On injectors of finite groups

Abstract

If $\mathscr{F}$ is a non-empty Fitting class, $\pi=\pi(\mathscr{F})$ and $G$ a group such that every chief factor of $G/G_{\mathscr{F}}$ is an $C_{\pi}^{s}$-group. Then $G$ has at least one $\mathscr{F}$-injector. This result is used to resolve an open problem and generalize some known results.