If two groups $G\bigoplus \mathbb{Z}$ and $H\bigoplus \mathbb{Z}$ are commensurable. Does it imply that the groups $G$ and $H$ are commensuarble?
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$[H\oplus \mathbb{Z}\cap G\oplus \mathbb{Z}: H\oplus \mathbb{Z}]=[H\cap G\oplus \mathbb{Z}: H\oplus \mathbb{Z}]=[H\cap G: H]$ – Pax Jul 07 '16 at 04:54
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@Kamina, are you writing indices the other way round? – Andreas Caranti Jul 07 '16 at 12:36
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2It's true when $G$ is finitely generated and nilpotent. – YCor Jul 07 '16 at 22:57
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How? can you give some hints? – Sunny Jul 08 '16 at 16:06
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http://math.stackexchange.com/questions/1122839/when-are-two-direct-products-of-groups-isomorphic/1123044#1123044, use Seirios' answer. – Moishe Kohan Jul 08 '16 at 19:29