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Correspondences of Characters for Relatively Prime Operator Groups

Published online by Cambridge University Press:  20 November 2018

George Glauberman
George Glauberman*
Affiliation:
University of Chicago, Chicago, Illinois
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Let G be a finite group and let A be a finite solvable operator group on G. Suppose that A and G have relatively prime orders. Let T be the fixed-point subgroup of G with respect to A. We say that A fixes a complex character ζ of G if ζ (gα) = ζ (g) for all g ∈ G and α ϵ A. Our aim in this paper is to define a one-to-one correspondence between the irreducible characters of T and those irreducible characters of G that are fixed by A, and to prove some properties of this correspondence that were mentioned in (8).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1465 - 1488
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Burnside, W., Theory of groups of finite order, 2nd ed. (Dover, New York, 1955).Google Scholar
2. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Wiley, New York, 1962).Google Scholar
3. Dixon, J. D., The fitting subgroup of a linear solvable group, J. Austral. Math. Soc. 7 (1967), 417–428.Google Scholar
4. Dixon, J. D., Normal p-subgroups of solvable linear groups, J. Austral. Math. Soc. 7 (1967), 545–551.Google Scholar
5. Feit, W., Characters of finite groups (Mimeographed notes, Yale University, 1965).Google Scholar
6. Feit, W. and Thompson, J. G., Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029.Google Scholar
7. Gallagher, P. X., Group characters and normal Hall subgroups, Nagoya Math. J. 21 (1962), 223–230.Google Scholar
8. Glauberman, G., Correspondence of characters in relatively prime automorphism groups, Notices Amer. Math. Soc. 11 (1964), 128–129.Google Scholar
9. Glauberman, G., Fixed points in groups with operator groups, Math. Z. 84 (1964), 120–125.Google Scholar
10. Hall, M., Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
11. Hall, P. and Higman, G., On the p-length of p-solvable groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. (3) 6 (1956), 1–42.Google Scholar
12. Schur, I., Uber eine Klasse von endlichen Gruppen linearer Substitutionen, S.-B. Preuss. Akad. Wiss. (1905), 77–91.Google Scholar
13. Speiser, A., Die Théorie der Gruppen von endlicher Ordnungt 4th éd. (Birkhâuser, Basel, Stuttgart, 1956).10.1007/978-3-0348-4153-5CrossRefGoogle Scholar
14. Thompson, J. G., Automorphisms of solvable groups, J. Algebra 1 (1964), 259–267.Google Scholar
15. van der, B. L. Waerden, Modern algebra, Volume I (Ungar, New York, 1953).Google Scholar
16. Zassenhaus, H., The theory of groups, 2nd Engl. ed. (Chelsea, New York, 1958).Google Scholar