I'm looking for a classification/reference/theory for groups with the following property: every finite index subgroup of $G$ lies in one of finitely many isomorphism classes. I would also be interested to see this as a straightforward consequence of some other more well known classification. A simple example, beyond the obvious abelian ones, would be the infinite dihedral group $D_\infty$. All the finite index subgroups are either isomorphic to $D_\infty$ or isomorphic to $\mathbb{Z}$.

Classes of examples that satisfy the property are also of interest.

  • $\begingroup$ Is the set of isomorphism classes the same for every group you consider here? $\endgroup$ – Qudit Mar 20 '15 at 15:28
  • $\begingroup$ That is of course interesting too, but it wasn't what I was thinking of. Just simply given $G$, there are only finitely many isomorphism classes for its finite index subgroups. $\endgroup$ – RMiles Mar 20 '15 at 15:39
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    $\begingroup$ I think that finitely generated virtually abelian groups, which include $D_\infty$ have that property, basically because there are only finitely many ways that a finite group can act on an $n$-generated abelian group and the associated cohomology groups are all finite. I don't know of any other examples. $\endgroup$ – Derek Holt Mar 20 '15 at 15:42
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    $\begingroup$ @RMiles Actually, I think $G \times \Bbb{Z}$ does the trick for any finite group $G$. $\endgroup$ – Qudit Mar 20 '15 at 16:10
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    $\begingroup$ Any group with no finite index proper subgroups, such as an infinite simple group, also satisfies the condition. $\endgroup$ – Derek Holt Mar 20 '15 at 17:40

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