Finite Non-Nilpotent Generalizations of Hamiltonian Groups

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 <TEX>A(G)</TEX> to be intersection of the normalizers of all non-cyclic subgroups of G. Set <TEX>A₀=1</TEX>. Define <TEX>Aᵢ₊₁(G)/Aᵢ(G)=A(G/Aᵢ(G))</TEX> for <TEX>i≥1</TEX>. By <TEX>A∞(G)</TEX> denote the terminal term of the ascending series. It is proved that if <TEX>G=A∞(G)</TEX>, then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in <TEX>A(G)</TEX>, then G' is also nilpotent.