Timeline for Name this pro-$p$ group
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Oct 22, 2017 at 17:30 | comment | added | YCor | This pro-$p$-presentation $G=\langle a,b\mid a^p,[a,a^b]\rangle$ clearly does not work. Indeed, first consider the subgroup of index $p$ kernel of the map to $C_p$ mapping $(a,b)\mapsto (1,0)$, and write $c=a^p$, $b_i=t^ibt^{-i}$. It has the presentation $\langle c,b_1,\dots, b_p\mid b_i^p,[b_i,b_{i+1}],[b_p,tb_1t^{-1}]\rangle$. The quotient by the generators $b_2,\dots,b_p$ yields the presentation $\langle c,b_1\mid b_1^p\rangle$ of the free pro-$p$ product of $C_p$ and $\mathbf{Z}_p$. So $G$ is certainly not metabelian. | |
| Feb 12, 2012 at 11:55 | vote | accept | Colin Reid | ||
| Jul 29, 2010 at 9:09 | comment | added | Colin Reid | Yes, that sounds about right. It certainly contains a dense copy of $C_p \wr \mathbb{Z}$ (restricted wreath product), but I'm not sure if it contains a copy of the unrestricted wreath product. | |
| Jul 28, 2010 at 17:00 | history | answered | Jonathan Kiehlmann | CC BY-SA 2.5 |