Timeline for Large gaps in the norm of a subgroup and its centraliser
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| May 21, 2020 at 21:48 | comment | added | YCor | Yes, in the lamplighter these are subgroups like this. | |
| May 21, 2020 at 21:46 | comment | added | ARG | Thanks, I am quite sure such an example should exist, but I'm don't know how to build one (the example I mention at the end with $C= Z_G(G)$ having large gaps is quite convoluted). I'll try to look at HNN-extensions... (I assume the lamplighter subgroups are some parts of the lamp group? By that I mean $\oplus_{i_n} \mathbb{Z}_2$ (as a subgroup of $\oplus_{\mathbb{Z}} \mathbb{Z}_2$ with a well-chose sequence $\{i_n\} \subset \mathbb{Z}$). | |
| May 21, 2020 at 21:39 | comment | added | YCor | For your question: first, I know finitely generated groups with a pair of infinite locally finite subgroups with your property (one can find such a pair in the lamplighter). But it's not a subgroup and its centralizer. I suspect an example how your require exists anyway, playing suitably with HNN extensions. | |
| May 21, 2020 at 21:34 | history | edited | ARG | CC BY-SA 4.0 |
added further missing adjectives "infinite"
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| May 21, 2020 at 21:32 | comment | added | ARG | Thanks (i just was not sure if there was nonetheless a new ingredient required). So for eventual readers (which are as slow as I am): (1) If a subgroup $S$ is infinite, $\mathrm{norm}(S)$ is unbounded (because $G$ is fin.gen.) (2) In a fin. gen. subgroup, the norm of the generating elements bound the size of the gap (again by the triangle inequality) (3) Hence the $N$ and $Z$ [in the question] may not have infinite finitely generated subgroups (which is to say that they are locally finite) | |
| May 21, 2020 at 21:23 | comment | added | YCor | I said "your argument... carries over". It's essentially the same argument but it yields a much stronger conclusion. | |
| May 21, 2020 at 21:18 | comment | added | ARG | Your remark on locally finiteness is essentially an "upgrade" on my remark on the fact that no element (of $N$ or $Z$) may have infinite order... or am I missing an important point? | |
| May 21, 2020 at 21:11 | history | edited | ARG | CC BY-SA 4.0 |
added the missing adjectives "infinite"
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| May 21, 2020 at 21:11 | comment | added | ARG | You are right: I forgot to add the adjective "infinite" before $N$ and $Z_G(N)$. | |
| May 21, 2020 at 21:08 | comment | added | YCor | These conditions are trivially fulfilled if $N$ or $Z$ is finite. Also your argument that this is "no" if $Z$ or $N$ is not torsion also carries over the case when $Z$ or $N$ is locally finite. That is, your conditions imply that $Z$ and $N$ are locally finite. | |
| May 21, 2020 at 20:42 | history | asked | ARG | CC BY-SA 4.0 |