On Groups with Certain Proper FC-Subgroups

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.