On Critical $\omega$-Fan Formations of Finite Groups

We consider finite groups only. Let $\omega$ be a nonempty subset of the set $P$ of all primes, and let $f\colon\omega\cup\{\omega'\}\to\{$formations of groups$\}$ and $\delta\colon P\to\{$nonempty Fitting formations of groups$\}$ be some functions. The formation consisting of all groups $G$ such that $G/O_\omega(G)\in f(\omega')$ and $G/G_{\delta(p)}\in f(p)$ for any $p\in\omega\cap\pi(G)$ is referred to as an $\omega$-fan formation with direction $\delta$. Let $\mathfrak H$ be some class of groups; an $\omega$-fan formation $\mathfrak F$ with direction $\delta$ is said to be an $\mathfrak H_{\omega\delta}$-critical formation if $\mathfrak F\nsubseteq\mathfrak H$ and any proper $\omega$-fan subformation with direction $\delta$ in $\mathfrak F$ is contained in the class $\mathfrak H$. In the paper, a description of the structure of the $\mathfrak H_{\omega\delta}$-critical formations is presented.