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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite solvable group. Assume that the degree graph of has exactly two connected components that do not contain . Suppose that one of these connected components contains the subset , where and are coprime when . Then the derived length of is less than or equal to .

**[1]**P. X. Gallagher,*Group characters and normal Hall subgroups*, Nagoya Math. J.**21**(1962), 223–230. MR**0142671****[2]**S. C. Garrison,*On groups with a small number of character degrees*, Ph.D. Thesis, Univ. of Wisconsin, Madison, 1973.**[3]**B. Huppert,*Endliche Gruppen. I*, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR**0224703****[4]**I. Martin Isaacs,*Character theory of finite groups*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR**0460423****[5]**I. M. Isaacs,*Coprime group actions fixing all nonlinear irreducible characters*, Canad. J. Math.**41**(1989), no. 1, 68–82. MR**996718**, 10.4153/CJM-1989-003-2**[6]**Mark L. Lewis,*Solvable groups having almost relatively prime distinct irreducible character degrees*, J. Algebra**174**(1995), no. 1, 197–216. MR**1332867**, 10.1006/jabr.1995.1124**[7]**M. L. Lewis,*Derived lengths of solvable groups satisfying the one-prime hypothesis*, Submitted to Comm. in Algebra.**[8]**Olaf Manz and Reiner Staszewski,*Some applications of a fundamental theorem by Gluck and Wolf in the character theory of finite groups*, Math. Z.**192**(1986), no. 3, 383–389. MR**845210**, 10.1007/BF01164012**[9]**Olaf Manz and Thomas R. Wolf,*Representations of solvable groups*, London Mathematical Society Lecture Note Series, vol. 185, Cambridge University Press, Cambridge, 1993. MR**1261638****[10]**Reiner Staszewski,*On 𝜋-blocks of finite groups*, Comm. Algebra**13**(1985), no. 11, 2369–2405. MR**807480**, 10.1080/00927878508823279

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

Email:
lewis@mcs.kent.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04391-3

Received by editor(s):
December 16, 1996

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1998
American Mathematical Society