Bulletin of the
Korean Mathematical Society BKMS

In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group $G$, we define the subgroup $\mathcal {A}(G)$ to be intersection of the normalizers of all non-cyclic subgroups of $G$. Set $\mathcal{A}_0=1$. Define $\mathcal {A}_{i+1}(G)/\mathcal {A}_i(G)=\mathcal {A}(G/\mathcal {A}_i(G))$ for $i\geq 1$. By $\mathcal {A}_{\infty}(G)$ denote the terminal term of the ascending series. It is proved that if $G=\mathcal {A}_{\infty}(G)$, then the derived subgroup $G'$ is nilpotent. Furthermore, if all elements of prime order or order $4$ of $G$ are in $\mathcal {A}(G)$, then $G'$ is also nilpotent.

Keywords: derived subgroup, meta-nilpotent group, solvable group, nilpotency class, fitting length

MSC numbers: 20D10, 20D15, 20D20, 20D30, 20F14, 20F19