18.8 (2014)

Let $\mathcal{X}$ be a class of finite simple groups such that $\pi(\mathcal{X}) = \text{char}(\mathcal{X})$. A formation of finite groups $\mathfrak{F}$ is said to be $\mathcal{X}$-saturated if a finite group $G$ belongs to $\mathfrak{F}$ whenever the factor group $G/\Phi(O_{\mathcal{X}}(G))$ is in $\mathfrak{F}$, where $O_{\mathcal{X}}(G)$ is the largest normal subgroup of $G$ whose composition factors are in $\mathcal{X}$. Is every $\mathcal{X}$-saturated formation $\mathcal{X}$-local in the sense of Förster? (See the definition in (P. Förster, Publ. Sec. Mat. Univ. Autònoma Barcelona, 29, no. 2–3 (1985), 39–76).)