Variants of Some of the Brauer-Fowler Theorems
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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. -
Group and Ring Theoretic Properties of Polycyclic Groups Algebra and Applications
Group and Ring Theoretic Properties of Polycyclic Groups Algebra and Applications Volume 10 Managing Editor: Alain Verschoren University of Antwerp, Belgium Series Editors: Alice Fialowski Eötvös Loránd University, Hungary Eric Friedlander Northwestern University, USA John Greenlees Sheffield University, UK Gerhard Hiss Aachen University, Germany Ieke Moerdijk Utrecht University, The Netherlands Idun Reiten Norwegian University of Science and Technology, Norway Christoph Schweigert Hamburg University, Germany Mina Teicher Bar-llan University, Israel Algebra and Applications aims to publish well written and carefully refereed mono- graphs with up-to-date information about progress in all fields of algebra, its clas- sical impact on commutative and noncommutative algebraic and differential geom- etry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C∗-algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science. B.A.F. Wehrfritz Group and Ring Theoretic Properties of Polycyclic Groups B.A.F. Wehrfritz School of Mathematical Sciences -
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. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. -
On Finitely Generated Profinite Groups, Ii 241
Annals of Mathematics, 165 (2007), 239–273 On finitely generated profinite groups, II: products in quasisimple groups By Nikolay Nikolov* and Dan Segal Abstract We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M(q) such that: if S is a finite quasisimple group with |S/Z(S)| >C, βj (j =1,... ,M) are any automorphisms of S, and qj (j =1,... ,M) are any divisors of q, then M qj there exist inner automorphisms αj of S such that S = 1 [S, (αjβj) ]. These results, which rely on the classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I. 1. Introduction The main theorems of Part I [NS] were reduced to two results about finite quasisimple groups. These will be proved here. A group S is said to be quasisimple if S =[S, S] and S/Z(S) is simple, where Z(S) denotes the centre of S. For automorphisms α and β of S we write −1 −1 α β Tα,β(x, y)=x y x y . Theorem 1.1. There is an absolute constant D ∈ N such that if S is a finite quasisimple group and α1,β1,... ,αD,βD are any automorphisms of S then ····· S = Tα1,β1 (S, S) TαD ,βD (S, S). -
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. -
Random Generation of Finite and Profinite Groups and Group
Annals of Mathematics 173 (2011), 769{814 doi: 10.4007/annals.2011.173.2.4 Random generation of finite and profinite groups and group enumeration By Andrei Jaikin-Zapirain and Laszl´ o´ Pyber Abstract We obtain a surprisingly explicit formula for the number of random ele- ments needed to generate a finite d-generator group with high probability. As a corollary we prove that if G is a d-generated linear group of dimension n then cd + log n random generators suffice. Changing perspective we investigate profinite groups F which can be generated by a bounded number of elements with positive probability. In response to a question of Shalev we characterize such groups in terms of certain finite quotients with a transparent structure. As a consequence we settle several problems of Lucchini, Lubotzky, Mann and Segal. As a byproduct of our techniques we obtain that the number of r-relator groups of order n is at most ncr as conjectured by Mann. 1. Introduction Confirming an 1882 conjecture of Netto [40], Dixon [13] proved in 1969 that two randomly chosen elements generate the alternating group Alt(n) with probability that tends to 1 as n ! 1. This was extended in [21] and [24] to arbitrary sequences of non-abelian finite simple groups. Such results form the basis of applying probabilistic methods to the solution of various problems concerning finite simple groups [50]. Interest in random generation of more general families of finite groups arose when it was realized that randomized algorithms play a critical role in handling matrix groups [4]. -
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. -
Finite Simple Groups Which Projectively Embed in an Exceptional Lie Group Are Classified!
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 1, January 1999, Pages 75{93 S 0273-0979(99)00771-5 FINITE SIMPLE GROUPS WHICH PROJECTIVELY EMBED IN AN EXCEPTIONAL LIE GROUP ARE CLASSIFIED! ROBERT L. GRIESS JR. AND A. J. E. RYBA Abstract. Since finite simple groups are the building blocks of finite groups, it is natural to ask about their occurrence “in nature”. In this article, we consider their occurrence in algebraic groups and moreover discuss the general theory of finite subgroups of algebraic groups. 0. Introduction Group character theory classifies embeddings of finite groups into classical groups. No general theory classifies embeddings into the exceptional complex al- gebraic group, i.e., one of G2(C), F4(C), E6(C), E7(C), E8(C). For exceptional groups, special methods seem necessary. Since the early 80s, there have been ef- forts to determine which central extensions of finite simple groups embed in an exceptional group. For short, we call this work a study of projective embeddings of finite simple groups into exceptional groups. Table PE on page 84 contains a summary. The classification program for finite subgroups of complex algebraic groups involves both existence of embeddings and their classification up to conjugacy. We have just classified embeddings of Sz(8) into E8(C) (there are three, up to E8(C)- conjugacy), thus settling the existence question for projective embeddings of finite simple groups into exceptional algebraic groups. The conjugacy part of the program is only partially resolved. The finite subgroups of the smallest simple algebraic group PSL(2; C)(upto conjugacy) constitute the famous list: cyclic, dihedral, Alt4, Sym4, Alt5. -
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].