Timeline for The meaning and purpose of "canonical''
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 27, 2020 at 20:20 | comment | added | Carl-Fredrik Nyberg Brodda | @RyleeLyman That it is undecidable whether a finite presentation defines the trivial group is one of the consequences of the Adian-Rabin theorem, which (very informally) says that "all we can really say about a group given by a finite presentation is that it is finitely presented". But it is important to note that your "canonical presentation" is a finite presentation iff $G$ is finite, so Adian-Rabin is a bit of a sideline. | |
| Feb 20, 2020 at 10:24 | comment | added | user44143 | @RyleeLyman, I agree about the theorem, and the resulting lack of algorithm for canonical group presentations. Meanwhile the construction you describe starts from an underlying set $G$ and a subset of $G^3$ representing $gh=k$ — isn’t that already a description without making any choices? | |
| Feb 20, 2020 at 7:19 | comment | added | Martin Brandenburg | More generally, when $T$ is a monad and $(X,h)$ is a $T$-algebra, then $(X,h)$ has a canonical presentation $T^2(X) \rightrightarrows T(X) \to X$. The two parallel maps on the left are $T(h)$ and $\mu_X$, the map on the right is $h$. I would not call this presentation silly, it is quite useful at least in the general theory. | |
| Feb 20, 2020 at 4:20 | comment | added | Robbie Lyman | @MattF. I believe that it is a theorem that there is no general algorithm to determine whether a given finite group presentation presents the trivial group, so I think there is no hope for an arbitrary group to have a canonical presentation in the way you describe. I choose to call it canonical because the presentation can be described without making any choices involving $G$ or its elements. | |
| Feb 20, 2020 at 3:18 | comment | added | user44143 | I want a canonical presentation of a group to allow an easy determination of when two groups are isomorphic — this does not do that, so I would not call it canonical. | |
| Feb 20, 2020 at 0:01 | history | answered | Robbie Lyman | CC BY-SA 4.0 |