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Journal articles on the topic 'Polycyclic groups][Algebraic groups'

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Published: 4 June 2021
Last updated: 15 February 2022

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1

Sautoy, Marcus Du. "Polycyclic Groups, Analytic Groups and Algebraic Groups." Proceedings of the London Mathematical Society 85, no. 1 (July 2002): 62–92. http://dx.doi.org/10.1112/plms/85.1.62.

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2

MEYEROVITCH, TOM. "Pseudo-orbit tracing and algebraic actions of countable amenable groups." Ergodic Theory and Dynamical Systems 39, no. 9 (January 24, 2018): 2570–91. http://dx.doi.org/10.1017/etds.2017.126.

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Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.
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3

LICHTMAN, A. I. "RESTRICTED LIE ALGEBRAS OF POLYCYCLIC GROUPS." Journal of Algebra and Its Applications 05, no. 05 (October 2006): 571–627. http://dx.doi.org/10.1142/s0219498806001892.

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We consider some classes of polycyclic groups which have a p-series such that the restricted graded Lie algebra associated to this p-series is free abelian. We also study p-series and restricted Lie algebras associated to them in arbitrary polycyclic groups.
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4

Assmann, Björn, and Bettina Eick. "Computing polycyclic presentations for polycyclic rational matrix groups." Journal of Symbolic Computation 40, no. 6 (December 2005): 1269–84. http://dx.doi.org/10.1016/j.jsc.2005.05.003.

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5

Sinanan, S. K., and D. F. Holt. "Algorithms for polycyclic-by-finite groups." Journal of Symbolic Computation 79 (March 2017): 269–84. http://dx.doi.org/10.1016/j.jsc.2016.02.008.

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6

Nikolaev, Andrey, and Alexander Ushakov. "Subset sum problem in polycyclic groups." Journal of Symbolic Computation 84 (January 2018): 84–94. http://dx.doi.org/10.1016/j.jsc.2017.03.007.

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7

Franciosi, Silvana, Francesco Giovanni, and Martin L. Newell. "Groups with Polycyclic Non-Normal Subgroups." Algebra Colloquium 7, no. 1 (March 2000): 33–42. http://dx.doi.org/10.1007/s10011-000-0033-1.

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8

Ostheimer, Gretchen. "Practical Algorithms for Polycyclic Matrix Groups." Journal of Symbolic Computation 28, no. 3 (September 1999): 361–79. http://dx.doi.org/10.1006/jsco.1999.0287.

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9

Asri, M. S. M., K. B. Wong, and P. C. Wong. "Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups." Algebra Colloquium 28, no. 01 (January 20, 2021): 119–30. http://dx.doi.org/10.1142/s1005386721000110.

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We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form [Formula: see text], where [Formula: see text] has infinite order and [Formula: see text] is finite.
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10

Garreta, Albert, Alexei Miasnikov, and Denis Ovchinnikov. "Diophantine problems in solvable groups." Bulletin of Mathematical Sciences 10, no. 01 (February 21, 2020): 2050005. http://dx.doi.org/10.1142/s1664360720500058.

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We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc.), which satisfy some natural “non-commutativity” conditions. For each group [Formula: see text] in one of these classes, we prove that there exists a ring of algebraic integers [Formula: see text] that is interpretable in [Formula: see text] by finite systems of equations ([Formula: see text]-interpretable), and hence that the Diophantine problem in [Formula: see text] is polynomial time reducible to the Diophantine problem in [Formula: see text]. One of the major open conjectures in number theory states that the Diophantine problem in any such [Formula: see text] is undecidable. If true this would imply that the Diophantine problem in any such [Formula: see text] is also undecidable. Furthermore, we show that for many particular groups [Formula: see text] as above, the ring [Formula: see text] is isomorphic to the ring of integers [Formula: see text], so the Diophantine problem in [Formula: see text] is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups [Formula: see text]. Then, we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups [Formula: see text].
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11

De Falco, Maria, Francesco de Giovanni, and Carmela Musella. "Groups with Finitely Many Normalizers of Non-polycyclic Subgroups." Algebra Colloquium 17, no. 02 (June 2010): 203–10. http://dx.doi.org/10.1142/s1005386710000210.

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The structure of locally graded groups with finitely many normalizers of non-polycyclic subgroups is investigated. In particular, it is proved that such groups either are polycyclic or have Černikov commutator subgroups.
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12

Kim, Seung Won, and Jong Bum Lee. "Universal factorization property of certain polycyclic groups." Journal of Pure and Applied Algebra 204, no. 3 (March 2006): 555–67. http://dx.doi.org/10.1016/j.jpaa.2005.06.006.

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13

Endimioni, Gérard. "On the Nilpotent Length of Polycyclic Groups." Journal of Algebra 203, no. 1 (May 1998): 125–33. http://dx.doi.org/10.1006/jabr.1997.7321.

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14

Kuhn, Norbert, Klaus Madlener, and Friedrich Otto. "Computing presentations for subgroups of polycyclic groups and of context-free groups." Applicable Algebra in Engineering, Communication and Computing 5, no. 5 (September 1994): 287–316. http://dx.doi.org/10.1007/bf01225643.

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15

Wehrfritz, B. A. F. "On hypercentre-by-polycyclic-by-nilpotent groups." Rendiconti del Seminario Matematico della Università di Padova 141 (June 6, 2019): 155–64. http://dx.doi.org/10.4171/rsmup/19.

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16

Wehrfritz, B. A. F. "On a theorem of Rhemtulla on polycyclic groups." Journal of Pure and Applied Algebra 214, no. 10 (October 2010): 1898–900. http://dx.doi.org/10.1016/j.jpaa.2010.02.004.

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17

Marubayashi, H. "Divtsorially graded rings by polycyclic-by-finite groups." Communications in Algebra 17, no. 9 (January 1989): 2135–77. http://dx.doi.org/10.1080/00927878908823842.

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18

Baumslag, Gilbert, Frank B. Cannonito, Derek J. S. Robinson, and Dan Segal. "The algorithmic theory of polycyclic-by-finite groups." Journal of Algebra 142, no. 1 (September 1991): 118–49. http://dx.doi.org/10.1016/0021-8693(91)90221-s.

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19

Lasserre, Clément. "Polycyclic-by-finite groups and first-order sentences." Journal of Algebra 396 (December 2013): 18–38. http://dx.doi.org/10.1016/j.jalgebra.2013.08.008.

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20

Lorenz, Martin. "On the cohomology of polycyclic-by-finite groups." Journal of Pure and Applied Algebra 40 (1986): 87–98. http://dx.doi.org/10.1016/0022-4049(86)90031-9.

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21

Liu, Heguo, Fang Zhou, and Tao Xu. "On some polycyclic groups with small Hirsch length." Journal of Algebra and Its Applications 16, no. 12 (November 20, 2017): 1750237. http://dx.doi.org/10.1142/s0219498817502371.

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A polycyclic group [Formula: see text] is called an [Formula: see text]-group if every normal abelian subgroup of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, the structure of [Formula: see text]-groups and [Formula: see text]-groups is determined.
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22

Liu, Heguo, Fang Zhou, and Tao Xu. "On some polycyclic groups with small Hirsch length II." Journal of Algebra and Its Applications 18, no. 09 (July 17, 2019): 1950169. http://dx.doi.org/10.1142/s021949881950169x.

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A polycyclic group [Formula: see text] is called a [Formula: see text]-group ([Formula: see text]-group) if every normal abelian subgroup (abelian subgroup) of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, we describe the structures of [Formula: see text]-groups and [Formula: see text]-groups, and bound the number of generators of [Formula: see text]-groups and the derived lengths of [Formula: see text]-groups, which is a continuation of [H. G. Liu, F. Zhou and T. Xu, On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710].
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23

Cannon, John J., Bettina Eick, and Charles R. Leedham-Green. "Special polycyclic generating sequences for finite soluble groups." Journal of Symbolic Computation 38, no. 5 (November 2004): 1445–60. http://dx.doi.org/10.1016/j.jsc.2004.05.003.

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24

Kołodziejczyk, Danuta. "Polyhedra with virtually polycyclic fundamental groups have finite depth." Fundamenta Mathematicae 197 (2007): 229–38. http://dx.doi.org/10.4064/fm197-0-10.

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25

Hsu, Tim, and Daniel T. Wise. "Ascending HNN extensions of polycyclic groups are residually finite." Journal of Pure and Applied Algebra 182, no. 1 (July 2003): 65–78. http://dx.doi.org/10.1016/s0022-4049(02)00310-9.

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26

Kurdachenko, Leonid A., Javier Otal, and Panagiotis Soules. "Groups with Polycyclic-by-Finite Conjugate Classes of Subgroups." Communications in Algebra 32, no. 12 (December 31, 2004): 4769–84. http://dx.doi.org/10.1081/agb-200036758.

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27

Dembegioti, Fotini. "On the Zeroeth Complete Cohomology of Certain Polycyclic Groups." Communications in Algebra 36, no. 5 (May 15, 2008): 1927–41. http://dx.doi.org/10.1080/00927870801941622.

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28

Metaftsis, V., E. Raptis, and D. Varsos. "On the Hopficity of HNN-Extensions of Polycyclic Groups." Communications in Algebra 39, no. 3 (March 16, 2011): 751–56. http://dx.doi.org/10.1080/00927871003591991.

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29

Rosset, Shmuel. "The Euler-Goldie rank of some virtually polycyclic groups." Journal of Algebra 103, no. 2 (October 1986): 656–61. http://dx.doi.org/10.1016/0021-8693(86)90158-4.

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30

Assmann, Björn, and Stephen Linton. "Using the Mal'cev correspondence for collection in polycyclic groups." Journal of Algebra 316, no. 2 (October 2007): 828–48. http://dx.doi.org/10.1016/j.jalgebra.2007.01.028.

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31

Zhou, Wei, and Goansu Kim. "Abelian Subgroup Separability of Certain Generalized Free Products of Groups." Algebra Colloquium 27, no. 04 (November 5, 2020): 651–60. http://dx.doi.org/10.1142/s1005386720000541.

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We prove that generalized free products of certain abelian subgroup separable groups are abelian subgroup separable. Applying this, we show that tree products of polycyclic-by-finite groups, amalgamating central subgroups or retracts are abelian subgroup separable.
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32

Wehrfritz, B. A. F. "Examples of polycyclic groups with regular automorphisms of order 4." Journal of Algebra 400 (February 2014): 78–81. http://dx.doi.org/10.1016/j.jalgebra.2013.10.027.

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33

Wauters, P. "Primitive Localizations of Group Algebras of Polycyclic-by-Finite Groups." Journal of Algebra 214, no. 2 (April 1999): 448–57. http://dx.doi.org/10.1006/jabr.1998.7720.

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34

Lo, Eddie H., and Gretchen Ostheimer. "A Practical Algorithm for Finding Matrix Representations for Polycyclic Groups." Journal of Symbolic Computation 28, no. 3 (September 1999): 339–60. http://dx.doi.org/10.1006/jsco.1999.0286.

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35

Ramezan-Nassab, Mojtaba, and Dariush Kiani. "Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups." Journal of Algebra 399 (February 2014): 269–76. http://dx.doi.org/10.1016/j.jalgebra.2013.09.042.

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36

AZEVEDO, A. C. P., G. G. BASTOS, and S. O. JURIAANS. "EXTENSION OF AUTOMORPHISMS OF SUBGROUPS OF ABELIAN AND POLYCYCLIC-BY-FINITE GROUPS." Journal of Algebra and Its Applications 06, no. 02 (April 2007): 315–22. http://dx.doi.org/10.1142/s0219498807002235.

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We classify the abelian groups G, for which the following property holds: for every subgroup H, every ϕ ∈ Aut (H) has an extension ψ ∈ Aut (G). We also classify the infinite polycyclic-by-finite groups and the finite nilpotent 2′-groups having this property. Fuchs, Bertholf, Walls and Tomkinson did similar work for groups which have the property that homomorphisms of its subgroups extend to the whole group.
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37

Eick, Bettina, and Gretchen Ostheimer. "On the orbit-stabilizer problem for integral matrix actions of polycyclic groups." Mathematics of Computation 72, no. 243 (February 3, 2003): 1511–30. http://dx.doi.org/10.1090/s0025-5718-03-01493-5.

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38

Jespers, E., and P. F. Smith. "Integral group rings of torsion-free polycyclic-by- finite groups are maximal orders." Communications in Algebra 13, no. 3 (January 1985): 669–80. http://dx.doi.org/10.1080/00927878508823184.

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39

Endimioni, Gérard, and Carmela Sica. "Centralizer of Engel Elements in a Group." Algebra Colloquium 17, no. 03 (September 2010): 487–94. http://dx.doi.org/10.1142/s1005386710000465.

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In this paper we show that some finiteness properties on a centralizer of a particular subgroup can be inherited by the whole group. Among other things, we prove the following characterization of polycyclic groups: a soluble group G is polycyclic if and only if it contains a finitely generated subgroup H, formed by bounded left Engel elements, whose centralizer CG(H) is polycyclic. In the context of Černikov groups we obtain a more general result: a radical group is a Černikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Černikov group. The aforementioned results generalize a theorem by Onishchuk and Zaĭtsev about the centralizer of a finitely generated subgroup in a nilpotent group.
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40

Rejali, A., and A. Yousofzadeh. "Group Properties Characterized by Two-sided Configurations." Algebra Colloquium 17, no. 04 (December 2010): 583–94. http://dx.doi.org/10.1142/s1005386710000568.

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In this paper, we define a new type of configurations as two-sided configurations, and investigate which group properties can be characterized by them. It is proved that for polycyclic torsion free groups, having the same finite quotient sets does not imply the (two-sided) configuration equivalence. We show that isomorphisms and configuration equivalences coincide for some free products of groups and a class of nilpotent groups.
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41

Bouchelaghem, Mounia, and Nadir Trabelsi. "Groups whose proper subgroups of infinite rank have minimax conjugacy classes." Journal of Algebra and Its Applications 16, no. 01 (January 2017): 1750003. http://dx.doi.org/10.1142/s0219498817500037.

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If [Formula: see text] is a class of groups, then a group [Formula: see text] is said to be a [Formula: see text]-group, if [Formula: see text] is a [Formula: see text]-group for all [Formula: see text]. This is a generalization of the familiar property of being an [Formula: see text]-group. In the present paper we consider a class [Formula: see text] of soluble-by-finite minimax groups such that [Formula: see text] is a subgroup closed class and if [Formula: see text] is a non-[Formula: see text]-group whose proper subgroups of infinite rank are [Formula: see text]-groups, then there exists a prime [Formula: see text] such that every finite homomorphic image of [Formula: see text] is a cyclic [Formula: see text]-group. Our main result states that if [Formula: see text] is a locally (soluble-by-finite) group of infinite rank which has no simple factor group of infinite rank and if all proper subgroups of [Formula: see text] of infinite rank are [Formula: see text]-groups, then so are all proper subgroups of [Formula: see text]. One can take for [Formula: see text] the class of finite, polycyclic-by-finite, Chernikov, reduced minimax or soluble-by-finite minimax groups.
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42

Malekan, Meisam Soleimani, and Ali Rejali. "Two-sided configuration equivalence and isomorphism." Journal of Algebra and Its Applications 19, no. 10 (October 11, 2019): 2050189. http://dx.doi.org/10.1142/s0219498820501893.

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The concept of configuration was first introduced by Rosenblatt and Willis to give a characterization for the amenability of groups. Then Rejali and Yousofzadeh introduced the notion of two-sided configuration to study the normal subsets of a group. In [A, Abdollahi, A. Rejali and G. A. Willis, Group properties characterised by configurations, Illinois J. Math. 48(3) (2004) 861–873.], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if [Formula: see text] and [Formula: see text] have same two-sided configuration sets and [Formula: see text] is a normal subgroup of [Formula: see text] with polycyclic or FC quotient, then there is a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text]. Also, we show that if [Formula: see text] and [Formula: see text] are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic.
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43

Linnell, Peter A., Gena Puninski, and Patrick Smith. "Idempotent ideals and nonfinitely generated projective modules over integral group rings of polycyclic-by-finite groups." Journal of Algebra 305, no. 2 (November 2006): 845–58. http://dx.doi.org/10.1016/j.jalgebra.2005.12.010.

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44

GOFFA, ISABEL, ERIC JESPERS, and JAN OKNIŃSKI. "PRIMES OF HEIGHT ONE AND A CLASS OF NOETHERIAN FINITELY PRESENTED ALGEBRAS." International Journal of Algebra and Computation 17, no. 07 (November 2007): 1465–91. http://dx.doi.org/10.1142/s0218196707004347.

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Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type.
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45

Fel'shtyn, Alexander, and Evgenij Troitsky. "Geometry of Reidemeister classes and twisted Burnside theorem." Journal of K-theory 2, no. 3 (March 4, 2008): 463–506. http://dx.doi.org/10.1017/is008001006jkt028.

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AbstractThe purpose of the present mostly expository paper (based mainly on [17, 18, 40, 16, 11]) is to present the current state of the following conjecture of A. Fel'shtyn and R. Hill [13], which is a generalization of the classical Burnside theorem.Let G be a countable discrete group, φ one of its automorphisms, R(φ) the number of φ-conjugacy (or twisted conjugacy) classes, and S(φ) = #Fix the number of φ-invariant equivalence classes of irreducible unitary representations. If one of R(φ) and S(φ) is finite, then it is equal to the other.This conjecture plays a important role in the theory of twisted conjugacy classes (see [26], [10]) and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Noncommutative Harmonic Analysis.First we prove this conjecture for finitely generated groups of type I and discuss its applications.After that we discuss an important example of an automorphism of a type II1 group which disproves the original formulation of the conjecture.Then we prove a version of the conjecture for a wide class of groups, including almost polycyclic groups (in particular, finitely generated groups of polynomial growth). In this formulation the role of an appropriate dual object plays the finite-dimensional part of the unitary dual. Some counter-examples are discussed.Then we begin a discussion of the general case (which also needs new definition of the dual object) and prove the weak twisted Burnside theorem for general countable discrete groups. For this purpose we prove a noncommutative version of Riesz-Markov-Kakutani representation theorem.Finally we explain why the Reidemeister numbers are always infinite for Baumslag-Solitar groups.
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46

Brion, Michel, Jens Carsten Jantzen, and Zinovy Reichstein. "Algebraic Groups." Oberwolfach Reports 10, no. 2 (2013): 1025–85. http://dx.doi.org/10.4171/owr/2013/17.

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47

De Concini, Corrado, Peter Littelmann, and Zinovy Reichstein. "Algebraic Groups." Oberwolfach Reports 14, no. 2 (April 27, 2018): 1281–347. http://dx.doi.org/10.4171/owr/2017/21.

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48

Galaktionova, E. "algebraic groups." Duke Mathematical Journal 77, no. 1 (January 1995): 63–69. http://dx.doi.org/10.1215/s0012-7094-95-07703-5.

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49

Platonov, V. P., and A. S. Rapinchuk. "Algebraic groups." Journal of Soviet Mathematics 31, no. 3 (November 1985): 2939–73. http://dx.doi.org/10.1007/bf02106806.

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50

Yagita, Nobuaki. "Witt Groups of Algebraic Groups." Publications of the Research Institute for Mathematical Sciences 50, no. 1 (2014): 113–51. http://dx.doi.org/10.4171/prims/126.

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