Let $G$ be an infinite countable group having a core-free subgroup $H$ such that the interval $[H,G]$ in the subgroup lattice $\mathcal{L}(G)$ is [ACC](https://en.wikipedia.org/wiki/Ascending_chain_condition) of infinite length, and for every $K \in (H,G]$, $G$ is generated by a single $K$-coset (i.e. there is $g \in G$ with $\langle Kg \rangle = G$). 


**Question**: Is $G$ generated by a single $H$-coset?


It is an exercise to reformulate as: let $[H,G]$ be an ACC interval of groups and $(K\_i)\_{i \in I}$ its [coatoms](https://en.wikipedia.org/wiki/Atom_(order_theory)#Coatoms), i.e. the maximal elements in $[H,G)$. Assume that $I$ is an infinite countable set, and for every *finite* subset $J \subset I$ we have $\bigcap\_{j \in J} (G \setminus K\_j) \neq \emptyset.$ *Question*: Is it true that $\bigcap\_{i \in I} (G \setminus K\_i) \neq \emptyset$?


*Examples*: for $G = \mathbb{Z}$ and $H = \{0\}$, the ACC is satisfied, the coatoms are $(p\mathbb{Z})\_{p \in \mathbb{P}}$ and $\bigcap\_{p \in \mathbb{P}} (\mathbb{Z} \setminus p\mathbb{Z}) = \{-1,1 \} \neq \emptyset$. For $G = \mathbb{Z} \rtimes C\_2$ and $H = C\_2$, it works as well.  

Any other example (with $H$ core-free) is welcome!


**Application**: [Generalization of a theorem of Øystein Ore in group theory: the infinite case](https://mathoverflow.net/q/323887/34538)



Infinite group generated by a single coset
The answer is "no". Let $G$ be a Tarski torsion-free monster, $H$ be the trivial subgroup. Then $(H,G]$ consists of $G$ and all cyclic subgroups of $G$, it has ACC and infinite length. If $K$ is a cyclic subgroup and $a$ is not in the cyclic centralizer $C(K)$, then the coset $aK$ contains two non-commuting elements, so it generates $G$ but no coset of $H$ generates $G$.


