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

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?

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.

• 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? – A.Messab Nov 21 '18 at 9:51
• 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. – Matthias Klupsch Nov 21 '18 at 10:12
• 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 – A.Messab Nov 21 '18 at 10:20
• I am glad that I could help you. If you are satisfied with my answer, you might consider accepting it. – Matthias Klupsch Nov 21 '18 at 10:23
• Sorry to didn´t that from the first, I was so excited! – A.Messab Nov 21 '18 at 10:27