Gabriela Olteanu WEDDERBURN DECOMPOSITION OF GROUP ALGEBRAS AND APPLICATIONS Gabriela Olteanu Wedderburn Decomposition of Group Algebras and Applications EDITURA FUNDAT¸IEI PENTRU STUDII EUROPENE Cluj-Napoca 2008 Editura Fundat¸iei pentru Studii Europene Str. Emanuel de Martonne, Nr. 1 Cluj–Napoca, Romania Director: Ion Cuceu c 2008 Gabriela Olteanu Descrierea CIP a Bibliotecii Nat¸ionale a Romˆaniei OLTEANU, GABRIELA Wedderburn decomposition of group algebras and applications / Gabriela Olteanu. - Cluj-Napoca : Editura Fundat¸iei pentru Studii Europene, 2008 Bibliogr. Index ISBN 978-973-7677-98-3 519.6 To my family Contents Preface 1 Notation 13 1 Preliminaries 15 1.1 Number fields and orders . 15 1.2 Group algebras and representations . 20 1.3 Units . 26 1.4 Crossed products . 29 1.5 Brauer groups . 32 1.6 Local fields . 43 1.7 Simple algebras over local fields . 49 1.8 Simple algebras over number fields . 52 1.9 Schur groups . 57 2 Wedderburn decomposition of group algebras 63 2.1 Strongly monomial characters . 64 2.2 An algorithmic approach of the Brauer–Witt Theorem . 68 2.3 A theoretical algorithm . 75 3 Implementation: the GAP package wedderga 79 3.1 A working algorithm . 80 3.2 Examples . 86 4 Group algebras of Kleinian type 93 4.1 Schur algebras of Kleinian type . 96 4.2 Group algebras of Kleinian type . 100 4.3 Groups of units . 106 i ii CONTENTS 5 The Schur group of an abelian number field 113 5.1 Factor set calculations . 114 5.2 Local index computations . 123 5.3 Examples and applications . 130 6 Cyclic cyclotomic algebras 133 6.1 Ring isomorphism of cyclic cyclotomic algebras . 133 6.2 The subgroup generated by cyclic cyclotomic algebras ............. 138 Conclusions and perspectives 151 Bibliography 153 Index 163 Preface Group rings are algebraic structures that have attracted the attention of many mathe- maticians since they combine properties of both groups and rings and have applications in many areas of Mathematics. Their study often requires techniques from Representa- tion Theory, Group Theory, Ring Theory and Number Theory and, in some cases, the use of properties of central simple algebras or local methods. By the Maschke Theo- rem, if G is a finite group and F is a field of characteristic not dividing the order of the group G, then the group algebra FG is semisimple artinian. In this case, the structure of FG is quite easy, but the explicit computation of the Wedderburn decomposition of the group algebra knowing the group G and the field F is not always an easy prob- lem. On the other hand, the explicit knowledge of the Wedderburn decomposition has applications to different problems. The Wedderburn decomposition of a semisimple group algebra FG is the decom- position of FG as a direct sum of simple algebras, that is, minimal two-sided ideals. Our main motivation for the study of the Wedderburn decomposition of group algebras is given by its applications. The main applications that we are interested in are the study of the groups of units of group rings with coefficients of arithmetic type and of the Schur groups of abelian number fields. Other applications of the Wedderburn decomposition that, even if not extensively studied in this book we have had in mind during its preparation, are the study of the automorphism group of group algebras, of the Isomorphism Problem for group algebras and of the error correcting codes with ideal structure in a finite group algebra, known as group codes. We start by presenting with more details the first application, which is the com- putation of units of group rings relying on the Wedderburn decomposition of group algebras. It is well known that the integral group ring ZG is a Z-order in the rational group algebra QG and it has been shown that a good knowledge and understanding of QG is an essential tool for the study of U(ZG). For example, some results of E. Jespers and G. Leal and of J. Ritter and S.K. Sehgal [JL, RitS2] show that, under some 1 2 PREFACE hypotheses, the Bass cyclic units and the bicyclic units (see Section 1.3. for definitions) generate a subgroup of finite index in U(ZG). These hypotheses are usually expressed in terms of the Wedderburn decomposition of the rational group algebra QG. Their theorems were later used by E. Jespers, G. Leal and C. Polcino Milies [JLPo] to char- acterize the groups G that are a semidirect product of a cyclic normal subgroup and a subgroup of order 2 such that the Bass cyclic units and the bicyclic units generate a subgroup of finite index in U(ZG). This latter characterization also had as starting point the computation of the Wedderburn decomposition of QG for these groups. As a consequence of a result of B. Hartley and P.F. Pickel [HP], if the finite group G is neither abelian nor isomorphic to Q8 × A, for Q8 the quaternion group with 8 elements and A an elementary abelian 2-group, then U(ZG) contains a non-abelian free group. The finite groups for which U(ZG) has a non-abelian free subgroup of finite index were characterized by E. Jespers [Jes]. Furthermore, the finite groups such that U(ZG) has a subgroup of finite index which is a direct product of free groups were classified by E. Jespers, G. Leal and A.´ del R´ıoin a series of articles [JLdR, JL, LdR]. In order to obtain the classification, they used the characterization of these groups in terms of the Wedderburn components of the corresponding rational group algebra. Furthermore, for every such group G, M. Ruiz and A.´ del R´ıoexplicitly constructed a subgroup of U(ZG) that had the desired structure and minimal index among the ones that are products of free groups [dRR]. Again, a fundamental step in the used arguments is based on the knowledge of the Wedderburn decomposition of the rational group algebra. The use of the methods of Kleinian groups in the study of the groups of units was started by M. Ruiz [Rui], A. Pita, A.´ del R´ıo and M. Ruiz [PdRR] and led to the notion of algebra of Kleinian type and of finite group of Kleinian type. The classification of the finite groups of Kleinian type has been done by E. Jespers, A. Pita, A.´ del R´ıo and P. Zalesski, by using again useful information on the Wedderburn components of the corresponding rational group algebras [JPdRRZ]. One of the first applications of the present book is a generalization of these results, obtaining a classification of the group algebras of Kleinian type of finite groups over number fields [OdR3]. This is explained in detail in Chapter 4 of the present book, dedicated to the applications of the Wedderburn decomposition to the study of the group algebras of Kleinian type. Another important application is to the study of the automorphism group of a semisimple group algebra. The automorphism group of a semisimple algebra can be computed by using the automorphism groups of the simple components of its Wedder- burn decomposition. By the Skolem–Noether Theorem, the automorphism group of PREFACE 3 every simple component S can be determined by using the automorphism group of the center of S and the group of inner automorphisms of S. These ideas were developed by S. Coelho, E. Jespers, C. Polcino Milies, A. Herman, A. Olivieri, A.´ del R´ıoand J.J. Sim´onin a series of articles [CJP, Her3, OdRS2], where the automorphism group of group algebras of finite groups with rational coefficients is studied. The same type of considerations shows that the Isomorphism Problem for semisimple group algebras can be reduced to the computation of the Wedderburn decomposition of such algebras and to the study of the existence of isomorphisms between the simple components. On the other hand, the knowledge of the Wedderburn decomposition of a group algebra FG allows one to compute explicitly all two-sided ideals of FG. This has direct applications to the study of error correcting codes, in the case when F is a finite field, since the majority of the most used codes in practice are ideals of group rings. For example, this is the case of cyclic codes, which are exactly the ideals of the group algebras of cyclic groups [PH]. In the last years, some authors have investigated families of group codes having in mind the applications to Coding Theory (see for the example the survey [KS]). The problem of computing the Wedderburn decomposition of a group algebra FG naturally leads to the problem of computing the primitive central idempotents of FG. The classical method used to do this starts by calculating the primitive central idempo- tents e(χ) of CG associated to the irreducible characters of G, for which there is a well known formula, and continues by summing up all the primitive central idempotents of the form e(σ ◦ χ) with σ ∈ Gal(F (χ)/F ) (see for example Proposition 1.24, Section 1.2). An alternative method to compute the primitive central idempotents of QG, for G a finite nilpotent group, that does not use the character table of G has been intro- duced by E. Jespers, G. Leal and A. Paques [JLPa]. A. Olivieri, A.´ del R´ıoand J.J. Sim´on[OdRS1] pointed out that this method relies on the fact that nilpotent groups are monomial and, using a theorem of Shoda [Sho], they gave an alternative presentation. In this way, the method that shows how to produce the primitive central idempotents of QG, for G a finite monomial group, depends on certain pairs of subgroups (H, K) of G and it was simplified in [OdRS1].