On Isaacs' three character degrees theorem
Author:
Yakov Berkovich
Journal:
Proc. Amer. Math. Soc. 125 (1997), 669677
MSC (1991):
Primary 20C15
MathSciNet review:
1376750
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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.
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 Y. Berkovich, Generalizations of Mgroups, Proc. Amer. Math. Soc. 123, 11 (1995), 32633268. CMP 95:16
 [Ber2]
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 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), 955959. MR 92j:20006
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 Y. Berkovich, D. Chillag, and E. Zhmud', Finite groups in which all nonlinear irreducible characters have three values, Houston Math. J. 21 (1) (1995), 1728. MR 96i:20005
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 Y. Berkovich and L. Kazarin, Finite groups in which only two nonlinear irreducible characters have equal degrees, J. of Algebra 184 (1996), 538560.
 [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), 263278.
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 S.C. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math 109 (1983), 263285. MR 85e:20009
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 D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, New York, 1982. MR 84j:20002
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 [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), 223232. MR 89b:20034
 [Sei]
 G. Seitz, Finite groups having only one irreducible representation of degree greater than one, Proc. Amer. Math. Soc. 19 (1968), 459461. MR 36:5212
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Additional Information
Yakov Berkovich
Affiliation:
Department of Mathematics and Computer Science, University of Haifa, Haifa 31905, Israel
DOI:
http://dx.doi.org/10.1090/S0002993997037908
PII:
S 00029939(97)037908
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
Article copyright:
© Copyright 1997
American Mathematical Society
