On Groups with Certain Proper FC-Subgroups
| Autor: | Aynur Arıkan, Ahmet Arikan, Og~uz Alkış |
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| Rok vydání: | 2021 |
| Předmět: |
Normal subgroup
Group (mathematics) General Mathematics 010102 general mathematics Sylow theorems 0211 other engineering and technologies Locally nilpotent 021107 urban & regional planning 02 engineering and technology 01 natural sciences Centralizer and normalizer Prime (order theory) Combinatorics Subgroup Order (group theory) 0101 mathematics Mathematics |
| Zdroj: | Algebras and Representation Theory. 25:953-961 |
| ISSN: | 1572-9079 1386-923X |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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