Critical $\Omega$ -Fiber Formations of Finite Groups

Abstract

Let $\mathfrak{H}$ be a class of finite groups. An $\Omega$-fiber formation $\mathfrak{F}$ of finite groups with direction $\varphi$ is said to be a minimal $\Omega$-fiber non-$\mathfrak{H}$-formation with direction $\varphi$, or briefly an $\mathfrak{H}_\Omega$-critical formation, if $\mathfrak{F} \nsubseteq \mathfrak{H}$, but any proper $\Omega$-fiber subformation with direction $\varphi$ in $\mathfrak{F}$ belongs to the class $\mathfrak{H}$. In the paper, a complete description of the structure of minimal $\Omega$-fiber non-$\mathfrak{H}$-formations of finite groups of two different directions is given.