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Torsion-free virtually-Z is Z

Let $G$ be a torsion-free group with a subgroup $H$ of finite index isomorphic to $\mathbb{Z}$. Is $G$ isomorphic to $\mathbb{Z}$?

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marked as duplicate by Arturo Magidin, t.b., Asaf Karagila, Sasha, Jonas Teuwen Nov 23 '11 at 23:58

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

If this is a homework question, see… – Dimitrije Kostic Nov 21 '11 at 16:47
It is not a homework. I just met torsion-free groups and I tried to find some basic properties of it. I was unable to solve this question. – student Nov 21 '11 at 16:51
The question has been asked there:… – PseudoNeo Nov 21 '11 at 16:58
Okay. I got it. – student Nov 21 '11 at 17:05
up vote 1 down vote accepted

Generally, we say $G$ is "virtually $\mathcal{P}$" if it has a subgroup $H$ that has property $\mathcal{P}$ and is of finite index in $G$. When the property $\mathcal{P}$ is inherited to subgroups (if $H$ has property $\mathcal{P}$, and $K$ is a subgroup of $H$, then $K$ has property $\mathcal{P}$), then in fact $G$ is virtually $\mathcal{P}$ if and only if it has a normal subgroup with property $\mathcal{P}$ that is of finite index (since every subgroup of $G$ of finite index contains a normal subgroup of $G$ of finite index).

So your question is whether a torsion-free virtually cyclic group must be cyclic.

The answer is "yes": in fact, a virtually cyclic group must be either finite (every finite group is virtually cyclic), finite by infinite cyclic, or finite by infinite-dihedral. For a proof of the case you are interested in, see this previous question.

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