Abstract
Let M be a subgroup of a finite group G, and suppose that M normalizes the nilpotent residual \(H^\infty \) of every non-subnormal subgroup H of G. We show that M must also normalize the nilpotent residuals of the subnormal subgroups of G. We also prove a similar result for the solvable residual.
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L. Gong, L. Zhao, and X. Guo.: On the generalized norm of a finite group, J. Algebra Appl. 15, 1–9 (2016)
B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967.
I. M. Isaacs, Finite group theory, Amer. Math. Soc., Providence, R.I., 2008.
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The research of the first author was supported by: The National Natural Science Foundation of China (11601245,11526114).
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Gong, L., Isaacs, I.M. Normalizers of nilpotent residuals. Arch. Math. 108, 1–7 (2017). https://doi.org/10.1007/s00013-016-0995-y
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DOI: https://doi.org/10.1007/s00013-016-0995-y