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I am looking for examples of the following situation:

$G$ is an infinite group. Every two finite index proper subgroups of $G$ are isomorphic.

The only examples that I have now are (1) $\mathbb{Z}^n$ for $n$ any cardinal number, (2) groups that have at most one proper infinite subgroup of finite index (thanks to Sasha Anan'in's and Derek Holt's comments), (3) direct product of groups in (1) and (2).

  1. Are these the only examples for the above situation?

  2. Is there a classification for groups with this property?

  3. Can we find examples with various properties? For example, $G$ is not amenable, etc.

Any related reference will be appreciated.

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    $\begingroup$ Take any simple infinite group. $\endgroup$ – Sasha Anan'in Feb 27 '14 at 22:30
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    $\begingroup$ More generally, perhaps you should discard all groups with at most one proper finite index subgroups. But you still get examples like direct products of infinite simple groups with ${\mathbb Z}^n$. $\endgroup$ – Derek Holt Feb 27 '14 at 22:40
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    $\begingroup$ The word "proper" in the question is quite unnatural, it would be more natural to ask about groups with all finite index subgroups isomorphic to avoid nonsense discussion. $\endgroup$ – YCor Feb 27 '14 at 22:45
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    $\begingroup$ Lamplighters and solvable Baumslag-solvable groups have lots of finite index subgroups isomorphic to the whole group using the automaton representation. $\endgroup$ – Benjamin Steinberg Feb 27 '14 at 23:04
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    $\begingroup$ The variable $u$ makes the module of infinite rank over $\mathbf{Q}[t^{\pm 1}]$, with basis $(u^n)_{n\ge 0}$, so that this does not change when replacing $\mathbf{Z}$ by a finite index subgroup. $\endgroup$ – YCor Feb 27 '14 at 23:14

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