|
On Isaacs' three character degrees theorem
Author(s):
Yakov
Berkovich
Journal:
Proc. Amer. Math. Soc.
125
(1997),
669-677.
MSC (1991):
Primary 20C15
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
Isaacs has proved that a finite group  is solvable whenever there are at most three characters of pairwise distinct degrees in  (Isaacs' three character degrees theorem). In this note, using Isaacs' result and the classification of the finite simple groups, we prove the solvability of  whenever  contains at most three monolithic characters of pairwise distinct degrees. §2 contains some additional results about monolithic characters.
References:
- [Ber1]
- Y. Berkovich, Generalizations of M-groups, Proc. Amer. Math. Soc. 123, 11 (1995), 3263-3268. CMP 95:16
- [Ber2]
- Y. Berkovich, Finite groups with small sums of some non-linear irreducible characters, J. Algebra 171 (1995), 426-443. MR 96c:20015
- [Ber3]
- Y. Berkovich, Finite groups with few nonlinear irreducible characters, Izv. Severo-Kavkazskogo Tzentra Vyschei Shkoly, estestvennye nauki 1 (1987), 8-13 (Russian). MR 88k:20021
- [Ber4]
- Y. Berkovich, Finite groups with few nonlinear irreducible characters, Problems in group theory and homological algebra, Yaroslav. Gos. Univ., Yaroslavl, 1990, pp. 97-107 (Russian). MR 93d:20015
- [BCH]
- Y. Berkovich, D. Chillag, and M. Herzog, Finite groups in which the degrees of the nonlinear irreducible characters are distinct, Proc. Amer. Math. Soc. 115 (1992), 955-959. MR 92j:20006
- [BCZ]
- Y. Berkovich, D. Chillag, and E. Zhmud', Finite groups in which all nonlinear irreducible characters have three values, Houston Math. J. 21 (1) (1995), 17-28. MR 96i:20005
- [BK]
- Y. Berkovich and L. Kazarin, Finite groups in which only two nonlinear irreducible characters have equal degrees, J. of Algebra 184 (1996), 538-560.
- [BZ1]
- Y. Berkovich and E. Zhmud', Characters of Finite Groups, 2, Amer. Math. Soc. (to appear).
- [BZ2]
- Y. Berkovich and E. Zhmud', On monolithic characters, Houston Math. J. 22 (1996), 263-278.
- [Gag]
- S.C. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math 109 (1983), 263-285. MR 85e:20009
- [Gor]
- D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, New York, 1982. MR 84j:20002
- [Hup]
- B. Huppert, Endliche Gruppen, Bd. 1, Springer, Berlin, 1967. MR 37:302
- [Isa]
- I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. MR 57:417
- [LPS]
- M.W. Liebeck, C.E. Praeger, and J. Saxl, The Maximal Factorizations of the Finite Simple Groups and Their Automorphism Groups, Memoirs of the American Mathematical Society, no. 432,, Providence, RI, 1990.
- [Mic]
- G.O. Michler, Modular representation theory and the classification of finite simple groups, Proc. Symp. Pure Math. 47 (1987), 223-232. MR 89b:20034
- [Sei]
- G. Seitz, Finite groups having only one irreducible representation of degree greater than one, Proc. Amer. Math. Soc. 19 (1968), 459-461. MR 36:5212
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:
Yakov
Berkovich
Affiliation:
Department of Mathematics and Computer Science, University of Haifa, Haifa 31905, Israel
DOI:
10.1090/S0002-9939-97-03790-8
PII:
S 0002-9939(97)03790-8
Keywords:
Monolith,
monolithic character,
automorphism group,
classification of finite simple groups
Received by editor(s):
September 5, 1995
Additional Notes:
The author was supported in part by the Ministry of Absorption of Israel
Communicated by:
Ronald M. Solomon
Copyright of article:
Copyright
1997,
American Mathematical Society
|
Comments: webmaster@ams.org