Bounds Of The Nilpotent Lengths Of Finite Groups With Given Sylow Normalizers

The realionships between the subgroups of a finite group G and the group G itself have been extensively studied in the literature. It is well known that the normalizers of Sylow subgroups of a finite group play a crucial role in the investigation of finite groups. Let χ be a class of groups. Then,we denote by Nχ the class of groups whose normalizers of all Sylow subgroups are in χ. In 1986, Bianchi, Mauri and Hauck first proved thatNN Nholds for the class of all nilpotent, then G is nilpotent[3]. On the other hand, Fedri and Serens have observed in [5\ that Nv v, for the class of supersoluble groups. Now, recall that a class Fof groups is a formation if it is closed under homomorphic images and subdirect products. A formation φ is called a S- formation if every minimal non-φ-group is either a Schmidt group or a group of prime order. In recent years, Guo has given a criterion fora group G to be in the an S- formation φ when the Sylow normalizers of Gare in φ [7.Ch.3]. A series of applications of this result were also given byhim. In particular, Bryce, Fedri, Serens and Guo( [2]and[8]) have studied thenilpotent length of the groups in NF \ φ, where φ is the class Y of allsupersoluble groups or the class Nr of groups with nilpotent length ≤r. We note that in all the above papers, the groups are finite groups whose normalizers of Sylow subgroups have some given internal properties. On the other hand, one can also study the finite groups whose normalizers of Sylow subgroups have some given external properties. In this aspect, Kondrat'ev [ll] has shown that a group G is 2- nilpotent if the normalizer of each Sylow subgroup of G is of odd index in G. In 1995, Zhang [13Jhas proved that a group G is soluble if the index of the normalizer of every Sylow subgroup in G is a prime power. Later on, Chigira [4] proved that a group G is p-nilpotent if p≠3 and (G: NG(Gr),p)=1 for every r(G). In 1996, Guo [9]has further proved that the index of the normalizer of every Sylow subgroup of G is an odd number or a prime if and only if G is a soluble group and G = KH, where K and H are the Hall subgroups of G , K is a nilptent subgroup which is normal in a2'-nilpotent. In this paper, we shall study the nilpotent length of finite groups whose Sylow normalizer indices are of prime powers. In particular, the bounds of the nilpotent lengths of such groups are found. As a consquence, some interesting properties of finite groups are derived.