DOCSLIB.ORG

7. Minimal Normal Subgroups and the Socle Throughout This Section G Is A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
7. Minimal Normal Subgroups and the Socle Throughout This Section G Is A 34 NICK GILL 7. Minimal normal subgroups and the socle Throughout this sectionG is a nontrivial group. A minimal normal subgroup ofG is a normal subgroup K= 1 ofG which does not contain any other nontrivial normal subgroup ofG. The socle, soc(G) is the subgroup� generated by the set of all minimal normal subgroups (ifG has no minimal normal subgroups, then we set soc(G) = 1). (E7.1)Find the socle ofD 10,A 4,S 4,S 4 Z. × (E7.2)Give an example of a group that has no minimal normal subgroup. (E7.3)IfG is the direct product of a (possibly infinite) number of finite simple groups, then what is soc(G)? (E7.4)Give a characteri ation of almost simple groups in terms of their socle. Clearly soc(G) is a characteristic subgroup ofG. 26 Theorem 7.1. LetG be a nontrivial finite group. (1) IfK is a minimal normal subgroup ofG, andL is any normal subgroup ofG, then eitherK L or K,L =K L. 27 ≤ (2) There� exist� minimal× normal subgroupsK ,...,K ofG such that soc(G)=K K . 1 m 1 × · · · m (3) Every minimal normal subgroupK ofG is a direct productK=T 1 T k where theT i are simple normal subgroups ofK which are conjugate inG. × · · · (4) If the subgroupsK i in (2) are all nonabelian, thenK 1,...,Km are the only minimal normal sub- groups ofG. Proof. (1) #inceK L�G, the minimality ofK implies that eitherK L orK L= 1 . $n the latter case, since∩ K andL are both normal, we have that K,L =≤KL=K∩L. { } (2) %ecauseG is &nite we can &nd a setS= K ,...,K � of� minimal normal× subgroups which is { 1 m} ma'imal with respect to the property thatH (= S =K 1 K m. )e must show thatH contains all minimal normal subgroups ofG, and� so� is equal to× soc(· · · ×G). This follows immediately from (1). (3) +etT be a minimal normal subgroup ofK and observe that all conjugatesT g, forg G, are also minimal normal subgroups ofK. Choose a setS= T ,...,T of these conjugates∈ which is { 1 m} ma'imal with respect to the property thatL (= S =T 1 T m. Then, arguing $ala (2), we see thatL contains all of the conjugates ofT under� G� and so×L · ·G ·. × %ut, since 1 <L K and � { } ≤ K is minimal normal, we conclude thatK=L=T 1 T m. Note, &nally, that, forT i to be minimal normal inT T , we must haveT simple.× · · · × 1 × · · · × m i (4) +etK be a minimal normal subgroup ofG that is distinct fromK 1,...,Km, then, applying (1) withL=K , we &nd that K,L =K L and, in particular,K centralizes each of theK . Thus i � � × i K Z(soc(G)). However if eachK i is nonabelian, then (3) implies thatZ(K i) = 1 and we have a contradiction,≤ as required. { } � 0bserve that if a minimal normal subgroupK is abelian, thenK is an elementary-abelianp-group for some primep. 28 (E7.5)Suppose thatG is elementary abelian. "ow many minimal (resp. maximal) normal sub- groups doesG have? 26i.e. soc(G) is invariant under any automorphism ofG. This is because any automorphism ofG must permute the minimal normal subgroups ofG. 27I am considering the internal direct product here, a special case of the internal semidirect product that we have already seen. 28Recall that an elementary-abelianp-group is defined to be a group that is isomorphic toC C , wheren is finite, p × · · · × p n p is a prime, andC p is a cyclic group of orderp. � �� � FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 35 7.1. Finite groups with elementary abelian socle. In this sectionG is a &nite group with an elementary-abelian socle. )e write V (= soc(G)=C C . p × · · · × p d 0bserve thatV has a natural structure as a vector space� over�� F p�, the &eld of orderp. This allows us to ma2e the following assertion. Lemma 7.2. G/CG(V) is isomorphic to a subgroup of 3+(V), and G/C G(V) acts onV via multiplication (on the right) by matrices. Proof. Consider the action ofGonV by conjugation. +emma !. implies that this induces a homomorphism φ(G Aut(V ) where we viewV is an ob,ect from Group. 4urthermoreC G(V) is the 2ernel ofφ and, in particular,→ it is a normal subgroup ofG. Now, sinceC G(V ) is the 2ernel of the conjugation action, the &rst isomorphism theorem of groups implies that G/V is isomorphic to a subgroup of Aut(V ). Now the result follows from (E7.6) below.� (E7.6) $etV be an elementary#abelianp#group. (1) $etG be a group of automorphisms ofV as an object from Group. &rove thatG acts linearly onV , i.e. prove thatG is a group of automorphisms ofV as an object from Vect Fp . (2) $etG be a group of automorphisms ofV as an object from Vect Fp . Prove thatG is a group of automorphisms ofV as an object from Group. (3) Conclude that Aut(V ) # $L(V), whether we consider it an object of Group or of Vect Fp . +et us strengthen our supposition( let us suppose thatG splits overV , i.e. that there e'ists a subgroup H < G, such thatG=V�H. To state the structure result in this case, we need a definition. Given a vector spaceV over a &eldK, define A3+(V)(= (g,v) v K d, g 3+(V) =K d � 3+(V). { | ∈ ∈ } Here we writeK d to mean the additive group whose elements ared-tuples with entries fromK. Multipli1 cation is defined in the usual way for a semidirect product( g2 (g1, v1)(g2, v2) (= (g1, g2, v1 v2) n g where, forv K andg 3+ d(K), we definev to be the product ofv (thought of as a row vector) with g, a matri'.∈ ∈ Note that, just as with 3+(V ), we will write A3+d(K) as a pseudonym for A3+(V ). 4urthermore, if K=F p is &nite, then we will write A3+d(p) as a synonym for A3+d(Fp). Proposition 7.3. Suppose thatG is a finite group with socleV of orderp d for some primep. IfG splits overV , thenG is isomorphic to a subgroup of A3+ d(p). Proof. +etH be a subgroup ofG such thatG=V�H. Consider the action ofH onV by con,ugation. )e claim thatC H (V ) is a normal subgroup ofG8 indeed it is enough to prove that it is a normal subgroup ofH, sinceG=VH andC H (V ) is certainly normalized byV. To prove the claim, ta2eh C H (V),h 1 H andv V . Now observe that ∈ ∈ h ∈ (h 1 ) h1 1 h1 v = (h )− v(h ) 1 1 1 = (h1− hh1)− v(h1− hh1) 1 1 1 =h 1− (h− (h1vh1− )h)h1 1 1 =h 1− (h1vh1− )h1 =v. h1 Thush C H (V ) and the claim is proved. But now, sinceH V= 1 , we know thatC H (V) V= 1 and so, since∈ V is the socle, we conclude thatC (V)= 1 . ∩ { } ∩ { } H { } Now considerC G(V ). Ifg G, theng=vh for a uni*uev V andh H andg C G(V) if and only ifh C (V ). In particular,∈ ∈ ∈ ∈ ∈ G CG(V)= V.C H (V)=V. 36 NICK GILL Now +emma 7. implies that G/V is isomorphic to a subgroup of 3+(V ). Moreover the groupH ∼= G/V acts onV by right multiplication by matrices; in other wordsV�H is isomorphic to a subgroup of A3+ d(p) as required. � (E7.7)Suppose thatK is finite. Prove that soc("$% (K)) # Kd and, moreover, that soc("$% (K)) is a minimal normal • d ∼ d subgroup of "$%d(K). Suppose that soc("$%d(K) G "$% d(K). (nder what conditions is soc("$%d(K))a • minimal normal subgroup of≤G?≤ (E7.8)Describe the structure of "$%1(p) for a primep. 7.2. The socle of a primitive permutation group. Theorem 7.4. IfG is a finite primitive subgroup of #ym(n), andK is a minimal normal subgroup ofG, then exactly one of the following holds: (1) for some primep and some integerd,K is a regular elementary abelian group of orderp d, and soc(G) =K=C G(K). (2)K is a regular non-abelian group,C G(K) is a minimal normal subgroup ofG which is permutation isomorphic toK, and soc(G) =K C G(K). (3)K is non-abelian and soc(G)=K.× Proof. +etC=C G(K). #inceC�G, eitherC=1orC is transitive. #inceK is transitive, Lemma !.7 implies thatC is semiregular, and hence eitherC=1orC is regular. #uppose thatC = 1. ClearlyK is non-abelian. 4urthermore Theorem 7.1 (1) implies thatK is the only minimal normal subgroup ofG and so soc(G)=K and conclusion (3) of the result holds. #uppose instead thatC is regular. Then (5!. !) implies thatC (C) is regular. #inceK Sym(Ω) ≤ CSym(Ω)(C) andK is transitive, we conclude thatK=C Sym(Ω)(C). Similarly,C=C Sym(Ω)(K) and, by (5!. "),C andK are permutation isomorphic. Now, sinceC is regular, we conclude thatC is a minimal normal subgroup ofG (since any proper subgroup ofC is intransitive). Now Theorem 7.1 (1) states that every minimal normal subgroup ofG distinct fromK is contained inC. Thus soc(G) =KC which e*ualsK orK C depending on whether C K or not. × ≤IfC K, thenC=K,K is abelian, soc(G)=K and conclusion (1) holds. IfC K, then conclusion ≤ �≤ (2) holds. � (E7.9) Suppose thatG is a maximal primitive subgroup of &ym(n).
Load more

Recommended publications
  • Math 602, Fall 2002. BIII (A). We First Prove the Following Proposition
    Homework I (due September 30), Math 602, Fall 2002. BIII (a). We first prove the following proposition: Proposition 1.1 Given a group G, for any finite normal subgroup, H, of G and any p-Sylow subgroup P of H, we have G = NG(P )H. Proof . For every g ∈ G, clearly, g−1P g is a subgroup of g−1Hg = H, since H is normal in G. Since |g−1P g| = |P |, the subgroup g−1P g is also a p-Sylow subgroup of H. By Sylow II, g−1P g is some conjugate of P in H, i.e., g−1P g = hP h−1 for some h ∈ H. −1 −1 Thus, ghP h g = P , which implies that gh ∈ NG(P ), so g ∈ NG(P )H. Since the reasoning holds for every g ∈ G, we get G = NG(P )H. Now, since H is normal in G, by the second homomorphism theorem, we know that ∼ G/H = (NG(P )H)/H = NG(P )/(NG(P ) ∩ H). Moreover, it is clear that NG(P ) ∩ H = ∼ NH (P ), so G/H = NG(P )/NH (P ), as desired. (b) We shall prove that every p-Sylow subgroup of Φ(G) is normal in Φ(G) (and in fact, in G). From this, we will deduce that Φ(G) has property (N). Indeed, if we inspect the proof of the proposition proved in class stating that if G is a finite group that has (N), then G is isomorphic to the product of its p-Sylow subgroups, we see that this proof only depends on the fact that every p-Sylow subgroup of G is normal in G.
    [Show full text]
  • A CHARACTERISTIC SUBGROUP of a P-GROUP
    Pacific Journal of Mathematics A CHARACTERISTIC SUBGROUP OF A p-GROUP CHARLES RAY HOBBY Vol. 10, No. 3 November 1960 A CHARACTERISTIC SUBGROUP OF A p-GROVP CHARLES HOBBY If x, y are elements and H, K subsets of the p-group G, we shall denote by [x, y] the element y~px~p(xy)p of G, and by [H, K] the sub- group of G generated by the set of all [h, k] for h in H and k in K. We call a p-group G p-abelίan if (xy)p = xpyp for all elements x, y of G. If we let Θ(G) — [G, G] then #(G) is a characteristic subgroup of G and Gjθ{G) is p-abelian. In fact, Θ(G) is the minimal normal subgroup N of G for which G/AΓ is p-abelian. It is clear that Θ(G) is contained in the derived group of G, and G/Θ(G) is regular in the sense of P. Hall [3] Theorem 1 lists some elementary properties of p-abelian groups. These properties are used to obtain a characterization of p-groups G (for p > 3) in which the subgroup generated by the pth powers of elements of G coincides with the Frattini subgroup of G (Theorems 2 and 3). A group G is said to be metacyclic if there exists a cyclic normal sub- group N with G/N cyclic. Theorem 4 states that a p-group G, for p > 2, is metacyclic if and only if Gjθ(G) is metacyclic.
    [Show full text]
  • REPRESENTATIONS of ELEMENTARY ABELIAN P-GROUPS and BUNDLES on GRASSMANNIANS
    REPRESENTATIONS OF ELEMENTARY ABELIAN p-GROUPS AND BUNDLES ON GRASSMANNIANS JON F. CARLSON∗, ERIC M. FRIEDLANDER∗∗ AND JULIA PEVTSOVA∗∗∗ Abstract. We initiate the study of representations of elementary abelian p- groups via restrictions to truncated polynomial subalgebras of the group alge- p p bra generated by r nilpotent elements, k[t1; : : : ; tr]=(t1; : : : ; tr ). We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several gen- eral constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples. Contents 1. The r-rank variety Grass(r; V)M 4 2. Radicals and Socles 10 3. Modules of constant radical and socle rank 15 4. Modules from quantum complete intersections 20 5. Radicals of Lζ -modules 27 6. Construction of Bundles on Grass(r; V) 35 6.1. A local construction of bundles 36 6.2. A construction by equivariant descent 38 7. Bundles for GLn-equivariant modules. 43 8. A construction using the Pl¨ucker embedding 51 9. APPENDIX (by J. Carlson). Computing nonminimal 2-socle support varieties using MAGMA 57 References 59 Quillen's fundamental ideas on applying geometry to the study of group coho- mology in positive characteristic [Quillen71] opened the door to many exciting de- velopments in both cohomology and modular representation theory. Cyclic shifted subgroups, the prototypes of the rank r shifted subgroups studied in this paper, were introduced by Dade in [Dade78] and quickly became the subject of an intense study.
    [Show full text]
  • Fusion Systems and Group Actions with Abelian Isotropy Subgroups
    FUSION SYSTEMS AND GROUP ACTIONS WITH ABELIAN ISOTROPY SUBGROUPS OZG¨ UN¨ UNL¨ U¨ AND ERGUN¨ YALC¸IN Abstract. We prove that if a finite group G acts smoothly on a manifold M so that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly n1 n on M × S × · · · × S k for some positive integers n1; : : : ; nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres with trivial action on homology. 1. Introduction and statement of results Madsen-Thomas-Wall [7] proved that a finite group G acts freely on a sphere if (i) G has no subgroup isomorphic to the elementary abelian group Z=p × Z=p for any prime number p and (ii) has no subgroup isomorphic to the dihedral group D2p of order 2p for any odd prime number p. At that time the necessity of these conditions were already known: P. A. Smith [12] proved that the condition (i) is necessary and Milnor [8] showed the necessity of the condition (ii). As a generalization of the above problem we are interested in the problem of character- izing those finite groups which can act freely and smoothly on a product of k spheres for a given positive integer k. In the case k = 1, Madsen-Thomas-Wall uses surgery theory and considers the unit spheres of linear representations of subgroups to show that certain surgery obstructions vanish.
    [Show full text]
  • Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere
    Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere By Sacha Breton, B.Sc. A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of Master of Science McMaster University c Copyright by Sacha Breton, August 30th 2011 Master of Science (2011) McMaster University Hamilton, Ontario Department of Mathematics and Statistics TITLE: Finite Group Actions on the Four-Dimensional Sphere AUTHOR: Sacha Breton B.Sc. (McGill University) SUPERVISOR: Ian Hambleton NUMBER OF PAGES: 41 + v ii Abstract Smith theory provides powerful tools for understanding the geometry of singular sets of group actions on spheres. In this thesis, tools from Smith theory and spectral sequences are brought together to study the singular sets of elementary abelian groups acting locally linearly on S4. It is shown that the singular sets of such actions are homeomorphic to the singular sets of linear actions. A short review of the literature on group actions on S4 is included. iii Acknowledgements I would like to thank Professor Ian Hambleton for his suggestion of thesis topic, for the breadth and depth of his advice, and for his patience and encouragement. I would also like to thank Val´erieTremblay, Barnaby and Sue Ore, my parents, and my friends for their continued support. iv Contents Descriptive Note ii Abstract iii Acknowledgements iv 1 Introduction 1 2 Introductory Material 1 2.1 Basic definitions and conventions . .1 2.2 The Projection and the Transfer homomorphisms . .4 2.3 Smith Theory . .7 2.4 Group cohomology .
    [Show full text]
  • ELEMENTARY ABELIAN P-SUBGROUPS of ALGEBRAIC GROUPS
    ROBERT L. GRIESS, JR ELEMENTARY ABELIAN p-SUBGROUPS OF ALGEBRAIC GROUPS Dedicated to Jacques Tits for his sixtieth birthday ABSTRACT. Let ~ be an algebraically closed field and let G be a finite-dimensional algebraic group over N which is nearly simple, i.e. the connected component of the identity G O is perfect, C6(G °) = Z(G °) and G°/Z(G °) is simple. We classify maximal elementary abelian p-subgroups of G which consist of semisimple elements, i.e. for all primes p ~ char K. Call a group quasisimple if it is perfect and is simple modulo the center. Call a subset of an algebraic group total if it is in a toms; otherwise nontoral. For several quasisimple algebraic groups and p = 2, we define complexity, and give local criteria for whether an elementary abelian 2-subgroup of G is total. For all primes, we analyze the nontoral examples, include a classification of all the maximal elementary abelian p-groups, many of the nonmaximal ones, discuss their normalizers and fusion (i.e. how conjugacy classes of the ambient algebraic group meet the subgroup). For some cases, we give a very detailed discussion, e.g. p = 3 and G of type E6, E 7 and E 8. We explain how the presence of spin up and spin down elements influences the structure of projectively elementary abelian 2-groups in Spin(2n, C). Examples of an elementary abelian group which is nontoral in one algebraic group but toral in a larger one are noted. Two subsets of a maximal torus are conjugate in G iff they are conjugate in the normalizer of the torus; this observation, with our discussion of the nontoral cases, gives a detailed guide to the possibilities for the embedding of an elementary abelian p-group in G.
    [Show full text]
  • Automorphism Groups of Free Groups, Surface Groups and Free Abelian Groups
    Automorphism groups of free groups, surface groups and free abelian groups Martin R. Bridson and Karen Vogtmann The group of 2 × 2 matrices with integer entries and determinant ±1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups GL(n, Z), the groups Out(Fn) of outer automorphisms of free groups of rank n ≥ 2, and the map- ± ping class groups Mod (Sg) of orientable surfaces of genus g ≥ 1. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous, and should have many properties in common. This program is occasionally derailed by uncooperative facts but has in general proved to be a success- ful strategy, leading to fundamental discoveries about the structure of these groups. In this article we will highlight a few of the most striking similar- ities and differences between these series of groups and present some open problems motivated by this philosophy. ± Similarities among the groups Out(Fn), GL(n, Z) and Mod (Sg) begin with the fact that these are the outer automorphism groups of the most prim- itive types of torsion-free discrete groups, namely free groups, free abelian groups and the fundamental groups of closed orientable surfaces π1Sg. In the ± case of Out(Fn) and GL(n, Z) this is obvious, in the case of Mod (Sg) it is a classical theorem of Nielsen.
    [Show full text]
  • The M\" Obius Number of the Socle of Any Group
    Contents 1 Definitions and Statement of Main Result 2 2 Proof of Main Result 3 2.1 Complements ........................... 3 2.2 DirectProducts.......................... 5 2.3 The Homomorphic Images of a Product of Simple Groups . 6 2.4 The M¨obius Number of a Direct Power of an Abelian Simple Group ............................... 7 2.5 The M¨obius Number of a Direct Power of a Nonabelian Simple Group ............................... 8 2.6 ProofofTheorem1........................ 9 arXiv:1002.3503v1 [math.GR] 18 Feb 2010 1 The M¨obius Number of the Socle of any Group Kenneth M Monks Colorado State University February 18, 2010 1 Definitions and Statement of Main Result The incidence algebra of a poset P, written I (P ) , is the set of all real- valued functions on P × P that vanish for ordered pairs (x, y) with x 6≤ y. If P is finite, by appropriately labeling the rows and columns of a matrix with the elements of P , we can see the elements of I (P ) as upper-triangular matrices with zeroes in certain locations. One can prove I (P ) is a subalgebra of the matrix algebra (see for example [6]). Notice a function f ∈ I(P ) is invertible if and only if f (x, x) is nonzero for all x ∈ P , since then we have a corresponding matrix of full rank. A natural function to consider that satisfies this property is the incidence function ζP , the characteristic function of the relation ≤P . Clearly ζP is invertible by the above criterion, since x ≤ x for all x ∈ P . We define the M¨obius function µP to be the multiplicative inverse of ζP in I (P ) .
    [Show full text]
  • Arxiv:1310.6548V1 [Math.GR] 24 Oct 2013 Every Hwta H Iuto O Ocmatmnflsi Ut Di Quite Is 3
    FINITE SUBGROUPS OF DIFFEOMORPHISM GROUPS ∗ VLADIMIR L. POPOV Abstract. We prove (1) the existence of a noncompact smooth four-di- mensional manifold M such that Diff(M) contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of Diff(N) for compact manifolds N. 1. In [Po10, Sect. 2] was introduced the definition of abstract Jordan group and initiated exploration of the following two problems on automorphism groups and birational self-map groups of algebraic varieties: Problem A. Describe algebraic varieties X for which Aut(X) is Jordan. Problem B. The same with Aut(X) replaced by Bir(X). Recall from [Po10, Def. 2.1] that Jordaness of a group is defined as follows: Definition 1. A group G is called a Jordan group if there exists a positive integer dG, depending on G only, such that every finite subgroup F of G contains a normal abelian subgroup whose index in F is at most dG. Informally Jordaness means that all finite subgroups of G are “almost” abelian in the sense that they are extensions of abelian groups by finite groups taken from a finite list. Since the time of launching the exploration program of Problems A and B a rather extensive information on them has been obtained (see [Po132]), but, for instance, at this writing (October 2013) is still unknown whether there exists X with non-Jordan group Aut(X) (see Question 1 in [Po132]; it is interesting to juxtapose it with Corollary 2 below). 2. In this note we explore the counterpart of Problem A, in which algebraic varieties X are replaced by connected smooth topological manifolds M, and arXiv:1310.6548v1 [math.GR] 24 Oct 2013 Aut(X) is replaced by Diff(M), the diffeomorphism group of M.
    [Show full text]
  • 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.
    [Show full text]
  • Kernels of Linear Representations of Lie Groups, Locally Compact
    Kernels of Linear Representations of Lie Groups, Locally Compact Groups, and Pro-Lie Groups Markus Stroppel Abstract For a topological group G the intersection KOR(G) of all kernels of ordinary rep- resentations is studied. We show that KOR(G) is contained in the center of G if G is a connected pro-Lie group. The class KOR(C) is determined explicitly if C is the class CONNLIE of connected Lie groups or the class ALMCONNLIE of almost con- nected Lie groups: in both cases, it consists of all compactly generated abelian Lie groups. Every compact abelian group and every connected abelian pro-Lie group occurs as KOR(G) for some connected pro-Lie group G. However, the dimension of KOR(G) is bounded by the cardinality of the continuum if G is locally compact and connected. Examples are given to show that KOR(C) becomes complicated if C contains groups with infinitely many connected components. 1 The questions we consider and the answers that we have found In the present paper we study (Hausdorff) topological groups. Ifall else fails, we endow a group with the discrete topology. For any group G one tries, traditionally, to understand the group by means of rep- resentations as groups of matrices. To this end, one studies the continuous homomor- phisms from G to GLnC for suitable positive integers n; so-called ordinary representa- tions. This approach works perfectly for finite groups because any such group has a faithful ordinary representation but we may face difficulties for infinite groups; there do exist groups admitting no ordinary representations apart from the trivial (constant) one.
    [Show full text]
  • Free and Linear Representations of Outer Automorphism Groups of Free Groups
    Free and linear representations of outer automorphism groups of free groups Dawid Kielak Magdalen College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2012 This thesis is dedicated to Magda Acknowledgements First and foremost the author wishes to thank his supervisor, Martin R. Bridson. The author also wishes to thank the following people: his family, for their constant support; David Craven, Cornelia Drutu, Marc Lackenby, for many a helpful conversation; his office mates. Abstract For various values of n and m we investigate homomorphisms Out(Fn) ! Out(Fm) and Out(Fn) ! GLm(K); i.e. the free and linear representations of Out(Fn) respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(Fn) we prove that each ho- momorphism Out(Fn) ! GLm(K) factors through the natural map ∼ πn : Out(Fn) ! GL(H1(Fn; Z)) = GLn(Z) whenever n = 3; m < 7 and char(K) 62 f2; 3g, and whenever n + 1 n > 5; m < 2 and char(K) 62 f2; 3; : : : ; n + 1g: We also construct a new infinite family of linear representations of Out(Fn) (where n > 2), which do not factor through πn. When n is odd these have the smallest dimension among all known representations of Out(Fn) with this property. Using the above results we establish that the image of every homomor- phism Out(Fn) ! Out(Fm) is finite whenever n = 3 and n < m < 6, and n of cardinality at most 2 whenever n > 5 and n < m < 2 .
    [Show full text]