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Journal articles on the topic 'Permutation groups. Finite groups'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

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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2

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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3

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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4

., Haci Aktas. "On Finite Topological Permutation Groups." Journal of Applied Sciences 2, no. 1 (December 15, 2001): 60–61. http://dx.doi.org/10.3923/jas.2002.60.61.

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5

Lucchini, Andrea. "Enumerating Transitive Finite Permutation Groups." Bulletin of the London Mathematical Society 30, no. 6 (November 1998): 569–77. http://dx.doi.org/10.1112/s0024609398004846.

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6

HOARE, A. H. M. "SUBGROUPS OF N.E.C. GROUPS AND FINITE PERMUTATION GROUPS." Quarterly Journal of Mathematics 41, no. 1 (1990): 45–59. http://dx.doi.org/10.1093/qmath/41.1.45.

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7

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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8

NEUMANN, PETER M., CHERYL E. PRAEGER, and SIMON M. SMITH. "SOME INFINITE PERMUTATION GROUPS AND RELATED FINITE LINEAR GROUPS." Journal of the Australian Mathematical Society 102, no. 1 (October 25, 2016): 136–49. http://dx.doi.org/10.1017/s1446788716000343.

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This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

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9

Praeger, Cheryl E., and Aner Shalev. "Bounds on finite quasiprimitive permutation groups." Journal of the Australian Mathematical Society 71, no. 2 (October 2001): 243–58. http://dx.doi.org/10.1017/s1446788700002895.

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AbstractA permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p-elements. We also pose some problems for further research.

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10

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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11

Xu, Jing. "On closures of finite permutation groups." Bulletin of the Australian Mathematical Society 74, no. 1 (January 2006): 153–54. http://dx.doi.org/10.1017/s0004972700047547.

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12

Praeger, Cheryl E., and Jan Saxl. "Closures of Finite Primitive Permutation Groups." Bulletin of the London Mathematical Society 24, no. 3 (May 1992): 251–58. http://dx.doi.org/10.1112/blms/24.3.251.

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13

Liebeck, Martin W., and Cheryl E. Praeger. "Relation Algebras and Finite Permutation Groups." Journal of the London Mathematical Society s2-45, no. 3 (June 1992): 433–45. http://dx.doi.org/10.1112/jlms/s2-45.3.433.

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14

Kovács, L. G. "Wreath decompositions of finite permutation groups." Bulletin of the Australian Mathematical Society 40, no. 2 (October 1989): 255–79. http://dx.doi.org/10.1017/s0004972700004366.

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There is a familiar construction with two finite, transitive permutation groups as input and a finite, transitive permutation group, called their wreath product, as output. The corresponding ‘imprimitive wreath decomposition’ concept is the first subject of this paper. A formal definition is adopted and an overview obtained for all such decompositions of any given finite, transitive group. The result may be heuristically expressed as follows, exploiting the associative nature of the construction. Each finite transitive permutation group may be written, essentially uniquely, as the wreath product of a sequence of wreath-indecomposable groups, amid the two-factor wreath decompositions of the group are precisely those which one obtains by bracketing this many-factor decomposition.If both input groups are nontrivial, the output above is always imprimitive. A similar construction gives a primitive output, called the wreath product in product action, provided the first input group is primitive and not regular. The second subject of the paper is the ‘product action wreath decomposition’ concept dual to this. An analogue of the result stated above is established for primitive groups with nonabelian socle.Given a primitive subgroup G with non-regular socle in some symmetric group S, how many subgroups W of S which contain G and have the same socle, are wreath products in product action? The third part of the paper outlines an algorithm which reduces this count to questions about permutation groups whose degrees are very much smaller than that of G.

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15

Qian, Guohua, and Yong Yang. "Permutation characters in finite solvable groups." Communications in Algebra 46, no. 1 (April 17, 2017): 167–75. http://dx.doi.org/10.1080/00927872.2017.1316856.

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16

Franchi, Clara. "On Permutation Groups of Finite Type." European Journal of Combinatorics 22, no. 6 (August 2001): 821–37. http://dx.doi.org/10.1006/eujc.2001.0508.

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17

Hickin, Kenneth. "Relatively homogeneous locally finite permutation groups." Mathematische Zeitschrift 194, no. 4 (December 1987): 495–504. http://dx.doi.org/10.1007/bf01161918.

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18

Hambleton, Ian, and Laurence R. Taylor. "Rational permutation modules for finite groups." Mathematische Zeitschrift 231, no. 4 (August 1999): 707–26. http://dx.doi.org/10.1007/pl00004749.

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19

CHAMBERLAIN, ROBERT. "MINIMAL EXCEPTIONAL -GROUPS." Bulletin of the Australian Mathematical Society 98, no. 3 (August 1, 2018): 434–38. http://dx.doi.org/10.1017/s0004972718000576.

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For a finite group $G$, denote by $\unicode[STIX]{x1D707}(G)$ the degree of a minimal permutation representation of $G$. We call $G$ exceptional if there is a normal subgroup $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$. To complete the work of Easdown and Praeger [‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc.38(2) (1988), 207–220], for all primes $p\geq 3$, we describe an exceptional group of order $p^{5}$ and prove that no exceptional group of order $p^{4}$ exists.

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20

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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21

PARIS, LUIS. "IRREDUCIBLE COXETER GROUPS." International Journal of Algebra and Computation 17, no. 03 (May 2007): 427–47. http://dx.doi.org/10.1142/s0218196707003779.

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We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that an indefinite irreducible Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. Let W be a Coxeter group. Write W = WX1 × ⋯ × WXb × WZ3, where WX1, … , WXb are non-spherical irreducible Coxeter groups and WZ3 is a finite one. By a classical result, known as the Krull–Remak–Schmidt theorem, the group WZ3 has a decomposition WZ3 = H1 × ⋯ × Hq as a direct product of indecomposable groups, which is unique up to a central automorphism and a permutation of the factors. Now, W = WX1 × ⋯ × WXb × H1 × ⋯ × Hq is a decomposition of W as a direct product of indecomposable subgroups. We prove that such a decomposition is unique up to a central automorphism and a permutation of the factors. Write W = WX1 × ⋯ × WXa × WZ2 × WZ3, where WX1, … , WXa are indefinite irreducible Coxeter groups, WZ2 is an affine Coxeter group whose irreducible components are all infinite, and WZ3 is a finite Coxeter group. The group WZ2 contains a finite index subgroup R isomorphic to ℤd, where d = |Z2| - b + a and b - a is the number of irreducible components of WZ2. Choose d copies R1, … , Rd of ℤ such that R = R1 × ⋯ × Rd. Then G = WX1 × ⋯ × WXa × R1 × ⋯ × Rd is a virtual decomposition of W as a direct product of strongly indecomposable subgroups. We prove that such a virtual decomposition is unique up to commensurability and a permutation of the factors.

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22

Kovács, László, and Cheryl Praeger. "Finite permutation groups with large abelian quotients." Pacific Journal of Mathematics 136, no. 2 (February 1, 1989): 283–92. http://dx.doi.org/10.2140/pjm.1989.136.283.

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23

Yoshizawa, M. "On locally finite k-homogeneous permutation groups." Archiv der Mathematik 79, no. 1 (July 2002): 1–4. http://dx.doi.org/10.1007/s00013-002-8277-2.

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24

Jabara, Enrico, and Pablo Spiga. "Abelian Carter subgroups in finite permutation groups." Archiv der Mathematik 101, no. 4 (September 11, 2013): 301–7. http://dx.doi.org/10.1007/s00013-013-0558-4.

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25

Macpherson, Dugald, and Anand Pillay. "Primitive Permutation Groups of Finite Morley Rank." Proceedings of the London Mathematical Society s3-70, no. 3 (May 1995): 481–504. http://dx.doi.org/10.1112/plms/s3-70.3.481.

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26

Stafford, Richard M. "Groups of Permutation Polynomials over Finite Fields." Finite Fields and Their Applications 4, no. 4 (October 1998): 450–52. http://dx.doi.org/10.1006/ffta.1998.0224.

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27

Cliff, Gerald, and Alfred Weiss. "Summands of permutation lattices for finite groups." Proceedings of the American Mathematical Society 110, no. 1 (January 1, 1990): 17. http://dx.doi.org/10.1090/s0002-9939-1990-1027091-5.

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28

FUMA, MICHITAKU, and YASUSHI NINOMIYA. "FINITE GROUPS WITH MULTIPLICITY-FREE PERMUTATION CHARACTERS." Journal of Algebra and Its Applications 04, no. 02 (April 2005): 187–94. http://dx.doi.org/10.1142/s021949880500106x.

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Let G be a finite group and H a subgroup of G. The Hecke algebra ℋ(G,H) associated with G and H is defined by the endomorphism algebra End ℂ[G]((ℂH)G), where ℂH is the trivial ℂ[H]-module and (ℂH)G = ℂH⊗ℂ[H] ℂ[G]. As is well known, ℋ(G,H) is a semisimple ℂ-algebra and it is commutative if and only if (ℂH)G is multiplicity-free. In [6], by a ring theoretic method, it is shown that if the canonical involution of ℋ(G,H) is the identity then ℋ(G,H) is commutative and, if there exists an abelian subgroup A of G such that G = AH then ℋ(G,H) is commutative. In this paper, by a character theoretic method, we consider the commutativity of ℋ(G,H).

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29

Yoshizawa, Mitsuo. "On multiply transitive locally finite permutation groups." Archiv der Mathematik 53, no. 5 (November 1989): 417–23. http://dx.doi.org/10.1007/bf01324716.

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30

Goren, Avi. "A measuring argument for finite permutation groups." Israel Journal of Mathematics 145, no. 1 (December 2005): 333–39. http://dx.doi.org/10.1007/bf02786698.

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31

Katsura, Takeshi. "Permutation presentations of modules over finite groups." Journal of Algebra 319, no. 9 (May 2008): 3653–65. http://dx.doi.org/10.1016/j.jalgebra.2008.01.023.

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32

Eick, Bettina, and Alexander Hulpke. "Computing Hall subgroups of finite groups." LMS Journal of Computation and Mathematics 15 (August 1, 2012): 205–18. http://dx.doi.org/10.1112/s1461157012001039.

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AbstractWe describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.

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33

EVANS, DAVID M. "SUBORBITS IN INFINITE PRIMITIVE PERMUTATION GROUPS." Bulletin of the London Mathematical Society 33, no. 5 (September 2001): 583–90. http://dx.doi.org/10.1112/s002460930100830x.

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34

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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35

JONES, GARETH A. "PRIMITIVE PERMUTATION GROUPS CONTAINING A CYCLE." Bulletin of the Australian Mathematical Society 89, no. 1 (July 18, 2013): 159–65. http://dx.doi.org/10.1017/s000497271300049x.

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AbstractThe primitive finite permutation groups containing a cycle are classified. Of these, only the alternating and symmetric groups contain a cycle fixing at least three points. This removes a primality condition from a classical theorem of Jordan. Some applications to monodromy groups are given, and the contributions of Jordan and Marggraff to this topic are briefly discussed.

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36

Kornyak, Vladimir. "Modeling Quantum Behavior in the Framework of Permutation Groups." EPJ Web of Conferences 173 (2018): 01007. http://dx.doi.org/10.1051/epjconf/201817301007.

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Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups.

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37

M.A., Pudovkina. "LINEAR STRUCTURES OF PERMUTATION GROUPS OVER FINITE MODULES." Prikladnaya diskretnaya matematika, no. 1 (June 1, 2008): 25–28. http://dx.doi.org/10.17223/20710410/1/5.

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38

Kovács, L. G., and Cheryl E. Praeger. "On minimal faithful permutation representations of finite groups." Bulletin of the Australian Mathematical Society 62, no. 2 (October 2000): 311–17. http://dx.doi.org/10.1017/s0004972700018797.

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The minimal faithful permutation degree μ(G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group Sn. Let N be a normal subgroup of a finite group G. We prove that μ(G/N) ≤ μ(G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate the structure of a (hypothetical) minimal counterexample to this conjecture.

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39

Easdown, David, and Cheryl E. Praeger. "On minimal faithful permutation representations of finite groups." Bulletin of the Australian Mathematical Society 38, no. 2 (October 1988): 207–20. http://dx.doi.org/10.1017/s0004972700027489.

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The minimal (faithful) degree μ(G) of a finite group G is the least positive integer n such that G ≲ Sn. Clearly if H ≤ G then μ(H) ≤ μ(G). However if N ◃ G then it is possible for μ(G/N) to be greater than μ(G); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.

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40

Korableva, V. V. "Primitive parabolic permutation representations for finite symplectic groups." Algebra and Logic 49, no. 3 (July 2010): 246–55. http://dx.doi.org/10.1007/s10469-010-9093-6.

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41

Praeger, Cheryl E. "Quotients and inclusions of finite quasiprimitive permutation groups." Journal of Algebra 269, no. 1 (November 2003): 329–46. http://dx.doi.org/10.1016/s0021-8693(03)00400-9.

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42

Liebeck, Martin W., and Jan Saxl. "The Finite Primitive Permutation Groups of Rank Three." Bulletin of the London Mathematical Society 18, no. 2 (March 1986): 165–72. http://dx.doi.org/10.1112/blms/18.2.165.

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43

Praeger, Cheryl E. "The Inclusion Problem for Finite Primitive Permutation Groups." Proceedings of the London Mathematical Society s3-60, no. 1 (January 1990): 68–88. http://dx.doi.org/10.1112/plms/s3-60.1.68.

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44

Liebeck, Martin W., Cheryl E. Praeger, and Jan Saxl. "On the 2-Closures of Finite Permutation Groups." Journal of the London Mathematical Society s2-37, no. 2 (April 1988): 241–52. http://dx.doi.org/10.1112/jlms/s2-37.2.241.

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45

Förster, P., and L. G. Kovács. "A Problem of Wielandt on Finite Permutation Groups." Journal of the London Mathematical Society s2-41, no. 2 (April 1990): 231–43. http://dx.doi.org/10.1112/jlms/s2-41.2.231.

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46

Pan, Jiangmin, and Cai Heng Li. "Finite permutation groups containing a transitive abelian subgroup." Communications in Algebra 46, no. 4 (September 2017): 1576–85. http://dx.doi.org/10.1080/00927872.2017.1350696.

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47

Kornyak, V. V. "Quantum mechanics and permutation invariants of finite groups." Journal of Physics: Conference Series 442 (June 10, 2013): 012050. http://dx.doi.org/10.1088/1742-6596/442/1/012050.

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48

Goren, Avi, and Marcel Herzog. "A general measuring argument for finite permutation groups." Proceedings of the American Mathematical Society 137, no. 10 (October 1, 2009): 3197. http://dx.doi.org/10.1090/s0002-9939-09-09993-6.

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49

Vasil'ev, V. A., and V. D. Mazurov. "Minimal permutation representations of finite simple orthogonal groups." Algebra and Logic 33, no. 6 (1995): 337–50. http://dx.doi.org/10.1007/bf00756348.

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50

Ito, Keiji. "Characters of finite permutation groups and Krein parameters." Journal of Algebra 514 (November 2018): 372–83. http://dx.doi.org/10.1016/j.jalgebra.2018.07.039.

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