2178/jsl/1231082300.

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It is generally known that infinite symmetric groups have few nontrivial normal subgroups (typically only the subgroups of bounded support) and none of small index. (We will explain later exactly what we mean by small). However the standard analysis relies heavily on the axiom of choice. By dint of a lot of combinatorics we have been able to dispense—largely—with the axiom of choice. Largely, but not entirely: our result is that if X is an infinite set with ∣X∣ = ∣X × X∣ then Symm(X) has no nontrivial normal subgroups of small index. Some condition like this is needed because of the work of Sam Tarzi who showed [4] that, for any finite group G, there is a model of ZF without AC in which there is a set X with Symm(X)/FSymm(X) isomorphic to G.The proof proceeds in two stages. We consider a particularly useful class of permutations, which we call the class of flexible permutations. A permutation of X is flexible if it fixes at least ∣X∣-many points. First we show that every normal subgroup of Symm(X) (of small index) must contain every flexible permutation. This will be theorem 4. Then we show (theorem 7) that the flexible permutations generate Symm(X).

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46

Gill, Nick, and Pablo Spiga. "Binary permutation groups: Alternating and classical groups." American Journal of Mathematics 142, no. 1 (2020): 1–43. http://dx.doi.org/10.1353/ajm.2020.0000.

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47

Banica, Teodor, Julien Bichon, and Sonia Natale. "Finite quantum groups and quantum permutation groups." Advances in Mathematics 229, no. 6 (April 2012): 3320–38. http://dx.doi.org/10.1016/j.aim.2012.02.012.

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48

BRYANT, R. M., L. G. KOVÁCS, and G. R. ROBINSON. "TRANSITIVE PERMUTATION GROUPS AND IRREDUCIBLE LINEAR GROUPS." Quarterly Journal of Mathematics 46, no. 4 (1995): 385–407. http://dx.doi.org/10.1093/qmath/46.4.385.

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49

Heath-Brown, D. R., Cheryl E. Praeger, and Aner Shalev. "Permutation groups, simple groups, and sieve methods." Israel Journal of Mathematics 148, no. 1 (December 2005): 347–75. http://dx.doi.org/10.1007/bf02775443.

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50

Praeger, Cheryl E. "Seminormal and subnormal subgroup lattices for transitive permutation groups." Journal of the Australian Mathematical Society 80, no. 1 (February 2006): 45–64. http://dx.doi.org/10.1017/s144678870001137x.

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AbstractVarious lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.

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