On one-generated and bounded totally $\omega$-composition formations of finite groups

All considered groups are finite. Let $G$ be a group. Then $c_{\infty}^\omega\mathrm{form}(G)$ denotes the intersection of all totally $\omega$-composition formations containing $G$. The formation $c_{\infty}^\omega\mathrm{form}(G)$ is called a totally $\omega$-composition formation generated by $G$ or a one-generated totally $\omega$-composition formation. A totally $\omega$-composition formation $\mathfrak{F}$ is called a bounded, if $\mathfrak{F}$ is a subformation of some one-generated totally $\omega$-composition formation, that is, $\mathfrak{F}\subseteq c_{\infty}^\omega\mathrm{form}(G)$ for some group $G$. In this paper, criteria for the one-generation (boundedness) of a totally $\omega$-composition formation are obtained.