Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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)

share|improve this question

1 Answer 1

up vote 8 down vote accepted

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.

share|improve this answer
2  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.