Sylow numbers and the structure of finite groups

Abstract

Suppose that the finite group $G=AB$ is a mutually permutable product of two subgroups $A$ and $B$. By using Sylow numbers of $A$ and $B$, we present some new bounds of the $p$-length $l_p(G)$ of a $p$-solvable group $G$ and the nilpotent length $F_l(G)$ and the derived length $dl(G/\Phi(G))$ of a solvable group $G$. Some known results of Zhang in J. Algebra 1995, 176 are extended.