Title: Sparse Subgroups of Profinite Groups

Abstract:

We construct for every set of natural numbers
$\{n_1,n_2,...\}$ a
profinite group $G$ and a closed subgroup $H$ of $G$ such that all the
subgroups
of $G$ containing $H$ are
$$G > H_1 > H_2 > H_3 >  ...> H$$
and $(H_{i-1}:H_i) = n_i$.