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Thread: A finitely generated group with a finite derived group.

  1. #1
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    A finitely generated group with a finite derived group.

    This is a problem from Theory and Problems of Group Theory by B.Baumslag and B.Chandler, McGraw-Hill, 1968, belonging to the Schaum's Outline Series. It is problem 7.57, page 244: $G$ is a finitely generated group every element of which has only a finite number of conjugates. Prove that $G'$, the derived group, is finite. (Hint: $\cap C(g_i)=Z(G)$ where the intersection is taken from $i=1$ to $i=n$ if $g_1, ..., g_n$ are the generators of $G$.)


    If I could show that $Z(G)$ has finite index then by theorem 7.8 in the book $G'$ is finite. If $C(g_i)$ is of finite index then the intersection, $Z(G)$ if of finite index too.

    Also if $G'$ is finitely generated and every element of $G'$ is of finite order then $G'$ is finite. Now $C(g)={x \in G: xgx^{-1}=g}$ and for $g$ fixed there is a finite number of $xgx^{-1}$. This is all I can see. How can I use the hint?
    Last edited by STF92; Sep 16th 2019 at 06:01 PM.
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  2. #2
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    Re: A finitely generated group with a finite derived group.

    Have you tried to prove that $G$ is finite? The derived subgroup would then be necessarily finite by containment as a subset.
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  3. #3
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    Re: A finitely generated group with a finite derived group.

    Thanks and sorry for the delay. If it were all about proving that G is finite the problem would have been stated another way: prove that G is finite.
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