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Let $G$ be a finitely generated nilpotent group of class $2$. Denote the center of $G$ by $Z$. Let $H\leq G$ be a subgroup such that $[G:HZ]<\infty$. Does it follow that $[G:H]<\infty$.

Equivalently, since $[G:H]=[G:HZ]\cdot[HZ:H]=[G:HZ]\cdot[Z:H\cap Z]$, does it follow that $[Z:H\cap Z]<\infty$?

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    No it doesn't follow. $H$ could be a direct factor of $G$, with the other factor infinite abelian. – 2017-01-31
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    @DerekHolt: Thanks. If I replace $Z$ by $[G,G]$, the statement does become true, right? – 2017-01-31
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    Yes, the statement becomes true if you replace $Z$ by $D =[G,G]$, basically because $D \le Z$ so $[HD,HD] = [H,H]$. – 2017-01-31
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    @DerekHolt: Isn't the statement for $[G,G]$ true for nilpotent groups of all classes, irregardless of whether $[G,G]\leq Z$? – 2017-02-01
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    Yes I believe it is! – 2017-02-01

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