Reducible τ-closed σ-local formations of finite groups with a given structure of subformations

Abstract

Let $\mathfrak{F}$ and $\mathfrak{H}$ be some $\tau$-closed $\sigma$-local formations of finite groups. By $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ we denote the lattice of all $\tau$-closed $\sigma$-local formations $\mathfrak{X}$ such that $\mathfrak{H}\cap\mathfrak{F}\subseteq \mathfrak{X}\subseteq \mathfrak{F}.$ The length of the lattice $\mathfrak{F}/^\tau_\sigma\mathfrak{H}\cap\mathfrak{F}$ is called the {\it $\mathfrak H^\tau_\sigma$-defect}, and for $\mathfrak{H}=(1)$ it is the formation of all identity groups, {\it $l^\tau_\sigma$-length} of $\mathfrak{F}$. The general properties of the $\mathfrak H^\tau_\sigma$-defect of $\tau$-closed $\sigma$-local formations are studied, and a description of the structural structure of reducible $\tau$-closed $\sigma$-local formations with $\mathfrak H^\tau_\sigma$-defect $\leq 2$ and $l^\tau_\sigma$-length $\leq 3$ is obtained.