Factorizations of One-Generated Composition Formations

Abstract

A non-empty formation ${\mathfrak{F}}$ of finite groups is said to be solubly saturated, or we call it a composition formation, if every finite group G having a normal subgroup N such that ${G \mathord{\left/{\vphantom {G {\Phi \left( N \right) \in }}} \right.\kern-\nulldelimiterspace} {\Phi \left( N \right) \in }}{\mathfrak{F}}$ belongs to ${\mathfrak{F}}$. An intersection of all composition formations containing a given group G is denoted cformG. Conditions are described under which ${\mathfrak{F}}$ has the form ${\mathfrak{F}} = {\mathfrak{M}}{\mathfrak{H}}$, where ${\mathfrak{M}} \ne {\mathfrak{F}} \ne {\mathfrak{H}}$.