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Common Transversals

Published online by Cambridge University Press:  20 January 2009

R. A. Rankin
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A well-known theorem in group theory [(8), p. 11, Satz 3] asserts that, when H is a subgroup of finite index in a group G, there exists a system of common representatives of the right cosets and the left cosets of H in G. Various proofs and generalisations, mainly involving combinatorial rather than grouptheoretical ideas, are known, and an excellent account of the subject is to be found in Chapter 5 of Ryser's book (6), where references to the literature are given. The purpose of the present paper is to use group-theoretical ideas to prove theorems of a similar nature. The motivation for this work comes from the theory of Hecke operators, and one of the main objects is to provide a simple proof of a result given by Petersson (4), which is needed in order to prove the normality of these operators.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 2 , December 1966 , pp. 147 - 154
Copyright
Copyright © Edinburgh Mathematical Society 1966

References

REFERENCES

(1) Hall, M., The Theory of Groups (New York, 1959).Google Scholar
(2) Macduffee, C. C., The Theory of Matrices (New York, 1946).Google Scholar
(3) Newman, M. and Smart, J. R., Modulary groups of t х t matrices, Duke Math. J. 30 (1963), 253–257.CrossRefGoogle Scholar
(4) Petersson, H., Konstruktion der sämtlichen Lösungen einer Riemannschen Funktionalgleichung durch Dirichlet-Reihen mit Eulerscher Produktentwicklung II, Math. Ann. 177 (1939), 39–64.Google Scholar
(5) Petersson, H., Über die Kongruenzgruppen der Stufe 4, J. Reine Angew. Math. 212 (1963), 63–72.CrossRefGoogle Scholar
(6) Ryser, H. J., Combinatorial Mathematics, Cams Mathematical Monographs, No. 14 (Menasha, Wisconsin, 1963).CrossRefGoogle Scholar
(7) Vivanti, G., Les Fonctions Polyédriques et Modulaires (Paris, 1910).Google Scholar
(8) Zassenhaus, H., Lehrbuch der Gruppentheorie (Leipzig, 1937).Google Scholar
(9) Mirsky, L. and Perfect, H., Systems of representatives, J. Math. Anal. Appl. 15 (1966), 520–569.CrossRefGoogle Scholar
(10) Ore, O., On coset representations in groups, Proc. Amer. Math. Soc. 9 (1958), 665–670.CrossRefGoogle Scholar