Let $G$ be a group, and let $H\leq G$ be a subnormal subgroup. Suppose there exist a cyclic series from $H$ to $G$, that is, a normal series $$H=H_0\lhd H_1\lhd\cdots\lhd H_k= G$$ of subgroups of $G$ such that each factor group $H_{i+1}/H_i$ is cyclic. Define the *relative Hirsch number* of the pair $(G,H)$ to be the number $h(G,H)$ of infinite cyclic factors.

**Questions**

- Under what conditions on the group $G$ and subgroup $H$ does such a cyclic series exist?
- Supposing existence of a cyclic series as above, is the relative Hirsch number a well-defined invariant (ie independent of the cyclic series chosen)?
- If $H$ is normal in $G$, then is $h(G,H)=h(G/H)$?
- Where is this notion to be found in the literature?

Sorry to be asking so many questions at once! If it helps, I am mainly interested in the case $G=\Gamma\times\Gamma$ and $H=\Gamma$ the diagonal subgroup, where $\Gamma$ is finitely generated, torsion-free nilpotent.

Thanks.