Timeline for Torsion - subgroup and quotient
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2011 at 9:38 | comment | added | Andrei Jaikin | @Michael: Yes, you prove the first claim by induction on the nilpotency class using that the factors of the (topological) lower central series are (topologically) finitely generated. The subgroup $\overline{\langle T\rangle }$ is topologically generated by a finite subset of $T$. Since these elements are torsion, then $\overline{\langle T\rangle }$ is finite. | |
| Jun 16, 2011 at 2:12 | comment | added | Michael | @Andrei: To prove your first claim, should I check that the factors of the (topological) lower central series are (topologically) finitely generated from basic commutators, or is there a simpler argument? About the second claim, how did you choose (topological) generators of $\overline{T}$ in $T$? Sorry about these questions, I'm just beginning to learn profinite groups. | |
| Jun 15, 2011 at 9:02 | comment | added | Andrei Jaikin | @Michael: If $G$ is finitely generated and nilpotent, then every closed subgroup is also finitely generated and a subgroup generated by a finite number of torsion elements is finite. Thus if $G$ is nilpotent and finitely generated, then $\langle T\rangle=T $ is finite. | |
| Jun 15, 2011 at 4:06 | comment | added | Michael | @Andrei: Many thanks! Though, I must say I need some time to fully understand your interesting example. Do you think it is possible to do something similar when the group is nilpotent? (the case I'm in fact interested in). Thanks again. | |
| Jun 14, 2011 at 13:16 | history | edited | Andrei Jaikin | CC BY-SA 3.0 |
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| Jun 14, 2011 at 12:50 | history | edited | Andrei Jaikin | CC BY-SA 3.0 |
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| Jun 14, 2011 at 12:42 | history | answered | Andrei Jaikin | CC BY-SA 3.0 |