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On Groups with Certain Proper FC-Subgroups

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Abstract

Let G be a group. If for every proper normal subgroup N and element x of G with N〈x〉≠G, N〈x〉 is an FC-group, but G is not an FC-group, then we call G an NFC-group. In the present paper we consider the NFC-groups. We prove that every non-perfect NFC-group with non-trivial finite images is a minimal non-FC-group. Also we show that if G is a non-perfect NFC-group having no nontrivial proper subgroup of finite index, then G is a minimal non-FC-group under the condition “every Sylow p-subgroup is an FC-group for all primes p”. In the perfect case, we show that there exist locally nilpotent perfect NFC-p-groups which are not minimal non-FC-groups and also that McLain groups \(M(\mathbb {Q},GF(p))\) for any prime p contain such groups. We give a characterization for torsion-free case. We also consider the p-groups such that the normalizer of every element of order p is an FC-subgroup.

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Acknowledgements

The Authors are grateful to the editor and reviewers for many crucial comments and feedback to improve the manuscript.

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Authors and Affiliations

  1. Science Faculty, Department of Mathematics, Gazi University, 06500 Teknikokullar, Ankara, Turkey

    Og~uz Alkış & Aynur Arıkan

  2. Gazi Education Faculty, Department of Mathematics and Science Education, Gazi University, 06500 Teknikokullar, Ankara, Turkey

    Ahmet Arıkan

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  1. Og~uz Alkış
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  2. Ahmet Arıkan
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  3. Aynur Arıkan
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Correspondence to Ahmet Arıkan.

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Presented by: Andrew Mathas

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Alkış, O., Arıkan, A. & Arıkan, A. On Groups with Certain Proper FC-Subgroups. Algebr Represent Theor 25, 953–961 (2022). https://doi.org/10.1007/s10468-021-10054-w

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