Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Derived lengths and character degrees

Author: Mark L. Lewis
Journal: Proc. Amer. Math. Soc. 126 (1998), 1915-1921
MSC (1991): Primary 20C15
MathSciNet review: 1452810
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a finite solvable group. Assume that the degree graph of $G$ has exactly two connected components that do not contain $1$. Suppose that one of these connected components contains the subset $\{ a_{1}, \dots , a_{n} \}$, where $a_{i}$ and $a_{j}$ are coprime when $i \not = j$. Then the derived length of $G$ is less than or equal to $|\operatorname{cd}(G)|-n+1$.

References [Enhancements On Off] (What's this?)

  • [1] P. X. Gallagher, Group characters and normal Hall subgroups, Nagoya Math. J. 21 (1962), 223-230. MR 26:240
  • [2] S. C. Garrison, On groups with a small number of character degrees, Ph.D. Thesis, Univ. of Wisconsin, Madison, 1973.
  • [3] B. Huppert, Endlichen gruppen I, Springer-Verlag, Berlin, 1967. MR 37:302
  • [4] I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976. MR 57:417
  • [5] I. M. Isaacs, Coprime group actions fixing all nonlinear irreducible characters, Can. J. Math. 41 (1989), 68-82. MR 90j:20038
  • [6] M. L. Lewis, Solvable groups having almost relatively prime distinct irreducible character degrees, J. Algebra 174 (1995), 197-216. MR 96d:20009
  • [7] M. L. Lewis, Derived lengths of solvable groups satisfying the one-prime hypothesis, Submitted to Comm. in Algebra.
  • [8] O. Manz and R. Staszewski, Some applications of a fundamental theorem by Gluck and Wolf in the character theory of finite groups, Math. Z. 192 (1986), 383-389. MR 87i:20018
  • [9] O. Manz and T. R. Wolf, Representations of solvable groups, Cambridge University Press, Cambridge, 1993. MR 95c:20013
  • [10] R. Staszewski, On $\pi $-blocks of finite groups, Comm. in Algebra 13 (1985), 2369-2405. MR 86j:20011

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20C15

Retrieve articles in all journals with MSC (1991): 20C15

Additional Information

Mark L. Lewis
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242

Received by editor(s): December 16, 1996
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1998 American Mathematical Society