If $G=langle{a_i}_{i=1}^rrangle$ is nilpotent with $|a_i|=m_i$, show that $|G|$ divides a power of...












1














enter image description here



use this notation for the following $textbf{Theorem}$ -



$textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$



$textbf{Theorem}$- In a finitely generated nilpotent group, $cap{G^p}$, for any infinite set of primes, is finite,(follows by a paper of HIGMAN) does it help me to prove this question above. I don't think so.



Is it true, that a group generated by finite elements of finite order is finite. If it is true then to prove $G$ is finite in the problem is easy, and don not require nilpotency. But it should not be true, but what can be an example?










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  • 1




    Tarski monsters are examples of finitely generated groups whose generators have finite order.
    – Teri
    Sep 25 '14 at 17:41












  • so we need nilpotency to say $G$ is finite. How can i approach this problem?
    – Bhaskar Vashishth
    Sep 25 '14 at 17:43










  • Use induction on the nilpotence class, and consider the central quotient in the non-abelian case.
    – James
    Sep 25 '14 at 19:24












  • I can't use induction, until I show $G$ is finite
    – Bhaskar Vashishth
    Sep 25 '14 at 19:25






  • 1




    @BhaskarVashishth The induction is on nilpotence class, which is finite by assumption.
    – zibadawa timmy
    Sep 25 '14 at 19:33
















1














enter image description here



use this notation for the following $textbf{Theorem}$ -



$textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$



$textbf{Theorem}$- In a finitely generated nilpotent group, $cap{G^p}$, for any infinite set of primes, is finite,(follows by a paper of HIGMAN) does it help me to prove this question above. I don't think so.



Is it true, that a group generated by finite elements of finite order is finite. If it is true then to prove $G$ is finite in the problem is easy, and don not require nilpotency. But it should not be true, but what can be an example?










share|cite|improve this question




















  • 1




    Tarski monsters are examples of finitely generated groups whose generators have finite order.
    – Teri
    Sep 25 '14 at 17:41












  • so we need nilpotency to say $G$ is finite. How can i approach this problem?
    – Bhaskar Vashishth
    Sep 25 '14 at 17:43










  • Use induction on the nilpotence class, and consider the central quotient in the non-abelian case.
    – James
    Sep 25 '14 at 19:24












  • I can't use induction, until I show $G$ is finite
    – Bhaskar Vashishth
    Sep 25 '14 at 19:25






  • 1




    @BhaskarVashishth The induction is on nilpotence class, which is finite by assumption.
    – zibadawa timmy
    Sep 25 '14 at 19:33














1












1








1







enter image description here



use this notation for the following $textbf{Theorem}$ -



$textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$



$textbf{Theorem}$- In a finitely generated nilpotent group, $cap{G^p}$, for any infinite set of primes, is finite,(follows by a paper of HIGMAN) does it help me to prove this question above. I don't think so.



Is it true, that a group generated by finite elements of finite order is finite. If it is true then to prove $G$ is finite in the problem is easy, and don not require nilpotency. But it should not be true, but what can be an example?










share|cite|improve this question















enter image description here



use this notation for the following $textbf{Theorem}$ -



$textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$



$textbf{Theorem}$- In a finitely generated nilpotent group, $cap{G^p}$, for any infinite set of primes, is finite,(follows by a paper of HIGMAN) does it help me to prove this question above. I don't think so.



Is it true, that a group generated by finite elements of finite order is finite. If it is true then to prove $G$ is finite in the problem is easy, and don not require nilpotency. But it should not be true, but what can be an example?







group-theory finite-groups nilpotent-groups






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share|cite|improve this question













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edited Nov 28 '18 at 18:58









Shaun

8,805113680




8,805113680










asked Sep 25 '14 at 17:35









Bhaskar Vashishth

7,58712053




7,58712053








  • 1




    Tarski monsters are examples of finitely generated groups whose generators have finite order.
    – Teri
    Sep 25 '14 at 17:41












  • so we need nilpotency to say $G$ is finite. How can i approach this problem?
    – Bhaskar Vashishth
    Sep 25 '14 at 17:43










  • Use induction on the nilpotence class, and consider the central quotient in the non-abelian case.
    – James
    Sep 25 '14 at 19:24












  • I can't use induction, until I show $G$ is finite
    – Bhaskar Vashishth
    Sep 25 '14 at 19:25






  • 1




    @BhaskarVashishth The induction is on nilpotence class, which is finite by assumption.
    – zibadawa timmy
    Sep 25 '14 at 19:33














  • 1




    Tarski monsters are examples of finitely generated groups whose generators have finite order.
    – Teri
    Sep 25 '14 at 17:41












  • so we need nilpotency to say $G$ is finite. How can i approach this problem?
    – Bhaskar Vashishth
    Sep 25 '14 at 17:43










  • Use induction on the nilpotence class, and consider the central quotient in the non-abelian case.
    – James
    Sep 25 '14 at 19:24












  • I can't use induction, until I show $G$ is finite
    – Bhaskar Vashishth
    Sep 25 '14 at 19:25






  • 1




    @BhaskarVashishth The induction is on nilpotence class, which is finite by assumption.
    – zibadawa timmy
    Sep 25 '14 at 19:33








1




1




Tarski monsters are examples of finitely generated groups whose generators have finite order.
– Teri
Sep 25 '14 at 17:41






Tarski monsters are examples of finitely generated groups whose generators have finite order.
– Teri
Sep 25 '14 at 17:41














so we need nilpotency to say $G$ is finite. How can i approach this problem?
– Bhaskar Vashishth
Sep 25 '14 at 17:43




so we need nilpotency to say $G$ is finite. How can i approach this problem?
– Bhaskar Vashishth
Sep 25 '14 at 17:43












Use induction on the nilpotence class, and consider the central quotient in the non-abelian case.
– James
Sep 25 '14 at 19:24






Use induction on the nilpotence class, and consider the central quotient in the non-abelian case.
– James
Sep 25 '14 at 19:24














I can't use induction, until I show $G$ is finite
– Bhaskar Vashishth
Sep 25 '14 at 19:25




I can't use induction, until I show $G$ is finite
– Bhaskar Vashishth
Sep 25 '14 at 19:25




1




1




@BhaskarVashishth The induction is on nilpotence class, which is finite by assumption.
– zibadawa timmy
Sep 25 '14 at 19:33




@BhaskarVashishth The induction is on nilpotence class, which is finite by assumption.
– zibadawa timmy
Sep 25 '14 at 19:33










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