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.

令G为一组。如果为每一个适当的正常子群Ñ和元件X的ģ与Ñ < X >≠ ģ，Ñ < X >是FC -基，但ģ不是FC -基，则称ģ一个NFC -基团。在本文中，我们考虑了NFC组。我们证明，每个具有非平凡有限图像的非完美NFC组都是最小非FC组。我们还表明，如果G是非完美的NFC-群没有有限索引的非平凡的适当子群，那么在“每个Sylow p-子群是所有素数p的FC-群”的条件下，G是最小的非FC-群。在理想情况下，我们表明，存在局部幂零完美NFC - p -基团不属于最小的非FC -groups同时又有麦克莱恩组\（M（\ mathbb {Q}，GF（P））\）的任何素数p都包含此类基团。我们给出了无扭转情况的特征。我们也考虑p -基团，使得订单的每一个元素的归一化p是FC-子组。