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A Constructive Brauer-Witt Theorem for Certain Solvable Groups

Published online by Cambridge University Press:  20 November 2018

Allen Herman
Allen Herman*
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2 e-mail: aherman@math. uregina. ca
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Abstract

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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.

Keywords

20C1516S34
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 6 , 01 December 1996 , pp. 1196 - 1209
Copyright
Copyright © Canadian Mathematical Society 1996

References

[C] -Martin Cram, G., On the field of character values of finite solvable groups, Arch. Math. 51(1988), 294–296.Google Scholar
[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] Fontaine, Jean-Marc, Surla decomposition des algebres de groups, Ann. Scient. Ec. Norm. Sup. (4)4(1971), 121–180.Google Scholar
[G] Gallagher, P. X., Group characters and normal Hall subgroups, Nagoya Math J. 21(1962), 223–230.Google Scholar
[Her] Herman, Allen, Metabelian groups and the Brauer-Witt theorem, Comm. Alg. (11)23(1995), 4073-4086.Google Scholar
[Hup] Huppert, Bertram, Enliche Gruppen I, Springer-Verlag, Berlin, 1967.Google Scholar
[I] Isaacs, I. M., The Character Theory of Finite Groups, Academic Press, New York, 1976.Google Scholar
[I2] Isaacs, I. M., Extensions of group representations over arbitrary fields, J. Algebra 68(1981), 54–74.Google Scholar
[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] Janusz, G. J., Some remarks on Clifford's Theorem and the Schur index, Pacific J. Math. 32(1970), 119–125.Google Scholar
[L] Lorenz, Falko, Über die Berechnung der Schurshen indizes von Charakteren endlicher Gruppen, J. Number Theory 3(1971), 60–103.Google Scholar
[MW] Manz, Olaf and Wolf, Thomas R., Representations of Solvable Groups, London Mathematical Society Lecture Note Series 185, Cambridge University Press, Cambridge, 1993.Google Scholar
[P] Passman, D. S., The Algebraic Structure of Group Rings, Wiley and Sons, New York, 1977.Google Scholar
[Roq] Roquette, P., Arithmetische Untersuchung des Charakterringes einer endlichen Gruppen, J. Reine Angew. Math. 190(1952), 148–168.Google Scholar
[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] Schmidt, Peter, Schur indices and Schur groups, J. Algebra 169(1994), 226–247.Google Scholar
[Sc] Scott, W. R., Group Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.Google Scholar
[Sh] Shirvani, M., The structure of simple rings generated by finite metabelian groups, J. Algebra 169(1994), 686–712.Google Scholar
[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] Yamada, T., The Schur Subgroup of the Brauer Group, Lecture Notes in Mathematics 397, Springer-Verlag, Berlin, 1974.Google Scholar