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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
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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

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