Given a free group $F$ on $d$ generators and a normal subgroup $H$ of $F$ whose index is finite of prime power order, is there a systematic way to find the numbers of generators of $H/[H,F]$ and of $H/[H,F]H^p$?



Generators of sections of free groups
This is really a cohomological question and has a simple cohomological answer. Recall that if a group $G$ acts on an abelian group $M$, then $M\_G$ denotes the *coinvariants* of the action, that is, the quotient of $M$ by the subgroup generated by $\{\text{$m-g(m)$ $|$ $m \in $M, $g \in G$}\}$. The group $F$ acts on $H$ by conjugation, and thus there is an induced action of $F$ on $H\_1(H)$.


**Key Observation** : $H/[H,F] \cong (H\_1(H;\mathbb{Z}))\_F$ and $H/[H,F]H^p \cong (H\_1(H;\mathbb{Z}/p))\_F$.


Indeed, we have $H/[H,H] \cong H\_1(H;\mathbb{Z})$ and $H/[H,H]H^p \cong H\_1(H;\mathbb{Z}/p)$ by definition, and quotienting by $[H,F]$ just kills off the $F$-action.


The other needed ingredient is the 5-term exact sequence in group homology. Given a short exact sequence


$$1 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 1$$


of groups and a ring of coefficients $R$, this 5-term exact sequence takes the form


$$H\_2(B;R) \longrightarrow H\_2(C;R) \longrightarrow (H\_1(A;R))\_B \longrightarrow H\_1(B;R) \longrightarrow H\_1(C;R) \longrightarrow 0.$$


Letting $Q = F/H$, we will apply this to the short exact sequence


$$1 \longrightarrow H \longrightarrow F \longrightarrow Q \longrightarrow 1.$$


The key simplification that occurs is that $H\_2(F;R) = 0$ since $F$ is free. We thus get exact sequences


$$0 \longrightarrow H\_2(Q;\mathbb{Z}) \longrightarrow H/[F,H] \longrightarrow H\_1(F;\mathbb{Z}) \longrightarrow H\_1(Q;\mathbb{Z}) \longrightarrow 0$$


and


$$0 \longrightarrow H\_2(Q;\mathbb{Z}/p) \longrightarrow H/[F,H]H^p \longrightarrow H\_1(F;\mathbb{Z}/p) \longrightarrow H\_1(Q;\mathbb{Z}/p) \longrightarrow 0.$$


If you understand $Q$ enough to calculate its first and second homologies, these short exact sequences allow you to determine $H/[F,H]$ and $H/[F,H]H^p$.


