Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-02T23:46:05.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Character Degrees and Derived Length of a Solvable Group

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.

Let G be a finite group. (All groups considered here are finite). There exist several results which control the structure of G in terms of cd(G), the set of degrees of the irreducible complex characters of G. Here, we are concerned with the situation where only the cardinality of cd(G) is given. If |cd(G)| ≦ 3,, then it is known [9 ; 7] that G is solvable and the derived length dl (G) ≦ cd (G) |., If |cd(G)| = 4, then G need not be solvable (e.g., G = PSL(2, 2n))\ however [5], if G is solvable then dl(G) ≦4.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 1 , 01 February 1975 , pp. 146 - 151
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Dixon, J. D., The structure oj'linear groups (Van Nostrand-Reinhold, London, 1971).Google Scholar
2. Dornhoff, L., Group representation theory (Marcel Dekker, New York, 1972).Google Scholar
3. Huppert, B., Lineare auflôsbare Gruppen, Math. Z. 67 (1957), 479–518.Google Scholar
4. Huppert, B., Endliche Gruppen. I (Springer-Verlag, Berlin, 1967).Google Scholar
5. Garrison, S., On groups with a small number of character degrees, Ph.D. Thesis, Univ. of Wisconsin, 1973.Google Scholar
6. Isaacs, I. M., Extensions of group representations over nonalgebraically closed fields, Trans. Amer. Math. Soc. lp (1969), 211–228.Google Scholar
7. Isaacs, I. M., Groups having at most three irreducible character degrees, Proc. Amer. Math. Soc. 21 (1969), 185–188.Google Scholar
8. Isaacs, I. M. and Passman, D. S., Groups with representations of bounded degree, Can. J. of Math. 16 (1964), 299–309.Google Scholar
9. Isaacs, I. M. and Passman, D. S., A characterization of groups in terms of the degrees of their characters. II, Pacific J. of Math. 24 (1968), 487–510.Google Scholar