Hall subgroups and $p$-solvability

Abstract:

Let $G$ be a finite group and let $\pi (G) = \{ p,{q_1}, \ldots ,{q_r}\}$ be the set of all prime divisors of $G$. Suppose there is a $p’$-Hall subgroup $H$. If there are subgroups $P,{Q_1}, \ldots ,{Q_r}$ such that $P\epsilon {\operatorname {Syl} _p}(G),{Q_i}\epsilon {\operatorname {Syl} _{{q_i}}}(H)$, and ${L_i} = P{Q_i}$ is a subgroup, $i = 1, \ldots ,r$, then $G$ is $p$-solvable. Moreover, if the subgroup $H$ is solvable, then $G$ is solvable too.