Group and Ring Theoretic Properties of Polycyclic Groups Algebra and Applications
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Determining Group Structure from the Sets of Character Degrees
DETERMINING GROUP STRUCTURE FROM THE SETS OF CHARACTER DEGREES A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Kamal Aziziheris May 2011 Dissertation written by Kamal Aziziheris B.S., University Of Tabriz, 1999 M.A., University of Tehran, 2001 Ph.D., Kent State University, 2011 Approved by M. L. Lewis, Chair, Doctoral Dissertation Committee Stephen M. Gagola, Jr., Member, Doctoral Dissertation Committee Donald White, Member, Doctoral Dissertation Committee Brett D. Ellman, Member, Doctoral Dissertation Committee Anne Reynolds, Member, Doctoral Dissertation Committee Accepted by Andrew Tonge, Chair, Department of Mathematical Sciences Timothy S. Moerland, Dean, College of Arts and Sciences ii . To the Loving Memory of My Father, Mohammad Aziziheris iii TABLE OF CONTENTS ACKNOWLEDGEMENTS . v INTRODUCTION . 1 1 BACKGROUND RESULTS AND FACTS . 12 2 DIRECT PRODUCTS WHEN cd(Oπ(G)) = f1; n; mg . 22 3 OBTAINING THE CHARACTER DEGREES OF Oπ(G) . 42 4 PROOFS OF THEOREMS D AND E . 53 5 PROOFS OF THEOREMS F AND G . 62 6 EXAMPLES . 70 BIBLIOGRAPHY . 76 iv ACKNOWLEDGEMENTS This dissertation is due to many whom I owe a huge debt of gratitude. I would especially like to thank the following individuals for their support, encouragement, and inspiration along this long and often difficult journey. First and foremost, I offer thanks to my advisor Prof. Mark Lewis. This dissertation would not have been possible without your countless hours of advice and support. Thank you for modeling the actions and behaviors of an accomplished mathematician, an excellent teacher, and a remarkable person. -
PRONORMALITY in INFINITE GROUPS by F G
PRONORMALITY IN INFINITE GROUPS By F G Dipartimento di Matematica e Applicazioni, Universita' di Napoli ‘Federico II’ and G V Dipartimento di Matematica e Informatica, Universita' di Salerno [Received 16 March 1999. Read 13 December 1999. Published 29 December 2000.] A A subgroup H of a group G is said to be pronormal if H and Hx are conjugate in fH, Hxg for each element x of G. In this paper properties of pronormal subgroups of infinite groups are investigated, and the connection between pronormal subgroups and groups in which normality is a transitive relation is studied. Moreover, we consider the pronorm of a group G, i.e. the set of all elements x of G such that H and Hx are conjugate in fH, Hxg for every subgroup H of G; although the pronorm is not in general a subgroup, we prove that this property holds for certain natural classes of (locally soluble) groups. 1. Introduction A subgroup H of a group G is said to be pronormal if for every element x of G the subgroups H and Hx are conjugate in fH, Hxg. Obvious examples of pronormal subgroups are normal subgroups and maximal subgroups of arbitrary groups; moreover, Sylow subgroups of finite groups and Hall subgroups of finite soluble groups are always pronormal. The concept of a pronormal subgroup was introduced by P. Hall, and the first results about pronormality appeared in a paper by Rose [13]. More recently, several authors have investigated the behaviour of pronormal subgroups, mostly dealing with properties of pronormal subgroups of finite soluble groups and with groups which are rich in pronormal subgroups. -
Polynomial Structures on Polycyclic Groups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 9, September 1997, Pages 3597{3610 S 0002-9947(97)01924-7 POLYNOMIAL STRUCTURES ON POLYCYCLIC GROUPS KAREL DEKIMPE AND PAUL IGODT Abstract. We know, by recent work of Benoist and of Burde & Grunewald, that there exist polycyclic–by–finite groups G,ofrankh(the examples given were in fact nilpotent), admitting no properly discontinuous affine action on Rh. On the other hand, for such G, it is always possible to construct a prop- erly discontinuous smooth action of G on Rh. Our main result is that any polycyclic–by–finite group G of rank h contains a subgroup of finite index act- ing properly discontinuously and by polynomial diffeomorphisms of bounded degree on Rh. Moreover, these polynomial representations always appear to contain pure translations and are extendable to a smooth action of the whole group G. 1. Introduction. In 1977 ([18]), John Milnor asked if it was true that any torsion-free polycyclic– by–finite group G occurs as the fundamental group of a compact, complete, affinely flat manifold. This is equivalent to saying that G acts properly discontinuously and h G via affine motions on R ( ). (Throughout this paper, we will use h(G)todenotethe Hirsch length of a polycyclic–by–finite group G.) As there are several definitions of the notion “properly discontinuous action” we write down the definition we use: Definition 1.1. Let Γ be a group acting (on the left) on a Hausdorff space X.We say that the action of Γ on X is properly discontinuous if and only if γ 1. -
The Theory of Finite Groups: an Introduction (Universitext)
Universitext Editorial Board (North America): S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo This page intentionally left blank Hans Kurzweil Bernd Stellmacher The Theory of Finite Groups An Introduction Hans Kurzweil Bernd Stellmacher Institute of Mathematics Mathematiches Seminar Kiel University of Erlangen-Nuremburg Christian-Albrechts-Universität 1 Bismarckstrasse 1 /2 Ludewig-Meyn Strasse 4 Erlangen 91054 Kiel D-24098 Germany Germany [email protected] [email protected] Editorial Board (North America): S. Axler F.W. Gehring Mathematics Department Mathematics Department San Francisco State University East Hall San Francisco, CA 94132 University of Michigan USA Ann Arbor, MI 48109-1109 [email protected] USA [email protected] K.A. Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA [email protected] Mathematics Subject Classification (2000): 20-01, 20DXX Library of Congress Cataloging-in-Publication Data Kurzweil, Hans, 1942– The theory of finite groups: an introduction / Hans Kurzweil, Bernd Stellmacher. p. cm. — (Universitext) Includes bibliographical references and index. ISBN 0-387-40510-0 (alk. paper) 1. Finite groups. I. Stellmacher, B. (Bernd) II. Title. QA177.K87 2004 512´.2—dc21 2003054313 ISBN 0-387-40510-0 Printed on acid-free paper. © 2004 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. -
Abelian-By-Polycyclic Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 41, Number 2, December 1973 FINITE FRATTINI FACTORS IN FINITELY GENERATED ABELIAN-BY-POLYCYCLIC GROUPS JOHN C. LENNOX Abstract. A criterion for a group to be finite is used to prove that subgroups of finitely generated Abelian-by-polycyclic groups are finite if their Frattini factor groups are finite. In a previous paper [2] we considered finitely generated soluble groups and their subgroups in relation to the question whether they were finite if their Frattini factor groups were finite. Among other results we answered this question in the affirmative for subgroups of finitely generated Abelian- by-nilpotent groups [2, Theorem B]. The class of all finitely generated Abelian-by-nilpotent groups is of course a proper subclass of the class of all finitely generated Abelian-by-polycyclic groups, since finitely generated nilpotent groups are polycyclic, but the converse does not hold. Now, although our methods in [2] depended on properties of finitely generated Abelian-by-polycyclic groups (more accurately, they depended on prop- erties of Hall's class 23 of Abelian groups where B is in 23 if and only if B is an Abelian group which can be extended to a finitely generated group G such that G/B is polycyclic), they were not good enough to settle the question for the wider class of groups. Our object here is to close this gap by proving the Theorem. Subgroups of finitely generated Abelian-by-polycyclic groups are finite if their Frattini factor groups are finite. The information we need here, supplementary to that contained in [2], is given in the following result which generalises Lemma 3 of [2] in a way which may be of interest in itself. -
Formulating Basic Notions of Finite Group Theory Via the Lifting Property
FORMULATING BASIC NOTIONS OF FINITE GROUP THEORY VIA THE LIFTING PROPERTY by masha gavrilovich Abstract. We reformulate several basic notions of notions in nite group theory in terms of iterations of the lifting property (orthogonality) with respect to particular morphisms. Our examples include the notions being nilpotent, solvable, perfect, torsion-free; p-groups and prime-to-p-groups; Fitting sub- group, perfect core, p-core, and prime-to-p core. We also reformulate as in similar terms the conjecture that a localisation of a (transnitely) nilpotent group is (transnitely) nilpotent. 1. Introduction. We observe that several standard elementary notions of nite group the- ory can be dened by iteratively applying the same diagram chasing trick, namely the lifting property (orthogonality of morphisms), to simple classes of homomorphisms of nite groups. The notions include a nite group being nilpotent, solvable, perfect, torsion- free; p-groups, and prime-to-p groups; p-core, the Fitting subgroup, cf.2.2-2.3. In 2.5 we reformulate as a labelled commutative diagram the conjecture that a localisation of a transnitely nilpotent group is transnitely nilpotent; this suggests a variety of related questions and is inspired by the conjecture of Farjoun that a localisation of a nilpotent group is nilpotent. Institute for Regional Economic Studies, Russian Academy of Sciences (IRES RAS). National Research University Higher School of Economics, Saint-Petersburg. [email protected]://mishap.sdf.org. This paper commemorates the centennial of the birth of N.A. Shanin, the teacher of S.Yu.Maslov and G.E.Mints, who was my teacher. I hope the motivation behind this paper is in spirit of the Shanin's group ÒÐÝÏËÎ (òåîðèòè÷åñêàÿ ðàçðàáîòêà ýâðèñòè÷åñêîãî ïîèñêà ëîãè÷åñêèõ îáîñíîâàíèé, theoretical de- velopment of heuristic search for logical evidence/arguments). -
Daniel Gorenstein 1923-1992
Daniel Gorenstein 1923-1992 A Biographical Memoir by Michael Aschbacher ©2016 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. DANIEL GORENSTEIN January 3, 1923–August 26, 1992 Elected to the NAS, 1987 Daniel Gorenstein was one of the most influential figures in mathematics during the last few decades of the 20th century. In particular, he was a primary architect of the classification of the finite simple groups. During his career Gorenstein received many of the honors that the mathematical community reserves for its highest achievers. He was awarded the Steele Prize for mathemat- New Jersey University of Rutgers, The State of ical exposition by the American Mathematical Society in 1989; he delivered the plenary address at the International Congress of Mathematicians in Helsinki, Finland, in 1978; Photograph courtesy of of Photograph courtesy and he was the Colloquium Lecturer for the American Mathematical Society in 1984. He was also a member of the National Academy of Sciences and of the American By Michael Aschbacher Academy of Arts and Sciences. Gorenstein was the Jacqueline B. Lewis Professor of Mathematics at Rutgers University and the founding director of its Center for Discrete Mathematics and Theoretical Computer Science. He served as chairman of the universi- ty’s mathematics department from 1975 to 1982, and together with his predecessor, Ken Wolfson, he oversaw a dramatic improvement in the quality of mathematics at Rutgers. Born and raised in Boston, Gorenstein attended the Boston Latin School and went on to receive an A.B. -
Polycyclic, Metabelian Or Soluble of Type (FP) $ {\Infty} $ Groups With
POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN VITALY ROMAN’KOV Omsk State Dostoevsky University and Omsk State Technical University, 644077, Omsk, Russia, e-mail:[email protected] Abstract Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a boolean algebra. Then G is virtually abelian. Every soluble biautomatic group is virtually abelian. 1 Introduction The topic of this paper is two important concepts: rational sets and biauto- matic structure. We study finitely generated soluble groups G such that the class Rat(G) of all rational subsets of G is a boolean algebra. We conjecture that every such group is virtually abelian. Note that every finitely generated virtually abelian group satisfies to this property. We confirm this conjecture in the case where G is a polycyclic, metabelian or soluble group of type FP∞. This conjecture remains open in general case. It appeared that the notion FP∞ helps to prove by the way that every soluble biautomatic group is virtually abelian. Thus, we give answer to known question posed in [1]. We provide full proofs of four theorems attributed to Bazhenova, a former student of the author, stating that some natural assumptions imply that a sol- uble group is virtually abelian. The original proofs was given by her in collab- arXiv:1511.00223v1 [math.GR] 1 Nov 2015 oration with the author more than 14 years ago and had never been published. -
The Fitting Subgroup of a Linear Solvable Group
THE FITTING SUBGROUP OF A LINEAR SOLVABLE GROUP JOHN D. DIXON (Received 5 May 1966) 1. Introduction Let G be a group. The Fitting subgroup F(G) of G is defined to be the set union of all normal nilpotent subgroups of G. Since the product of two normal nilpotent subgroups is again a normal nilpotent subgroup (see [10] p. 238), F(G) is the unique maximal normal, locally nilpotent subgroup of G. In particular, if G is finite, then F(G) is the unique maximal normal nilpotent subgroup of G. If G is a nontrivial solvable group, then clearly F(G) * 1. The principal result of this paper is the following theorem. THEOREM 1. Let G be a completely reducible solvable subgroup of the general linear group GL (n, &) over an algebraically closed field 3F. Then \G:F(G)\ ^a-W where a = 2.3* = 2.88 • • • and b = 2.3? = 4.16 • • •. Moreover the bound is attained by some finite solvable group over the complex field whenever n = 2.4* (k = 0, 1, • • •). NOTE. When G is finite and SF is perfect, then the condition "alge- braically closed" is unnecessary since the field may be extended without affecting the complete reducibility. See [3] Theorem (70.15). A theorem of A. I. Mal'cev shows that there is a bound /?„ such that each completely reducible linear solvable group of degree n has a normal abelian subgroup of index at most /?„. (See [5].) The previous estimates for /?„ are by no means precise, but Theorem 1 allows us to give quite a precise estimate of the corresponding bound for a subnormal abelian subgroup. -
Solvable Groups Acting on the Line 403
TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 278, Number I. July 1983 SOLVABLEGROUPS ACTING ON THE LINE BY J. F. PLANTE1 Abstract. Actions of discrete groups on the real line are considered. When the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line. In the process of obtaining these results sufficient conditions are also de- termined for the existence of invariant (quasi-invariant) measures for abelian (solva- ble) groups acting on the line. It turns out, for example, that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi-invariant measure, and that any abelian group of C diffeomorphisms has an invariant measure. An example is given to show that some restrictions are necessary in order to obtain such conclusions. Some applications to the study of codimension one foliations are indicated. Introduction. If G is a group of homeomorphisms of the line it is reasonable to ask for a dynamic description of how G acts when reasonable restrictions are placed on G. For example, in [9] it is shown that if G is finitely generated and nilpotent then there is a G-invariant Borel measure on R which is finite on compact sets. This already says much about G. The proof is quite different from classical arguments which guarantee the existence of a finite invariant measure when an amenable group acts on a compact Hausdorff space [6] in that one uses pseudogroup properties of G rather than group properties (specifically, nonexponential growth of orbits rather than nilpotence of G). -
Arxiv:1705.06265V1 [Math.GR] 17 May 2017 Subgroup
GROUPS IN WHICH EVERY NON-NILPOTENT SUBGROUP IS SELF-NORMALIZING COSTANTINO DELIZIA, URBAN JEZERNIK, PRIMOZˇ MORAVEC, AND CHIARA NICOTERA Abstract. We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with the above property. Furthermore, we give some structural information in the infinite perfect case. 1. Introduction A long standing problem posed by Y. Berkovich [3, Problem 9] is to study the finite p-groups in which every non-abelian subgroup contains its centralizer. In [8], the finite p-groups which have maximal class or exponent p and satisfy Berkovich’s condition are characterized. Furthermore, the infi- nite supersoluble groups with the same condition are completely classi- fied. Although it seems unlikely to be able to get a full classification of finite p-groups in which every non-abelian subgroup is self-centralizing, Berkovich’s problem has been the starting point for a series of papers investigating finite and infinite groups in which every subgroup belongs to a certain family or is self-centralizing. For instance, in [6] and [7] locally finite or infinite supersoluble groups in which every non-cyclic arXiv:1705.06265v1 [math.GR] 17 May 2017 subgroup is self-centralizing are described. A more accessible version of Berkovich’s problem has been proposed by P. Zalesskii, who asked to classify the finite groups in which every non-abelian subgroup equals its normalizer. This problem has been solved in [9]. -
Product of Periodic Groups
ISSN 2664-4150 (Print) & ISSN 2664-794X (Online) South Asian Research Journal of Engineering and Technology Abbreviated Key Title: South Asian Res J Eng Tech | Volume-1 | Issue-1| Jun-Jul -2019 | Research Article Product of Periodic Groups Behnam Razzaghmaneshi* Assistant Professor, Department of Mathematics and Computer Science, Islamic Azad University Talesh Branch, Talesh, Iran *Corresponding Author Behnam Razzaghmaneshi Article History Received: 06.07.2019 Accepted: 12.07.2019 Published: 30.07.2019 Abstract: A group G is called radicable if for each element x of G and for each positive integer n there exists an element y of G such that x=yn. A group G is reduced if it has no non-trivial radicable subgroups. Let the hyper-((locally nilpotent) or finte) group G=AB be the product of two periodic hyper-(abelian or finite) subgroups A and B. Then the following hold: (i) G is periodic. (ii) If the Sylow p- subgroups of A and B are Chernikov (respectively: finite, trivial), then the p-component of every abelian normal section of G is Chernikov (respectively: finite, trivial). Keywords: Residual, factorize, minimal condition, primary subgroup 2000 Mathematics subject classification: 20B32, 20D10. INTRODUCTION In 1968 N. F. Sesekin proved that a product of two abelian subgroups with minimal condition satisfies also the minimal condition [1]. N. F. Seskin and B.Amberg independently obtained a similar result for maximal condition around 1972 [1, 2]. Moreover, a little later the later proved that a soluble product of two nilpotent subgroups with maximal condition likewise satisfies the minimal condition, and its Fitting subgroup inherits the factorization.