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A Remark on Orthogonality Relations in Finite Groups

Published online by Cambridge University Press:  22 January 2016

Kenzo Iizuka  and
Tadasi Nakayama
Kenzo Iizuka
Affiliation:
Tôhoku University and Kumamoto University
Tadasi Nakayama
Affiliation:
Nagoya University
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On the basis of Prof. R. Brauer’s fundamental work, certain orthogonality relations for characters of finite groups have recently been studied by Brauer himself, M. Osima, and one of the present writers; see Iizuka [7] and the references there. In the present short note some general remarks on orthogonality relations, dealing with “blocks” and “sections” of general type, are given first. They are of elementary, and often formal, nature and their proofs are merely combinations of known arguments. So, no deep significance is claimed on them, in comparison with the above alluded results based on deeper arithmetico-group-theoretical considerations. However, applied to blocks and sections of such deeper nature, our remarks give some rather useful informations on them. Thus, for instance, the “maximality” feature of 77-blocks is given a formulation (Prop. 5 below) finer than the one given in [71 Further, some new types of blocks and sections can be constructed, again in application of our remarks to such classical ones. These new blocks and sections give thus new orthogonality relations and we hope that some of them may turn to have some significance. There arize also several problems, which are stated at the end of the present note and to some of which we wish to come back elsewhere.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 20 , June 1962 , pp. 185 - 194
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1962

References

[1] Brauer, R., Zur Darstellungstheorie der Gruppen endlicher Ordnung, II, Math. Zeits. 72 (1959), 25–46.CrossRefGoogle Scholar
[2] Brauer, R. and Feit, W., On the number of irreducible characters of finite groups in a given block, Proc. Nat. Acad. Sci. USA 45 (1959), 361–365.CrossRefGoogle Scholar
[3] Brauer, R.-Robinson, G. de B., On a conjecture by Nakayama, Trans. Roy. Soc. Canada 3. S. 41 (1947), Sec. Ill, 11.Google Scholar
[4] Brauer, R.-Tuan, B. F., On simple groups of finite order, I, Bull. Amer. Math. Soc. 51 (1945), 756–766.CrossRefGoogle Scholar
[5] Iizuka, K., On the blocks and the sections of finite groups, Kumamoto J. of Sci. 5 (1960), 53–62.Google Scholar
[6] Iizuka, K., On Brauer’s theorem on sections in the theory of blocks of group characters, Math. Zeits. 75 (1961), 299–304.CrossRefGoogle Scholar
[7] Iizuka, K., Some studies on the orthogonality relations for group characters, Kumamoto J. of Sci. 5 (1960), 111–118.Google Scholar
[8] Osima, M., On the representations of groups of finite order, Math. J. Okayama Univ. 1 (1952), 33–61.Google Scholar
[9] Osima, M., Notes on blocks of group characters, Math. J. Okayama Univ. 4 (1955), 175–188.Google Scholar
[10] Osima, M., On some properties of group characters, Proc. Japan Acad. 36 (1960), 18–21.Google Scholar
[11] Roquette, P., Arithmetische Untersuchung des Charakterringes einer endlichen Gruppe, Jour, f.d. r.u.a. Math. 190 (1952), 148–168.Google Scholar