On Isaacs' three character degrees theorem

Author:
Yakov Berkovich

Journal:
Proc. Amer. Math. Soc. **125** (1997), 669-677

MSC (1991):
Primary 20C15

DOI:
https://doi.org/10.1090/S0002-9939-97-03790-8

MathSciNet review:
1376750

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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.

**[Ber1]**Y. Berkovich,*Generalizations of M-groups*, Proc. Amer. Math. Soc.**123, 11**(1995), 3263-3268. CMP**95:16****[Ber2]**Yakov Berkovich,*Finite groups with small sums of degrees of some non-linear irreducible characters*, J. Algebra**171**(1995), no. 2, 426–443. MR**1315905**, https://doi.org/10.1006/jabr.1995.1020**[Ber3]**Ya. G. Berkovich,*Finite groups with a small number of irreducible nonlinear characters*, Izv. Severo-Kavkaz. Nauchn. Tsentra Vyssh. Shkoly Estestv. Nauk.**1**(1987), 8–13, 142 (Russian). MR**907973****[Ber4]**Ya. G. Berkovich,*Finite groups with a small number of nonlinear irreducible characters*, Problems in group theory and homological algebra (Russian), Matematika, Yaroslav. Gos. Univ., Yaroslavl′, 1990, pp. 97–107 (Russian). MR**1169969****[BCH]**Yakov Berkovich, David Chillag, and Marcel Herzog,*Finite groups in which the degrees of the nonlinear irreducible characters are distinct*, Proc. Amer. Math. Soc.**115**(1992), no. 4, 955–959. MR**1088438**, https://doi.org/10.1090/S0002-9939-1992-1088438-9**[BCZ]**Yakov Berkovich, David Chillag, and Emmanuel Zhmud,*Finite groups in which all nonlinear irreducible characters have three distinct values*, Houston J. Math.**21**(1995), no. 1, 17–28. MR**1331241****[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]**Stephen M. Gagola Jr.,*Characters vanishing on all but two conjugacy classes*, Pacific J. Math.**109**(1983), no. 2, 363–385. MR**721927****[Gor]**Daniel Gorenstein,*Finite simple groups*, University Series in Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to their classification. MR**698782****[Hup]**B. Huppert,*Endliche Gruppen. I*, Die Grundlehren der Mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR**0224703****[Isa]**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****[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]**Gerhard O. Michler,*Modular representation theory and the classification of finite simple groups*, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 223–232. MR**933361****[Sei]**Gary Seitz,*Finite groups having only one irreducible representation of degree greater than one*, Proc. Amer. Math. Soc.**19**(1968), 459–461. MR**0222160**, https://doi.org/10.1090/S0002-9939-1968-0222160-X

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Additional Information

**Yakov Berkovich**

Affiliation:
Department of Mathematics and Computer Science, University of Haifa, Haifa 31905, Israel

DOI:
https://doi.org/10.1090/S0002-9939-97-03790-8

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

Article copyright:
© Copyright 1997
American Mathematical Society