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Contributions to the Integral Representation Theory of Groups Contributions to the integral representation theory of groups Martin Hertweck Habilitationsschrift, approved by the Faculty of Mathematics and Physics University of Stuttgart, 2004 Je sais bien que le lecteur n'a pas grand besoin de savoir tout cela, mais j'ai besoin, moi, de le lui dire. Jean-Jacques Rousseau Les Confessions { Livre I, 1782 2000 Mathematics Subject Classi¯cation. 20C05 Group rings of ¯nite groups and their modules 20C07 Group rings of in¯nite groups and their modules 20C10 Integral representations of ¯nite groups 16S34 Group rings 16U60 Units, groups of units 16U70 Center, normalizer (invariant elements) Keywords and Phrases. group, automorphism, unit group, integral representation theory, isomorphism problem, normalizer problem, Zassenhaus conjecture This thesis is available as an online publication at http://elib.uni-stuttgart.de/opus Contents Notation v Zusammenfassung (German summary) vii Summary 1 I. On the isomorphism problem for integral group rings 16 1. Local{global considerations . 16 2. Mazur's construction adapted to ¯nite groups . 20 3. Semilocal analysis of the counterexample . 25 4. A group-theoretical problem related to the isomorphism problem . 33 5. Automorphisms of group rings of abelian by nilpotent groups . 38 II. On the Zassenhaus conjecture 42 6. Some general observations . 42 7. A pullback diagram for integral group rings . 48 8. Semilocal counterexamples . 59 9. Group- and character table automorphisms of (Z=rZ) S . 64 o n III. Automorphisms of integral group rings: local{global considerations 74 10. General considerations and the groups of Blanchard . 74 11. Two groups lead to global counterexamples . 78 12. : : : but not the third one . 80 13. Bicyclic units and torsion . 99 IV. Some results on speci¯c automorphism groups 102 14. Class-preserving automorphisms . 102 15. Coleman automorphisms . 105 16. Subdirect products of ¯nite groups . 112 V. On the normalizer problem for in¯nite groups 126 17. Normalizers of group bases: general coe±cients . 126 18. Normalizers of group bases: G-adapted coe±cients . 134 iii iv Contents 19. Groups satisfying the normalizer property . 136 20. Trivial central units . 142 VI. Hypercentral units in integral group rings 145 21. Hypercentral units and Q¤-groups . 145 22. Groups with nontrivial intersection of their non-normal subgroups . 147 23. The hypercenter and unipotent elements . 150 24. Subgroups of a group basis which are normal in the unit group . 154 25. Non-central elements of the hypercenter . 157 VII. Finite conjugacy for orders in division rings 160 26. The ¯nite conjugacy center and the second center . 160 27. On the ¯nite conjugacy center . 163 28. Division rings of dimension greater than 4 over the center . 165 29. Division rings of dimension 2 over the center . 168 VIII. Central units in p-blocks 173 30. On Robinson's unit . 173 Bibliography 181 Notation Abbreviations Set Theory Aut automorphism group A number of elements in set A j j Inn inner automorphism group f restriction of function f to set A jA conj conjugation det determinant Miscellaneous dim dimension N natural numbers End endomorphism ring Z rational integers id identity Z localization a=b a; b Z; p - b (p) f j 2 g Irr irreducible Zp p-adic integers ker kernel Z¼ semilocalization p ¼ Z(p) Out outer automorphism group Q rational ¯eld 2 T supp support C complex ¯eld Tr trace Fq ¯nite ¯eld with q elements Linear Algebra Mat (R) ring of n n matrices over ring R n £ GL (R) group of invertible n n matrices over R n £ SL (R) X GL (R) det(X) = 1 n f 2 n j g char.pol(M) characteristic polynomial of matrix M diag(a1; : : : ) diagonal matrix with diagonal entries a1; : : : Group Theory H G subgroup inclusion · H < G proper subgroup inclusion H E G H is normal subgroup of G H C G H is proper normal subgroup of G G : H index of H in G j j x cyclic group generated by x h i Cn a cyclic group of order n x; y; : : : group generated by the elements x; y; : : : h i G G direct product of groups 1 £ 2 N o K semidirect product of normal subgroup N with subgroup K v vi Notation G=N factor group of G by normal subgroup N CG(H) centralizer of H in G NG(H) normalizer of H in G ¼(G) set of prime divisors of G y 1 j j x conjugation y¡ xy 1 1 [x; y] commutator x¡ y¡ xy Z(G) center of G Op(G) largest normal p-subgroup of G Op0 (G) largest normal p0-subgroup of G G;¹ G;~ : : : homomorphic images of G (\bar convention") F(G) Fitting subgroup of G ¤ F (G) generalized Fitting subgroup of G ¢(G) FC-center of G ¢+(G) set of torsion elements of ¢(G) Zn(G) n-th term of upper central series of G Z (G) hypercenter i Zi(G) 1 R(G) intersection of the non-normal subgroups of G S proj lim projective limit Autc(G) group of class-preserving automorphisms of G AutCol(G) group of Coleman automorphisms of G AutR(G) group of automorphisms inducing inner automorphisms of RG PAut(G) group of power automorphisms of G AutCT(G) automorphism group of character table of G Ring Theory ¤£ group of units of ring ¤ RG group ring of group G over commutative ring R R[G] R-span of G U(RG) group of units of RG V(RG) group of units of RG of augmentation 1 IR(G) augmentation ideal of RG I (N)G kernel of natural map RG RG=N R ! M^ sum of elements of set M in group ring 1 ²N idempotent N n N n 2 ´ idempotent 1j j ² N ¡PN Autn(RG) group of augmentation-preserving automorphisms of RG Autcent(¤) group of central automorphisms of ¤ Cl(¤) locally free class group of ¤ Picent(¤) Picard group of ¤ relative to the center Zusammenfassung (German summary) Longum iter est per praecepta, breve et e±cax per exempla. Lucius Annaeus Seneca Epistulae Morales ad Lucilium { Liber VI, 62{65 \Darstellungstheorie" ist, grob gesprochen, \Modultheorie". Eine der Hauptaufgaben der ganzzahligen Darstellungstheorie sollte die Konstruktion von unzerlegbaren Gittern ubÄ er Ordnungen sein. Ein prominentes Beispiel einer Z-Ordnung ist der ganzzahlige Gruppenring ZG einer endlichen Gruppe G. Ein grundlegendes Problem (mit dem wir es hier jedoch nicht zu tun haben werden) ist, einen vollstÄandigen Satz von Invarianten (unter Isomorphie) eines ZG-Gitters M zu ¯nden welcher die Isomorphieklasse von M eindeutig bestimmt. Man kann sich ZG, oder allgemeiner die ganzzahlige Darstellungstheorie, als ein Bindeglied zwischen gewÄohnlicher und modularer Darstellungstheorie vorstellen. (Diese \Allgemeinheit" lÄa¼t bereits erkennen, dass die KlÄarung der Struktur von ZG im all- gemeinen eine delikate Aufgabe ist.) Einen Schritt weitergehend kÄonnen wir uns ganz- zahlige Darstellungstheorie, im Sinne von Curtis und Reiner [28,27], als einen zentralen Kern vorstellen, welcher verschiedene Themen in gewÄohnlicher und modularer Darstel- lungstheorie, algebraischer Zahlentheorie, und algebraischer K-Theorie verbindet. Dieser Standpunkt wird in Kapitel III veranschaulicht, wo wir anhand eines Beispiels lokal{ globale Aspekte in Bezug auf Automorphismen von ganzzahligen Gruppenringen erÄortern werden. In der Darstellungstheorie ist es ublicÄ h ubÄ er ZG-Moduln zu sprechen und dabei die ausgezeichnete Gruppenbasis G ausdrucÄ klich im Auge zu haben. (Andernfalls, was sollte es bedeuten dass M ein Permutationsmodul ist?) Wir kÄonnen jedoch die verschiedenen MÄoglichkeiten wie G, als Gruppenbasis, in ZG eingebettet werden kann in Betracht zie- hen: Dies fuhrtÄ zu Fragen ubÄ er Ringautomorphismen von ZG, von denen die sogenannte \Zassenhaus-Vermutung" die beachtenswerteste ist. Wir kÄonnen auch fragen welche Ei- genschaften einer endlichen Gruppe G durch ihre ganzzahligen Darstellungen bestimmt sind. Ob die Gruppe G bis auf Isomorphie durch ihren ganzzahligen Gruppenring be- stimmt ist, ist das sogenannte \Isomorphieproblem furÄ ganzzahlige Gruppenringe". Die- se Fragestellungen sind sicherlich im Sinne einiger wohlbekannter Probleme die Richard vii viii Zusammenfassung (German summary) Brauer [18] zur Diskussion stellte, und sie waren Ende des vergangenen Jahrhunderts der Gegenstand vieler Forschung. Wir werden die \semilokale Version" des Isomor- phieproblems und der Zassenhaus-Vermutung in den ersten beiden Kapiteln eingehend besprechen. Dabei wird der Koe±zientenring der ganzen Zahlen Z ersetzt durch eine geeignete Semilokalisation Z¼ von Z (die gleich eingefuhrtÄ wird), so dass Fragen ubÄ er lokal freie Klassengruppen vermieden werden. Kenntnis der p-adischen Gruppenringe ZpG gibt Einblick wie G auf abelschen Gruppen operieren kann. Da die interessantesten arithmetischen Eigenschaften beim UbÄ ergang von ZG zu einer maximalen UbÄ erordnung in QG verloren gehen, sind wir versucht, die Semilokalisation Z¼ := Z(p) p G \jj j als angemessenen \ganzzahligen Koe±zientenring mit Bezug auf G" anzusehen. Dieser Ring ist \komfortabler" als Z da er nur endlich viele maximale Ideale besitzt, und das- selbe gilt furÄ den Gruppenring Z¼G. Dennoch hat Z¼G alle interessanten Quotienten: Zu einer die Ordnung von G teilenden Primzahl p, und einer naturlicÄ hen Zahl n, haben wir kanonische Ringhomomorphismen ZG , Z G ³ (Z=pnZ)G. ! ¼ Vorausgesetzt man interessiert sich furÄ die Eigenschaften von G welche durch die Modulkategorie ZG Mod bestimmt sind, ist dies sogar der bessere Rahmen, denn die folgenden Aussagen sind aquivÄ alent: Es gibt eine AquivÄ alenz ZG Mod ZH Mod von ' Modulkategorien; es gibt einen Isomorphismus Z¼G »= Z¼H von Ringen; es gibt eine AquivÄ alenz Z G Mod Z H Mod. ¼ ' ¼ In Kapitel III zeigen wir furÄ eine Gruppe G der Ordnung 96, dass der semilokale Gruppenring Z¼G einen Automorphismus besitzt, welcher mit keinem Automorphismus von ZG bis auf einen inneren Automorphismus von QG ubÄ ereinstimmt. Gewisse Gruppenautomorphismen treten in naturlicÄ her Weise bei Untersuchungen zur Zassenhaus-Vermutung und des Isomorphieproblems auf. Klassenerhaltende Au- tomorphismen zu studieren wurde neu motiviert durch Arbeiten von Roggenkamp und Kimmerle, die diese Automorphismen in Beziehung zur Zassenhaus-Vermutung setzten (Untersuchungen in diese Richtung begannen in [117]), und eine Beobachtung von Mazur verknupftÄ das Isomorphieproblem furÄ ganzzahlige Gruppenringe mit der Existenz be- stimmter nicht innerer klassenerhaltender Automorphismen, nÄamlich jenen welche innere Automorphismen des Gruppenrings induzieren.
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