Soluble products of connected subgroups

Abstract

The main result in the paper states the following: For a finite group $G=AB$, which is the product of the soluble subgroups $A$ and $B$, if $⟨a,b⟩$ is a metanilpotent group for all $a\in A$ and $b\in B$, then the factor groups $⟨a,b⟩F(G)/F(G)$ are nilpotent, $F(G)$ denoting the Fitting subgroup of $G$. A particular generalization of this result and some consequences are also obtained. For instance, such a group $G$ is proved to be soluble of nilpotent length at most $l+1$, assuming that the factors $A$ and $B$ have nilpotent length at most $l$. Also for any finite soluble group $G$ and $k\ge 1$, an element $g\in G$ is contained in the preimage of the hypercenter of $G/F_{k-1}(G)$, where $F_{k-1}(G)$ denotes the ($k-1$)th term of the Fitting series of $G$, if and only if the subgroups $⟨g,h⟩$ have nilpotent length at most $k$ for all $h\in G$.