Timeline for Group presentation in the category of finite group
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jun 6 at 6:48 | comment | added | Ian Agol | An even more elementary example of a non-residually finite group is provided by Baumslag-Solitar groups $BS(2,3)$. There is a surjective non-injective homomorphism $f:BS(2,3)\to BS(2,3)$. Then $ker(f)$ is contained in the kernel of any map to a finite group, hence the group is not residually finite. en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_group | |
| Jun 6 at 1:17 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
verify —> satisfies
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| Oct 16, 2021 at 7:22 | history | edited | hivert | CC BY-SA 4.0 |
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| Oct 15, 2021 at 16:58 | history | became hot network question | |||
| Oct 15, 2021 at 14:43 | comment | added | hivert | Fair point. Thanks. | |
| Oct 15, 2021 at 14:20 | comment | added | YCor | @hivert but the core of the issue in your question lies in presentations of the trivial group. So no, it's definitely not silly and let me give an example: generators $h_n=1$ ($n\ge 0$), with relators $h_{n+1}^2=h_n$ $\forall n\ge 0$ and $h_0^2=1$. Then such relators in a finite group force $h_0=1$ (and, by induction force $h_n=1$ for all $n$). But $h_0=1$ does not formally follow from these relators, since it is easy to find an infinite group with such elements with these relations, but $h_0\neq 1$. | |
| Oct 15, 2021 at 13:34 | comment | added | hivert | @YCor: I should have written one or two more finite in the rephrasing ! Anyway having an infinite presentation (either in the generator or the relations) for a finite group seems to be silly to me... | |
| Oct 15, 2021 at 13:07 | comment | added | YCor | I had missed that the set $I$ is finite. So the first question is indeed whether every finitely presented group is residually finite. But in the "rephrasing", OP doesn't assume $R$ nor the indexing set of the family $(h_i)$ to be finite, so it's indeed equivalent to asking whether every group is residually finite. | |
| Oct 15, 2021 at 11:22 | vote | accept | hivert | ||
| Oct 15, 2021 at 10:43 | answer | added | Derek Holt | timeline score: 12 | |
| Oct 15, 2021 at 9:56 | comment | added | Denis Nardin | @hivert In Derek Holt's comment $G$ is the trivial group. They're saying that there are finitely presented groups with no nontrivial finite quotients (I do not know much group theory, so I don't know any such example) | |
| Oct 15, 2021 at 9:48 | comment | added | hivert | @DerekHolt: Sorry I don't get it. In my definition I'm assuming that G itself is finite. | |
| Oct 15, 2021 at 9:38 | comment | added | Derek Holt | No, you seem to be asking whether every group is residually finite, which is false. According to your definition, a (standard) presentation of a nontrivial group with no nontrivial finite quotients would be a presentation of the trivial group. | |
| Oct 15, 2021 at 9:33 | comment | added | hivert | It seems that I've found what I need in groupprops.subwiki.org/wiki/Residually_finite_group | |
| Oct 15, 2021 at 9:18 | comment | added | hivert | @YCor: So I guess my question is : is any finite group residually finite. Wikipedia claims it is. Do you have any reference for that ? I'm coming from algebraic combinatorics and I'm mostly knowledgable in Coxeter groups where it goes to group theory. | |
| Oct 15, 2021 at 9:13 | history | edited | hivert | CC BY-SA 4.0 |
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| Oct 15, 2021 at 9:13 | comment | added | YCor | Your rephrased question is equivalent to whether every group is residually finite (take $H=1$). | |
| Oct 15, 2021 at 9:10 | comment | added | YCor | The groups satisfying such a property are the quotient of the group $H$ defined by this presentation (in the usual sense) by normal subgroups that are contained in the finite residual (=intersection of finite index subgroups of $H$). | |
| Oct 15, 2021 at 8:58 | history | asked | hivert | CC BY-SA 4.0 |