How to proof that $G$ and $G^*$ has the same number of generators?

Question

Let $G$ be a $d-$generator $p-$group, Let $F$ be the free group of rank $d$ freely generated by $a_1$ . . . . . $a_d$, and let $R$ be the kernel of a homomorphism $\theta$ from $F$ onto $G$; Define $R^*$ to be $[R, F]R^p$ and $G^*$ to be $F/R^*$. Does $G$ and $G^*$ has the same number of generators?

Answer 1

In the following I will assume that $\theta: F \to G$ is not just any homomorphism of groups but a surjective one, so that we have an isomorphism $F/R \cong G$ (or a presentation $\langle a_1, \dots, a_d\:|\: R \rangle$)

Similarly, $G^*$ admits - by definition - a surjective morphism $F \to G^*$ or the presentation $\langle a_1, \dots, a_d\:|\: [R,F]R^q \rangle$. In particular, $G^*$ is generated by $d$ elements.

It might also be noteworthy that if $G$ is not generated by less than $d$ elements, then this is true for $G^*$, too, since $G$ is a quotient of $G^*$.

Also note that the number of generators is not something which is well-defined per se as any element of a group may participate in a generating set and minimal generating sets might not have the same size (for example $\Bbb{Z}$ has minimal generating sets $\{1\}$ and $\{2,3\}$). You can ask if the sizes of smallest generating sets of $G$ and $G^*$ is identical and the above considerations show that this is actually the case.