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A Proof of Brauer’s Theorem on Generalized Decomposition Numbers

Published online by Cambridge University Press:  22 January 2016

Hirosi Nagao
Hirosi Nagao*
Affiliation:
Osaka City University
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In [3] R. Brauer gave a proof of his theorem on generalized decomposition numbers which was first announced in [1], and a simplification of it has been made by K. Iizuka [5]. In this note we shall show that the theorem may be proved from another point of view by using some results obtained by J. A. Green in [4].

After stating some results by Green and Osima in the first and second sections we first prove a theorem on characters (Theorem 1) and by using the theorem we prove Brauer’s theorem in the fourth section.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 22 , June 1963 , pp. 73 - 77
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

[1] Brauer, R., On blocks of characters of groups of finite order, II, Proc. Nat. Acad. Sci. U.S.A. 32, 215–219 (1946).CrossRefGoogle ScholarPubMed
[2] Brauer, R., Zur Darstellunstheorie der Gruppen endlicher Ordnung, Math. Z. 63, 406–444 (1956).CrossRefGoogle Scholar
[3] Brauer, R., Zur Darstellugstherie der Gruppen endlicher Ordnung, II, Math. Z. 72, 25–46 (1959).Google Scholar
[4] Green, J. A., Brocks of modular representations, Math. Z. 79, 100–115 (1962).Google Scholar
[5] Iizuka, K., On Brauer’s theorem on sections in the theory of blocks of group characters, Math. Z. 75, 299–304 (1961).Google Scholar
[6] Osima, M., Notes on blocks of group characters, Math. J. Okayama Univ. 4, 175–188 (1955).Google Scholar