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Journal articles on the topic 'General theory for finite groups'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

Hansen, Vagn Lundsgaard, and Peter Petersen. "Groups, Coverings and Galois Theory." Canadian Journal of Mathematics 43, no. 6 (1991): 1281–93. http://dx.doi.org/10.4153/cjm-1991-073-0.

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AbstractFinite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

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2

Hanaki, Akihide, Masahiko Miyamoto, and Daisuke Tambara. "Quantum Galois theory for finite groups." Duke Mathematical Journal 97, no. 3 (1999): 541–44. http://dx.doi.org/10.1215/s0012-7094-99-09720-x.

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3

Levi, Ran. "On finite groups and homotopy theory." Memoirs of the American Mathematical Society 118, no. 567 (1996): 0. http://dx.doi.org/10.1090/memo/0567.

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4

Goyal, Pallav. "Invariant theory of finite general linear groups modulo Frobenius powers." Communications in Algebra 46, no. 10 (2018): 4511–29. http://dx.doi.org/10.1080/00927872.2018.1448842.

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5

Meierfrankenfeld, U., G. Parmeggiani, and B. Stellmacher. "Baumann-components of finite groups of characteristic p, general theory." Journal of Algebra 515 (December 2018): 19–51. http://dx.doi.org/10.1016/j.jalgebra.2018.08.014.

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6

Hall, J. I. "The general theory of 3-transposition groups." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 2 (1993): 269–94. http://dx.doi.org/10.1017/s0305004100071589.

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A set D of 3-transpositions in the group G is a normal set of elements of order 2 such that, for all d and e in D, the order of the product de is 1, 2, or 3. If G is generated by the conjugacy class D of 3-transpositions, we say that (G, D) is a 3-transposition group or (loosely) that G is a 3-transposition group. The study of 3-transposition groups was instituted by Bernd Fischer [6, 7, 8] who classified all 3-transposition groups which are finite and have no non-trivial normal solvable subgroups. Recently the present author and H. Cuypers[5] extended Fischer's result to include all 3-transposition groups with trivial centre. For this classification the present paper provides the extension of Fischer's paper [8] where he gave two basic reductions, the Normal Subgroup Theorem and the Transitivity Theorem stated below. Other results of help in the classification are also presented here.

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7

Lubotzky, Alexander, and Avinoam Mann. "Residually finite groups of finite rank." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 3 (1989): 385–88. http://dx.doi.org/10.1017/s0305004100068110.

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The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:Theorem 1. A residually finite group of finite rank is virtually locally soluble.

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8

Isaacs, I. M. "Book Review: Character theory of finite groups." Bulletin of the American Mathematical Society 36, no. 04 (1999): 489–93. http://dx.doi.org/10.1090/s0273-0979-99-00789-2.

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9

Holt, Derek F. "Book Review: Theory of finite simple groups." Bulletin of the American Mathematical Society 46, no. 1 (2008): 151–56. http://dx.doi.org/10.1090/s0273-0979-08-01215-9.

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10

Bruner, R. R., та J. P. C. Greenlees. "The connective 𝐾-theory of finite groups". Memoirs of the American Mathematical Society 165, № 785 (2003): 0. http://dx.doi.org/10.1090/memo/0785.

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11

Howard, Eric. "Theory of groups and symmetries: Finite groups, Lie groups and Lie algebras." Contemporary Physics 60, no. 3 (2019): 275. http://dx.doi.org/10.1080/00107514.2019.1663933.

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12

Strunkov, S. P. "On the theory of equations in finite groups." Izvestiya: Mathematics 59, no. 6 (1995): 1273–82. http://dx.doi.org/10.1070/im1995v059n06abeh000057.

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13

Borovik, A. V. "Sylow theory for groups of finite Morley rank." Siberian Mathematical Journal 30, no. 6 (1990): 873–77. http://dx.doi.org/10.1007/bf00970907.

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14

Isaacs, I. M. "The π-character theory of solvable groups". Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 57, № 1 (1994): 81–102. http://dx.doi.org/10.1017/s1446788700036077.

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AbstractThere is a deeper structure to the ordinary character theory of finite solvable groups than might at first be apparent. Mauch of this structure, which has no analog for general finite gruops, becomes visible onyl when the character of solvable groups are viewes from the persepective of a particular set π of prime numbers. This purely expository paper discusses the foundations of this πtheory and a few of its applications. Included are the definitions and essential properties of Gajendragadkar's π-special characters and their connections with the irreducible πpartial characters and their associated Fong characters. Included among the consequences of the theory discussed here are applications to questions about the field generated by the values of a character, about extensions of characters of subgroups and about M-groups.

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15

Koberda, Thomas, and Alexander I. Suciu. "Residually finite rationally p groups." Communications in Contemporary Mathematics 22, no. 03 (2019): 1950016. http://dx.doi.org/10.1142/s0219199719500160.

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In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.

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16

Waterhouse, William C. "Two generators for the general linear groups over finite fields." Linear and Multilinear Algebra 24, no. 4 (1989): 227–30. http://dx.doi.org/10.1080/03081088908817916.

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17

Tulunay, I. "Restriction of a cuspidal module for finite general linear groups." Journal of Algebra 273, no. 1 (2004): 60–87. http://dx.doi.org/10.1016/j.jalgebra.2003.02.005.

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18

Crisp, Tyrone, Ehud Meir, and Uri Onn. "Principal series for general linear groups over finite commutative rings." Communications in Algebra 49, no. 11 (2021): 4857–68. http://dx.doi.org/10.1080/00927872.2021.1931264.

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19

Ceccherini-Silberstein, T., F. Scarabotti, and F. Tolli. "Representation theory of wreath products of finite groups." Journal of Mathematical Sciences 156, no. 1 (2008): 44–55. http://dx.doi.org/10.1007/s10958-008-9256-3.

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20

Kazarin, Lev. "Factorizations of Finite Groups and Related Topics." Algebra Colloquium 27, no. 01 (2020): 149–80. http://dx.doi.org/10.1142/s1005386720000139.

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This is a survey on the recent progress in the theory of finite groups with factorizations and around it, done by the author and his co-authors, and this has no pretensions to cover all topics in this wide area of research. In particular, we only touch the great consequences of the fundamental paper of Liebeck, Praeger and Saxl on maximal factorizations of almost simple finite groups for the theory of groups with factorizations. In each case the reader can find additional references at the end of Section 1. Some of the methods of investigation can be used to obtain information about finite groups in general, nilpotent algebras and related nearrings.

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21

Durakov, B. E., and A. I. Sozutov. "On Periodic Groups Saturated with Finite Frobenius Groups." Bulletin of Irkutsk State University. Series Mathematics 35 (2021): 73–86. http://dx.doi.org/10.26516/1997-7670.2021.35.73.

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A group is called weakly conjugate biprimitively finite if each its element of prime order generates a finite subgroup with any of its conjugate elements. A binary finite group is a periodic group in which any two elements generate a finite subgroup. If $\mathfrak{X}$ is some set of finite groups, then the group $G$ saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in a subgroup of $G$, isomorphic to some group from $\mathfrak{X}$. A group $G = F \leftthreetimes H$ is a Frobenius group with kernel $F$ and a complement $H$ if $H \cap H^f = 1$ for all $f \in F^{\#}$ and each element from $G \setminus F$ belongs to a one conjugated to $H$ subgroup of $G$. In the paper we prove that a saturated with finite Frobenius groups periodic weakly conjugate biprimitive finite group with a nontrivial locally finite radical is a Frobenius group. A number of properties of such groups and their quotient groups by a locally finite radical are found. A similar result was obtained for binary finite groups with the indicated conditions. Examples of periodic non locally finite groups with the properties above are given, and a number of questions on combinatorial group theory are raised.

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22

Berrick, A. J., and C. F. Miller. "Strongly torsion generated groups." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 2 (1992): 219–29. http://dx.doi.org/10.1017/s0305004100075319.

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It has long been known that the integral homology of a non-trivial finite group must be non-zero in infinitely many dimensions 17. Recent work on the Sullivan conjecture in homotopy theory has made it possible to extend this result to locally finite groups. For more general groups with torsion it becomes more difficult to make such a strong statement. Nevertheless we prove that when a non-perfect group is generated by torsion elements its integral homology must also be non-zero in infinitely many dimensions. Remarkably, this result is best possible, in that for perfect torsion generated groups all (finite or infinite) sequences of abelian groups are shown below to be attainable as higher homology groups.

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23

Glasby, Stephen Peter. "Computational approaches to the theory of finite soluble groups." Bulletin of the Australian Mathematical Society 38, no. 1 (1988): 157–58. http://dx.doi.org/10.1017/s0004972700027386.

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24

Fleischmann, Peter. "The Noether Bound in Invariant Theory of Finite Groups." Advances in Mathematics 156, no. 1 (2000): 23–32. http://dx.doi.org/10.1006/aima.2000.1952.

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25

Smith, Larry. "On the invariant theory of finite pseudo reflection groups." Archiv der Mathematik 44, no. 3 (1985): 225–28. http://dx.doi.org/10.1007/bf01237854.

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26

Guo, Wenbin, and Evgeny P. Vdovin. "Number of Sylow subgroups in finite groups." Journal of Group Theory 21, no. 4 (2018): 695–712. http://dx.doi.org/10.1515/jgth-2018-0010.

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AbstractDenote by {\nu_{p}(G)} the number of Sylow p-subgroups of G. It is not difficult to see that {\nu_{p}(H)\leqslant\nu_{p}(G)} for {H\leqslant G}, however {\nu_{p}(H)} does not divide {\nu_{p}(G)} in general. In this paper we reduce the question whether {\nu_{p}(H)} divides {\nu_{p}(G)} for every {H\leqslant G} to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof of Navarro’s theorem.

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27

Miller, G. A. "Book Review: Theory of groups of a finite order." Bulletin of the American Mathematical Society 37, no. 01 (1999): 80——80. http://dx.doi.org/10.1090/s0273-0979-99-00807-1.

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28

Quinn, Frank. "Algebraic $K$-theory of poly-(finite or cyclic) groups." Bulletin of the American Mathematical Society 12, no. 2 (1985): 221–27. http://dx.doi.org/10.1090/s0273-0979-1985-15353-4.

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29

Horan, Katherine, and Peter Fleischmann. "The finite unipotent groups consisting of bireflections." Journal of Group Theory 22, no. 2 (2019): 191–230. http://dx.doi.org/10.1515/jgth-2018-0123.

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Abstract Let k be a field of characteristic p and V a finite-dimensional k-vector space. An element {g\in{\rm GL}(V)} is called a bireflection if it centralizes a subspace of codimension less than or equal to 2. It is known by a result of Kemper that if for a finite p-group {G\leq{\rm GL}(V)} the ring of invariants {{\rm Sym}(V^{*})^{G}} is Cohen–Macaulay, G is generated by bireflections. Although the converse is false in general, it holds in special cases e.g. for particular families of groups consisting of bireflections. In this paper we give, for {p>2} , a classification of all finite unipotent subgroups of {{\rm GL}(V)} consisting of bireflections. Our description of the groups is given explicitly in terms useful for exploring the corresponding rings of invariants. This further analysis will be the topic of a forthcoming paper.

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30

Østvær, Paul Arne. "K-groups with finite coefficients and arithmetic." MATHEMATICA SCANDINAVICA 93, no. 1 (2003): 41. http://dx.doi.org/10.7146/math.scand.a-14412.

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In this paper we prove rank formulas for the even K-groups of number rings and relate Leopoldt's conjecture to K-theory. These results follow from a computation of the higher K-groups with finite coefficients.

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31

Fuchs, L., and K. M. Rangaswamy. "Valuated Butler groups of finite rank." Journal of the Australian Mathematical Society 80, no. 3 (2006): 335–50. http://dx.doi.org/10.1017/s144678870001404x.

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AbstractValuated Butler groups of finite rank are investigated. The valuated B2-groups are both epic images and pure subgroups of completely decomposable valuated groups of finite rank (Theorem 3.1). However, there are fundamental changes in the theory of Butler groups when valuations are involved. We introduce valuated B1-groups and show that they are valuated B2-groups. Surprisingly, valuated B2-groups of rank greater than 1 need not be valuated B1 -groups, unless they carry a special kind valuation, see Theorem 7.1. Theorem 6.5 gives a full characterization of valuated B1 -groups of finite rank, generalizing Bican's characterization of Butler groups.

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32

Juan-Pineda, D., J. F. Lafont, S. Millan-Vossler, and S. Pallekonda. "Algebraic K-theory of virtually free groups." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 6 (2011): 1295–316. http://dx.doi.org/10.1017/s0308210510000417.

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We provide a general procedure for computing the algebraic K-theory of finitely generated virtually free groups. The procedure describes these groups in terms of the algebraic K-theory of various finite subgroups and various Farrell Nil groups. We illustrate this process by carrying out the computation for several interesting classes of examples. The first two classes serve as a check on the method and show that our algorithm recovers results that already exist in the literature. The last two classes of examples yield new computations.

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33

Curtis, Charles W. "Representation theory of finite groups: From frobenius to brauer." Mathematical Intelligencer 14, no. 4 (1992): 48–57. http://dx.doi.org/10.1007/bf03024474.

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34

Yagodovsky, P. V. "Duality in the theory of finite commutative multivalued groups." Journal of Mathematical Sciences 174, no. 1 (2011): 97–119. http://dx.doi.org/10.1007/s10958-011-0284-z.

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35

Shokuev, V. N. "The foundations of enumeration theory for finite nilpotent groups." Journal of Mathematical Sciences 83, no. 5 (1997): 673–79. http://dx.doi.org/10.1007/bf02434858.

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36

Broer, Abraham. "Invariant Theory of Abelian Transvection Groups." Canadian Mathematical Bulletin 53, no. 3 (2010): 404–11. http://dx.doi.org/10.4153/cmb-2010-044-6.

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AbstractLet G be a finite group acting linearly on the vector space V over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants, and the direct summand property holds if there is a surjective k[V]G-linear map π : k[V] → k[V]G.The following Chevalley–Shephard–Todd type theorem is proved. Suppose G is abelian. Then the action is coregular if and only if G is generated by pseudo-reflections and the direct summand property holds.

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37

Groves, J. R. J., and G. C. Smith. "Soluble groups with a finite rewriting system." Proceedings of the Edinburgh Mathematical Society 36, no. 2 (1993): 283–88. http://dx.doi.org/10.1017/s0013091500018381.

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We describe a class of soluble groups with a finite complete rewriting system which includes all the soluble groups known to have such a system. It is an open question, related to deep questions in the theory of groups, whether it includes all soluble groups with such a system.

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38

Giordano Bruno, Anna, and Flavio Salizzoni. "Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups." Journal of Group Theory 23, no. 5 (2020): 831–46. http://dx.doi.org/10.1515/jgth-2019-0096.

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AbstractAdditivity with respect to exact sequences is, notoriously, a fundamental property of the algebraic entropy of group endomorphisms. It was proved for abelian groups by using the structure theorems for such groups in an essential way. On the other hand, a solvable counterexample was recently found, showing that it does not hold in general. Nevertheless, we give a rather short proof of the additivity of algebraic entropy for locally finite groups that are either quasihamiltonian or FC-groups.

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39

Bonnafé, C., G. I. Lehrer, and J. Michel. "Twisted Invariant Theory for Reflection Groups." Nagoya Mathematical Journal 182 (June 2006): 135–70. http://dx.doi.org/10.1017/s0027763000026854.

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AbstractLet G be a finite reflection group acting in a complex vector space V = ℂr, whose coordinate ring will be denoted by S. Any element γ ∈ GL(V) which normalises G acts on the ring SG of G-invariants. We attach invariants of the coset Gγ to this action, and show that if G′ is a parabolic subgroup of G, also normalised by γ, the invariants attaching to G′γ are essentially the same as those of Gγ. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of G and G′ and secondly, we give a general criterion for an element of Gγ to be regular (in Springer’s sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of G which are themselves reflection groups.

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40

Kepka, T., and M. Niemenmaa. "On conjugacy classes in finite loops." Bulletin of the Australian Mathematical Society 38, no. 2 (1988): 171–76. http://dx.doi.org/10.1017/s000497270002743x.

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The rôle of the conjugacy relation is certainly important in the structure theory of groups. Here we study this relation in a considerably more general setting, namely in the theory of loops. We first recall some basic facts about quasigroups, their multiplication groups, their inner mapping groups and the conjugacy relation. After this we estimate the size and the number of the conjugacy classes and we study the structure of loops having only two conjugacy classes. Finally, the values of the centraliser function are discussed.

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41

Kessar, Radha. "Shintani descent and perfect isometries for blocks of finite general linear groups." Journal of Algebra 276, no. 2 (2004): 493–501. http://dx.doi.org/10.1016/j.jalgebra.2004.02.013.

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42

Evans, Anthony B. "On Elementary Abelian Cartesian Groups." Canadian Mathematical Bulletin 34, no. 1 (1991): 58–59. http://dx.doi.org/10.4153/cmb-1991-009-3.

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AbstractJ. Hayden [2] proved that, if a finite abelian group is a Cartesian group satisfying a certain "homogeneity condition", then it must be an elementary abelian group. His proof required the character theory of finite abelian groups. In this note we present a shorter, elementary proof of his result.

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43

Vasil'ev, A. F. "A problem in the theory of formations of finite groups." Mathematical Notes 62, no. 1 (1997): 44–49. http://dx.doi.org/10.1007/bf02356062.

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44

Uribe-Vargas, Ricardo. "Topology of dynamical systems in finite groups and number theory." Bulletin des Sciences Mathématiques 130, no. 5 (2006): 377–402. http://dx.doi.org/10.1016/j.bulsci.2005.07.003.

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45

Smith, Larry. "E. Noether's bound in the invariant theory of finite groups." Archiv der Mathematik 66, no. 2 (1996): 89–92. http://dx.doi.org/10.1007/bf01273338.

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46

Pushkarev, I. A. "On the representation theory of wreath products of finite groups and symmetric groups." Journal of Mathematical Sciences 96, no. 5 (1999): 3590–99. http://dx.doi.org/10.1007/bf02175835.

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47

GAN, WEE LIANG, and JOHN WATTERLOND. "STABLE DECOMPOSITIONS OF CERTAIN REPRESENTATIONS OF THE FINITE GENERAL LINEAR GROUPS." Transformation Groups 23, no. 2 (2017): 425–35. http://dx.doi.org/10.1007/s00031-017-9440-y.

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48

Ban, Dubravka, and Chris Jantzen. "The Langlands quotient theorem for finite central extensions of p-adic groups." Glasnik Matematicki 48, no. 2 (2013): 313–34. http://dx.doi.org/10.3336/gm.48.2.07.

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49

Shirvani, M. "The finite inner automorphism groups of division rings." Mathematical Proceedings of the Cambridge Philosophical Society 118, no. 2 (1995): 207–13. http://dx.doi.org/10.1017/s030500410007359x.

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Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

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50

MARKUS-EPSTEIN, L. "STALLINGS FOLDINGS AND SUBGROUPS OF AMALGAMS OF FINITE GROUPS." International Journal of Algebra and Computation 17, no. 08 (2007): 1493–535. http://dx.doi.org/10.1142/s0218196707003846.

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Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups. In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.

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