A generalization of Hall-Wielandt theorem

Abstract

Let Gbe a finite group and P∈Sylp(G). We denote the k’th term of the upper central series of Gby Zk(G)and the norm of Gby Z∗(G). In this article, we prove that if for every tame intersection P∩Qsuch that Zp−1(P) <P∩Q <P, the group NG(P∩Q)is p-nilpotent then NG(P) controls p-transfer inG. Fo r p =2, we sharpen our results by proving if for every tame intersection P∩Qsuch that Z∗(P) <P∩Q <P, the group NG(P∩Q)is p-nilpotent then NG(P) controls p-transfer in G. We also obtain several corollaries which give sufficient conditions for NG(P)to control p-transfer in Gas a generalization of some well known theorems, including Hall-Wielandt theorem and Fr o b e n i u s normal complement theorem