Elliptically embedded subgroups of polycyclic groups

Abstract

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