Hostname: page-component-84b7d79bbc-lrf7s Total loading time: 0 Render date: 2024-07-31T07:05:54.825Z Has data issue: false hasContentIssue false
  • English
  • Français

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

Published online by Cambridge University Press:  20 November 2018

I. M. Isaacs
I. M. Isaacs*
Affiliation:
University of Wisconsin, Madison, Wisconsin
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The main result of this paper is the following:

Theorem A. Let H and N be finite groups with coprime orders andsuppose that H acts nontrivially on N via automorphisms. Assume that Hfixes every nonlinear irreducible character of N. Then the derived subgroup ofN is nilpotent and so N is solvable of nilpotent length≦ 2.

Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group Gfrom a knowledge of the set

cd(G) = ﹛x(l)lx ∈ Irr(G) ﹜

of irreducible character degrees of G.Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 1 , 01 February 1989 , pp. 68 - 82
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Camina, A.R., Some conditions which almost characterize Frobenius Groups, Israel J. of Math. 31 (1978), 153–160.Google Scholar
2. Chillag, David and Macdonald, I.D., Generalized Frobenius groups, Israel J. of Math. 47(1984), 111–122.Google Scholar
3. Glauberman, G., Correspondence of characters for relatively prime operator groups, Can. J.Math. 20 (1968), 1465–1488.Google Scholar
4. Higman, G., Suzuki 2-groups, Illinois J. of Math. 7 (1963), 79–96.Google Scholar
5. Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin-Heidelberg-New York, 1967).Google Scholar
6. Isaacs, I.M. and Passman, D.S., Groups with relatively few nonlinear irreducible characters, Can. J. Math. 20 (1968), 1451–1458.Google Scholar
7. Isaacs, I.M., Character theory of finite groups (Academic Press, New York, 1976).Google Scholar
8. Isaacs, I.M., Character correspondences in solvable groups, Advances in Math. 43 (1982), 284–306.Google Scholar
9. Manz, O. and Staszewski, R., Some applications of a fundamental theorem by Gluck and Wolf in the character theory of finite groups, Math. Zeit. 192 (1986), 383–389.Google Scholar