Degrees, kernels and quasikernels of monolithic characters

Author:
Yakov Berkovich

Journal:
Proc. Amer. Math. Soc. **128** (2000), 3211-3219

MSC (1991):
Primary 20C15

Published electronically:
May 11, 2000

MathSciNet review:
1707506

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Abstract | References | Similar Articles | Additional Information

Abstract: Theorem 5 yields the condition sufficient for a group to be a direct product of a -group and an abelian -group. We also obtain characterizations of nilpotent groups, prime power groups, -nilpotent and -closed groups in the language of characters. Proofs of some results depend on the classification of finite simple groups. Some problems are posed and discussed.

**[B]**Yakov Berkovich,*On Isaacs’ three character degrees theorem*, Proc. Amer. Math. Soc.**125**(1997), no. 3, 669–677. MR**1376750**, 10.1090/S0002-9939-97-03790-8**[BIK]**Y. Berkovich, I. M. Isaacs and L. Kazarin, Distinct monolithic character degrees, J. Algebra**216**(1999), 448-480.**[BZ]**Ya. G. Berkovich and E. M. Zhmud′,*Characters of finite groups. Part 1*, Translations of Mathematical Monographs, vol. 172, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by P. Shumyatsky [P. V. Shumyatskiĭ] and V. Zobina. MR**1486039****[CCNPW]**J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,*Atlas of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR**827219****[I]**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****[W]**Wolfgang Willems,*Blocks of defect zero in finite simple groups of Lie type*, J. Algebra**113**(1988), no. 2, 511–522. MR**929777**, 10.1016/0021-8693(88)90176-7

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Additional Information

**Yakov Berkovich**

Affiliation:
Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

Email:
berkov@mathcs2.haifa.ac.il

DOI:
http://dx.doi.org/10.1090/S0002-9939-00-05550-7

Keywords:
$\pi$-closed and $\pi$-nilpotent groups,
kernel and quasikernel,
monolithic character,
Frobenius group,
classification of finite simple groups

Received by editor(s):
January 21, 1999

Published electronically:
May 11, 2000

Additional Notes:
The author was supported in part by the Ministry of Absorption of Israel.

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 2000
American Mathematical Society