# Commutator subgroup of a subgroup

Suppose $H \subset G$ is a subgroup of finite index (assume normal if necessary). Must it be the case that $[H, H] \subset [G,G]$ is of finite index? (i.e. the map $H^{ab} \rightarrow G^{ab}$ is of finite kernel)

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Let's take $G = D_{\infty} = \langle r, s : s^2 = 1, srs = r^{-1}\rangle$ and its subgroup $H = \langle r \rangle$. This must be a counterexample if I didn't make a mistake again.
Indeed, when $H=[G,G]$, one is asking if the first quotient of the derived series is finite, must the others all be as well. This is clearly false as infinite abelian groups can have automorphisms of finite order. –  Jack Schmidt Jun 17 '12 at 19:16