Relevant bibliographies by topics / Group theory and generalizations – Permutation groups – General theory for finite groups

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Academic literature on the topic 'Group theory and generalizations – Permutation groups – General theory for finite groups'

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Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Group theory and generalizations – Permutation groups – General theory for finite groups"

1

Tornier, Stephan. "Prime localizations of Burger–Mozes-type groups." Journal of Group Theory 21, no. 2 (2018): 229–40. http://dx.doi.org/10.1515/jgth-2017-0036.

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AbstractThis article concerns Burger–Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primespwe determine their localp-Sylow subgroups as well as subgroups of theirp-localization, which is identified as a group of the same type in certain cases.
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2

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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3

Niemenmaa, Markku. "Decomposition of Transformation Groups of Permutation Machines." Fundamenta Informaticae 10, no. 4 (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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4

Lulek, B., T. Lulek, J. Biel, and R. Chatterjee. "Racah–Wigner approach to standardization of permutation representations for finite groups." Canadian Journal of Physics 63, no. 8 (1985): 1065–73. http://dx.doi.org/10.1139/p85-174.

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The problem of equivalence of permutation representations of the finite groups is discussed in terms of transforms by bijections of the carrier sets and by group automorphisms. A formal description of a transformation between equivalent representations is given, and a standard form for an arbitrary permutation representation is proposed. The standardization is achieved through the canonical realization of transitive representations and of imprimitivity sets on the left cosets of the group with respect to an appropriate stability subgroup. The purpose of this paper is to pave the way for a systematic formulation of permutation representations, analogous to the Racah algebra of angular momentum theory, which will be useful to multicentre problems of quantum mechanics and statistical physics.
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5

Li, Chi-Kwong, and Wayne Whitney. "Closed Symmetric Overgroups of Sn in On." Canadian Mathematical Bulletin 39, no. 1 (1996): 83–94. http://dx.doi.org/10.4153/cmb-1996-011-7.

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AbstractA norm on ℝn is said to be permutation invariant if its value is preserved under permutation of the coordinates of a vector. The isometry group of such a norm must be closed, contain Sn and —I, and be conjugate to a subgroup of On, the orthogonal group. Motivated by this, we are interested in classifying all closed groups G such that 〈—I,Sn〉 < G < On. We use the theory of Lie groups to classify all possible infinite groups G, and use the theory of finite reflection groups to classify all possible finite groups G. In keeping with the original motivation, all groups arising are shown to be isometry groups. This completes the work of Gordon and Lewis, who studied the same problem and obtained the results for n ≥ 13.
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6

BAMBERG, JOHN, TOMASZ POPIEL, and CHERYL E. PRAEGER. "SIMPLE GROUPS, PRODUCT ACTIONS, AND GENERALIZED QUADRANGLES." Nagoya Mathematical Journal 234 (September 14, 2017): 87–126. http://dx.doi.org/10.1017/nmj.2017.35.

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The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group$G$preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on$G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that$G$cannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.
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7

ALP, MURAT, and CHRISTOPHER D. WENSLEY. "ENUMERATION OF CAT1-GROUPS OF LOW ORDER." International Journal of Algebra and Computation 10, no. 04 (2000): 407–24. http://dx.doi.org/10.1142/s0218196700000170.

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In this paper we describe a share package [Formula: see text] of functions for computing with finite, permutation crossed modules, cat1-groups and their morphisms, written using the [Formula: see text] group theory programming language. The category XMod of crossed modules is equivalent to the category Cat1 of cat1-groups and we include functions emulating the functors between these categories. The monoid of derivations of a crossed module [Formula: see text] , and the corresponding monoid of sections of a cat1-group [Formula: see text] , are constructed using the Whitehead multiplication. The Whitehead group of invertible derivations, together with the group of automorphisms of [Formula: see text] , are used to construct the actor crossed module of [Formula: see text] which is the automorphism object in XMod. We include a table of the 350 isomorphism classes of cat1-structures on groups of order at most 30.
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8

Smith, Larry. "Modular Vector Invariants of Cyclic Permutation Representations." Canadian Mathematical Bulletin 42, no. 1 (1999): 125–28. http://dx.doi.org/10.4153/cmb-1999-014-5.

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AbstractVector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, e.g., [1], [2], [4], [7], [11] and [12]. It is therefore all the more surprising that the unpleasant properties of these invariants may be derived from two unexpected, and remarkable, nice properties: namely for vector permutation invariants of the cyclic group of prime order in characteristic p the image of the transfer homomorphism is a prime ideal, and the quotient algebra is a polynomial algebra on the top Chern classes of the action.
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9

Zaid, Nurhidayah, Nor Haniza Sarmin, and Sanhan Muhammad Salih Khasraw. "THE APPLICATIONS OF ZERO DIVISORS OF SOME FINITE RINGS OF MATRICES IN PROBABILITY AND GRAPH THEORY." Jurnal Teknologi 83, no. 1 (2020): 127–32. http://dx.doi.org/10.11113/jurnalteknologi.v83.14936.

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Let R be a finite ring. The zero divisors of R are defined as two nonzero elements of R, say x and y where xy = 0. Meanwhile, the probability that two random elements in a group commute is called the commutativity degree of the group. Some generalizations of this concept have been done on various groups, but not in rings. In this study, a variant of probability in rings which is the probability that two elements of a finite ring have product zero is determined for some ring of matrices over integers modulo n. The results are then applied into graph theory, specifically the zero divisor graph. This graph is defined as a graph where its vertices are zero divisors of R and two distinct vertices x and y are adjacent if and only if xy = 0. It is found that the zero divisor graph of R is a directed graph.
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10

DOOLEY, A. H., and V. YA GOLODETS. "The geometric dimension of an equivalence relation and finite extensions of countable groups." Ergodic Theory and Dynamical Systems 29, no. 6 (2009): 1789–814. http://dx.doi.org/10.1017/s014338570800093x.

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AbstractWe say that the geometric dimension of a countable group G is equal to n if any free Borel action of G on a standard Borel probability space (X,μ), induces an equivalence relation of geometric dimension n on (X,μ) in the sense of Gaboriau. Let ℬ be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let 𝒜 be the subset of ℬ consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in 𝒜. The geometric dimension of any such group K is one: we prove that also geom-dim(Gf(K))=1 for any finite extension Gf(K) of K, applying the results of Stallings on finite extensions of free product groups, together with the results of Gaboriau and others in orbit equivalence theory. Using results of Karrass, Pietrowski and Solitar we extend these results to finite extensions of free groups. We also give generalizations and applications of these results to groups with geometric dimension greater than one. We construct a family of finitely generated groups {Kn}n∈ℕ,n>1, such that geom-dim(Kn)=n and geom-dim(Gf(Kn))=n for any finite extension Gf(Kn) of Kn. In particular, this construction allows us to produce, for each integer n>1, a family of groups {K(s,n)}s∈ℕ of geometric dimension n, such that any finite extension of K(s,n) also has geometric dimension n, but the finite extensions Gf(K(s,n)) are non-isomorphic, if s≠s′.
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