On Groups with Certain Proper FC-Subgroups


Alkis O., ARIKAN A. , ARIKAN A.

ALGEBRAS AND REPRESENTATION THEORY, 2021 (SCI İndekslerine Giren Dergi) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası:
  • Basım Tarihi: 2021
  • Doi Numarası: 10.1007/s10468-021-10054-w
  • Dergi Adı: ALGEBRAS AND REPRESENTATION THEORY

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.