Degrees, kernels and quasikernels of monolithic characters
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
- Y. Berkovich, On Isaacs' three character degrees theorem, Proc. Amer. Math. Soc. 125, 3 (1997), 669-677. MR 97i:20006
- Y. Berkovich, I. M. Isaacs and L. Kazarin, Distinct monolithic character degrees, J. Algebra 216 (1999), 448-480.
- Y. G. Berkovich and E.M. Zhmud', Characters of Finite Groups. Parts 1, 2, Translations of Mathematical Monographs 172, 181, American Mathematical Society, Providence, 1998. MR 98m:20011
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985. MR 88g:20025
- I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. MR 57:417
- W. Willems, Blocks of defect zero in finite simple groups, J. of Algebra 113 (1988), 511-522. MR 89c:20025
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Affiliation: Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
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