On Groups with Certain Proper FC-Subgroups

Alkis O., ARIKAN A., ARIKAN A.

ALGEBRAS AND REPRESENTATION THEORY, cilt.25, sa.4, ss.953-961, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 25 Sayı: 4
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1007/s10468-021-10054-w
  • Dergi Adı: ALGEBRAS AND REPRESENTATION THEORY
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Compendex, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.953-961
  • Anahtar Kelimeler: FC-group, NFC-group, Minimal non-FC-group, Finitary permutation group, Perfect group, Locally nilpotent group
  • Gazi Üniversitesi Adresli: Evet

Özet

Let G be a group. If for every proper normal subgroup N and element x of G with N < x > not equal 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(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.