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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 

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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