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











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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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    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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      up vote
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      down vote

      favorite









      up vote
      0
      down vote

      favorite











      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?










      share|cite|improve this 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?







      abstract-algebra finite-groups






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      asked Nov 21 at 9:19









      A.Messab

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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.






          share|cite|improve this answer





















          • 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 at 9:51










          • What is a presentation of a group? Writing $G = langle a_1, a_2, dots, a_d:|: Rrangle$ 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 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 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 at 10:23










          • Sorry to didn´t that from the first, I was so excited!
            – A.Messab
            Nov 21 at 10:27











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          1 Answer
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          1 Answer
          1






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          active

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          active

          oldest

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          up vote
          0
          down vote



          accepted










          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.






          share|cite|improve this answer





















          • 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 at 9:51










          • What is a presentation of a group? Writing $G = langle a_1, a_2, dots, a_d:|: Rrangle$ 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 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 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 at 10:23










          • Sorry to didn´t that from the first, I was so excited!
            – A.Messab
            Nov 21 at 10:27















          up vote
          0
          down vote



          accepted










          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.






          share|cite|improve this answer





















          • 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 at 9:51










          • What is a presentation of a group? Writing $G = langle a_1, a_2, dots, a_d:|: Rrangle$ 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 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 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 at 10:23










          • Sorry to didn´t that from the first, I was so excited!
            – A.Messab
            Nov 21 at 10:27













          up vote
          0
          down vote



          accepted







          up vote
          0
          down vote



          accepted






          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.






          share|cite|improve this answer












          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.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 21 at 9:34









          Matthias Klupsch

          6,1391227




          6,1391227












          • 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 at 9:51










          • What is a presentation of a group? Writing $G = langle a_1, a_2, dots, a_d:|: Rrangle$ 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 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 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 at 10:23










          • Sorry to didn´t that from the first, I was so excited!
            – A.Messab
            Nov 21 at 10:27


















          • 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 at 9:51










          • What is a presentation of a group? Writing $G = langle a_1, a_2, dots, a_d:|: Rrangle$ 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 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 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 at 10:23










          • Sorry to didn´t that from the first, I was so excited!
            – A.Messab
            Nov 21 at 10:27
















          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 at 9:51




          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 at 9:51












          What is a presentation of a group? Writing $G = langle a_1, a_2, dots, a_d:|: Rrangle$ 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 at 10:12






          What is a presentation of a group? Writing $G = langle a_1, a_2, dots, a_d:|: Rrangle$ 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 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 at 10:20




          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 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 at 10:23




          I am glad that I could help you. If you are satisfied with my answer, you might consider accepting it.
          – Matthias Klupsch
          Nov 21 at 10:23












          Sorry to didn´t that from the first, I was so excited!
          – A.Messab
          Nov 21 at 10:27




          Sorry to didn´t that from the first, I was so excited!
          – A.Messab
          Nov 21 at 10:27


















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