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Let $F$ be a free group on two generators, let $F^p$ denote the normal subgroup of $F$ generated by the $p$-th power, $p$ a prime. Do we know anything about the exponent of $[F,F]/[F^p,F]$.

Edit: $G=F/F^p$ is known to be infinite.

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  • $\begingroup$ If $p=2$, then $[F,F]/[F^2,F]$ is generated by the image of $[x,y]$ in the quotient group. So the quotient is cyclic if $p=2$. Why you think that the exponent must be finite in the case $p=2$? $x$ and $y$ are free generators of $F$. $\endgroup$
    – Alireza Abdollahi
    Commented Mar 8, 2016 at 10:03
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    $\begingroup$ The answer is $2$ for $p=2$ and $3$ for $p=3$. It would probably be possible to compute it for $p=4$ and $6$ (but was $p$ supposed to be prime?). I presume the question is mainly concerned with other values of $p$, when $F/F^p$ is either known to be infinite or at least not known to be finite. $\endgroup$
    – Derek Holt
    Commented Mar 8, 2016 at 10:15

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