A Constructive Brauer-Witt Theorem for Certain Solvable Groups

Herman, Allen

Canadian journal of mathematics, Tome 48 (1996) no. 6, pp. 1196-1209

Résumé

Division algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

Détail

DOI : 10.4153/CJM-1996-063-1

Mots-clés : 20C15, 16S34

Bibliographie

[C] [C] -Martin Cram, G., On the field of character values of finite solvable groups, Arch. Math. 51(1988), 294–296. Google Scholar

[CR] [CR] Curtis, Charles W. and Reiner, Irving, Methods of Representation Theory: With Applications to Finite Groups and Orders, Wiley-Interscience, New York, 1981. Google Scholar

[F] [F] Fontaine, Jean-Marc, Surla decomposition des algebres de groups, Ann. Scient. Ec. Norm. Sup. (4)4(1971), 121–180. Google Scholar

[G] [G] Gallagher, P. X., Group characters and normal Hall subgroups, Nagoya Math J. 21(1962), 223–230. Google Scholar

[Her] [Her] Herman, Allen, Metabelian groups and the Brauer-Witt theorem, Comm. Alg. (11)23(1995), 4073-4086. Google Scholar

[Hup] [Hup] Huppert, Bertram, Enliche Gruppen I, Springer-Verlag, Berlin, 1967. Google Scholar

[I] [I] Isaacs, I. M., The Character Theory of Finite Groups, Academic Press, New York, 1976. Google Scholar

[I2] [I2] Isaacs, I. M., Extensions of group representations over arbitrary fields, J. Algebra 68(1981), 54–74. Google Scholar

[J] [J] Janusz, G. J., Generators for the Schur group of local and global number fields, Pacific J. Math. 56(1975), 525–546. Google Scholar

[J2] [J2] Janusz, G. J., Some remarks on Clifford's Theorem and the Schur index, Pacific J. Math. 32(1970), 119–125. Google Scholar

[L] [L] Lorenz, Falko, Über die Berechnung der Schurshen indizes von Charakteren endlicher Gruppen, J. Number Theory 3(1971), 60–103. Google Scholar

[MW] [MW] Manz, Olaf and Wolf, Thomas R., Representations of Solvable Groups, London Mathematical Society Lecture Note Series Cambridge University Press, Cambridge, 1993. Google Scholar

[P] [P] Passman, D. S., The Algebraic Structure of Group Rings, Wiley and Sons, New York, 1977. Google Scholar

[Roq] [Roq] Roquette, P., Arithmetische Untersuchung des Charakterringes einer endlichen Gruppen, J. Reine Angew. Math. 190(1952), 148–168. Google Scholar

[Rit] [Rit] Ritter, Jürgen, Über eine Erweiterung der Witschen Reduktion bei der Bestimmung der Schurindizes von endlichen Gruppen über lohalen Körpen, Arch. Math. 29(1977), 78–91. Google Scholar

[Sch] [Sch] Schmidt, Peter, Schur indices and Schur groups, J. Algebra 169(1994), 226–247. Google Scholar

[Sc] [Sc] Scott, W. R., Group Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. Google Scholar

[Sh] [Sh] Shirvani, M., The structure of simple rings generated by finite metabelian groups, J. Algebra 169(1994), 686–712. Google Scholar

[W] [W] Witt, E., Die algebraische Struktur des Gruppenringes einer endlichen Gruppe über einem Zahlkörper, J. Reine Angew. Math. 190(1952), 231–245. Google Scholar

[Y] [Y] Yamada, T., The Schur Subgroup of the Brauer Group, Lecture Notes in Mathematics Springer-Verlag, Berlin, 1974. Google Scholar