Abstract
The groups having exactly one normalizer are Dedekind groups. All finite groups with exactly two normalizers were classified by Pérez-Ramos in 1988. In this paper we prove that every finite group with at most 26 normalizers of \(\{2,3,5\}\)-subgroups is soluble and we also show that every finite group with at most 21 normalizers of cyclic \(\{2,3,5\}\)-subgroups is soluble. These confirm Conjecture 3.7 of Zarrin (Bull Aust Math Soc 86:416–423, 2012).
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The author would like to thank the anonymous reviewers for their valuable comments and suggestions to shorten the paper.
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Malinowska, I.A. Finite groups with few normalizers or involutions. Arch. Math. 112, 459–465 (2019). https://doi.org/10.1007/s00013-018-1290-x
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DOI: https://doi.org/10.1007/s00013-018-1290-x