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Monomial Representations and Metabelian Groups1)

Published online by Cambridge University Press:  22 January 2016

B.G. Basmaji
B.G. Basmaji*
Affiliation:
University of Nebraska
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1. Introduction. In this paper we develop a method to find all the irreducible and inequivalent representations over the complex field C of a family of finite groups that includes the metabelian groups. The outline of the paper is as follows: In §2 we let P be a one-dimensional representation of a subgroup H of a finite group G and find a maximal subgroup K, H ⊆ K ⊆ G, such that an extension P̅ of P to K exists. We show that the induced representation P̅G is irreducible if K is normal in G. In §3 we give all the irreducible and inequivalent representations of the “generalized metabelian group” G, and in particular of the metabelian group, and decompose the group ring CG into its simple components. The convenience of this method is shown in §4 where we determine the representations of the metacyclic group and a metabelian group of order 2sp2, p an odd prime and s|p — l. The algorithm in §5 is supplementary and can be used to find representations of more general groups than the two in §4.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 35 , July 1969 , pp. 99 - 107
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1969

Footnotes

1)

This paper is essentially Chapter III of the author’s dissertation submitted to the University of Maryland for the Ph.D. degree. The author wishes to thank Professor Kuroda for his advice. Also, this research was partially supported by NSF grants GP-1837 and GP-6010.

References

[1] Berman, S. On the representations of the semi-direct product of two abelian groups. Dokl. Akad, Nauk. 98 (1954), 177–180.Google Scholar
[2] Blackburn, N. On prime power groups in which the derived group has two generators. Proc. Cambridge Philo. Soc. 23 (1957), 19–27.CrossRefGoogle Scholar
[3] Curtis, C. and Reiner, I. “Representation Theory of Finite Groups and Associative Algebras.” Interscience, New York, 1962.Google Scholar
[4] Itô, N. On monomial representations of finite groups. Osaka Math. J. 6 (1954), 119–127.Google Scholar
[5] Munkholm, H. Induced monomial representations, young elements, and metacyclic groups. Proc. Amer. Math. Soc. 19 (1968), 453–458.Google Scholar
[6] Shoda, K. Über die monomialen Darstellungen einer endlichen Gruppe. Proc. Phys. Math. Japan 15 (1933), 249–257.Google Scholar