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Fixed Points and Characters in Groups with Non-Coprime Operator Groups

Published online by Cambridge University Press:  20 November 2018

I. M. Isaacs
I. M. Isaacs*
Affiliation:
University of Chicago, Chicago, Illinois
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Let A and H be finite groups with A acting on H, i.e., there is a given, fixed homomorphism A → Aut(H). In this situation, A acts on the set of conjugacy classes of H and also on the set of irreducible characters of H. If A is cyclic, it follows from a lemma of Brauer (see, for instance, 1, 12.1) that the number of fixed points in these two actions are equal and therefore one can conclude that for any A, the number of orbits in the two actions are equal.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1315 - 1320
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Feit, W., Characters of finite groups (Benjamin, New York, 1967).Google Scholar
2. Glauberman, G., Fixed points in groups with operator groups, Math. Z. 84 (1964), 120–125.Google Scholar
3. Isaacs, I. M.. Extensions of certain linear groups, J. Algebra 4 (1966), 3–12.Google Scholar
4. Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, N. J., 1964).Google Scholar