A Generalization of Hall-Wielandt Theorem

Abstract

Let $G$ be a finite group and $P\in Sy{l}_p(G)$. We denote the $k$'th term of the upper central series of $G$ by $Z_k(G)$ and the norm of $G$ by $Z^{\ast }(G)$. In this article, we prove that if for every tame intersection $P\cap Q$ such that $Z_{p-1}(P)<P\cap Q<P$, the group $N_G(P\cap Q)$ is $p$-nilpotent then $N_G(P)$ controls $p$-transfer in $G$. For $p=2$, we sharpen our results by proving if for every tame intersection $P\cap Q$ such that $Z^{\ast }(P)<P\cap Q<P$, the group $N_G(P\cap Q)$ is $p$-nilpotent then $N_G(P)$ controls $p$-transfer in $G$. We also obtain several corollaries which give sufficient conditions for $N_G(P)$ to controls $p$-transfer in $G$ as a generalization of some well known theorems, including Hall-Wielandt theorem and Frobenius normal complement theorem.