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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?

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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.

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  • $\begingroup$ Many thanks for your answer, the fact is I understand that you used isomorphism theorem to establish those tow isomorphism relations! Yet, how did you commute between the assertion of isomorphic to the presentation? $\endgroup$ – A.Messab Nov 21 '18 at 9:51
  • $\begingroup$ What is a presentation of a group? Writing $G = \langle a_1, a_2, \dots, a_d\:|\: R\rangle$ is just another way of saying that there exists a surjective morphism $F \to G$ from the free group on $d$ generators and the kernel of this morphism is the smallest normal subgroup of $F$ containing $R$ (which is $R$ if $R$ is itself a normal subgroup). As an example you might consider what writing $D_{8} = \langle s,t \:|\: s^2, t^4, sts^{-1}t \rangle$ actually means. $\endgroup$ – Matthias Klupsch Nov 21 '18 at 10:12
  • $\begingroup$ Many many many thanks; I was reading polycyclic presentations to get a deep understanding for what so-called p-generating, your definition for presentation is the most "meaningful" that I "encounter" with. My best regards $\endgroup$ – A.Messab Nov 21 '18 at 10:20
  • $\begingroup$ I am glad that I could help you. If you are satisfied with my answer, you might consider accepting it. $\endgroup$ – Matthias Klupsch Nov 21 '18 at 10:23
  • $\begingroup$ Sorry to didn´t that from the first, I was so excited! $\endgroup$ – A.Messab Nov 21 '18 at 10:27

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