Abstract
Gagola and Lewis proved that a finite group G is nilpotent if and only if \(\chi (1)^2\) divides |G : \(\mathrm{Ker}\) \(\chi |\) for all irreducible characters \(\chi \) of G. In this paper, we prove the following generalization that a finite group G is nilpotent if and only if \(\chi (1)^2\) divides |G : \(\mathrm{Ker}\) \(\chi |\) for all monolithic characters \(\chi \) of G.
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References
S. M. Gagola, Jr., and M. L. Lewis, A character theoretic condition characterizing nilpotent groups, Commun. Algebra. 27 (1999), 1053–1056.
I. M. Isaacs, Large orbits in actions of nilpotent groups, Proc. Amer. Math. Soc. 127 (1999), 45-50.
I. M. Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006.
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Lu, J., Qin, X. & Liu, X. Generalizing a theorem of Gagola and Lewis characterizing nilpotent groups. Arch. Math. 108, 337–339 (2017). https://doi.org/10.1007/s00013-016-1001-4
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DOI: https://doi.org/10.1007/s00013-016-1001-4