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Finite groups with few normalizers or involutions

  • Izabela Agata MalinowskaEmail author
Open Access
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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).

Keywords

Normalizer Normalizer subgroup 

Mathematics Subject Classification

Primary 20D10 Secondary 20D20 
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Notes

Acknowledgements

The author would like to thank the anonymous reviewers for their valuable comments and suggestions to shorten the paper.

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Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of BiałystokBiałystokPoland

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