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Character Correspondences and π-Special Characters in π-Separable Groups

Published online by Cambridge University Press:  20 November 2018

Thomas R. Wolf
Thomas R. Wolf*
Affiliation:
Ohio University, Athens, Ohio
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Let π be a set of primes and let G be a π-separable group (all groups considered are finite). Two subsets Xπ(G) and Bπ(G) of the set Irr(G) of irreducible characters of G play an important role in the character theory of π-separable groups and particularly solvable groups. If p is prime and π is the set of all other primes, then the Bπ characters of G give a natural one-to-one lift of the Brauer characters of G into Irr(G). More generally, they have been used to define Brauer characters for sets of primes.

The π-special characters of G (i.e., Xπ(G)) restrict irreducibly and in a one-to-one fashion to a Hall-π-subgroup of G. If an irreducible character χ is quasi-primitive, it factors uniquely as a product of a π-special character an a π′-special character. This is a particularly useful tool in solvable groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 920 - 937
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Gajendragadkar, D., A characteristic class of finite π-separable groups, J. of Algebra 59 (1979), 237–259.Google Scholar
2. Iizuka, K., On the blocks and sections of finite groups, Kumamoto J. Sci. A5 (1960), 53.Google Scholar
3. Isaacs, I. M., Lifting Brauer characters of p-solvable groups, Pacific J. of Math. 53 (1974). 171–188.Google Scholar
4. Isaacs, I. M., Character theory of finite groups (Academic Press, New York, 1976).Google Scholar
5. Isaacs, I. M., Lifting Brauer characters of p-solvable groups II, J of Algebra 51 (1978), 476–490.Google Scholar
6. Isaacs, I. M., Characters of π-separable groups, J. of Algebra, to appear.Google Scholar
7. Robinson, G., Some topics in character theory, Master's thesis, Oxford (1977).Google Scholar
8. Slattery, M., π-blocks of π-separable groups, Ph.D. Thesis, University of Wisconsin-Madison (1983).Google Scholar
9. Slattery, M., π-blocks of π-separable groups I, J. of Algebra, to appear.CrossRefGoogle Scholar
10. Uno, K., Character correspondences in p-solvable groups, Osaka J. Math. 20 (1983), 713–725.Google Scholar
11. Wolf, T. R., Character correspondences in solvable groups, Illinois J. Math. 22 (1978), 327–340.Google Scholar
12. Wolf, T. R., Character correspondences induced by subgroups of operator groups, J. of Algebra 57 (1979), 502–521.Google Scholar