Elliptically embedded subgroups of polycyclic groups

Abstract:

A subgroup $H$ of a group $G$ is elliptically embedded in $G$ if for each subgroup $K$ of $G$ there is an integer $n = n(K)$ such that $\left \langle {H,K} \right \rangle = HK \cdots HK$, where the product has $2n$ factors. It is shown that a subgroup $H$ of a polycyclic by finite group $G$ is elliptically embedded in $G$ if and only if $H$ is subnormal in some subgroup of finite index in $G$.