Timeline for What is the standard 2-generating set of the symmetric group good for?
Current License: CC BY-SA 4.0
25 events
| when toggle format | what | by | license | comment | |
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| Jul 27 at 19:37 | history | edited | Matthieu Romagny | CC BY-SA 4.0 |
his/her --> them
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| Oct 4, 2021 at 19:50 | vote | accept | Matthieu Romagny | ||
| Oct 4, 2021 at 14:16 | comment | added | Roland Bacher | It can be used for building a cheap racetrack. A fast car can overtake a slower one only at a unique point in the track. (This is of course essentially the same answer as Robert Furber's.) | |
| Oct 4, 2021 at 12:59 | comment | added | HJRW | (Apologies if this answer is duplicated somewhere else. It's certainly similar in spirit to some others, such as the beautiful answer about Galois theory, but is more elementary.) A generating set for a group $G$ is very good for determining if a homomorhims $H\to G$ is surjective, since you just need to check if the generators are in the image! One can set lots of easy exercises that are variants on this. This is similar to Derek Holt's answer, except that because Derek focussed on homomorphisms from $G$, one run into the awkward fact that most $S_n$'s are almost simple. | |
| Oct 4, 2021 at 12:16 | comment | added | markvs | The answer is: "Absolutely nothing. Uh ha haa ha." | |
| Oct 4, 2021 at 10:57 | answer | added | JP McCarthy | timeline score: 0 | |
| Oct 3, 2021 at 2:48 | comment | added | Robert Furber | This isn't really a mathematical answer, but the way I like to think about this presentation is that I have a set of plates on a "lazy Susan", one of those wooden turntables you can put plates on in the middle of a table. One generator is rotating the lazy Susan, and the other is swapping the two plates directly in front of me. Using only these two operations, I can get any ordering of the plates that I like. | |
| Oct 2, 2021 at 15:16 | review | Close votes | |||
| Oct 2, 2021 at 19:29 | |||||
| Oct 2, 2021 at 12:51 | comment | added | Matthieu Romagny | I must say that I thought the answer would have interest for educators, but interesting answers would come mainly from researchers. | |
| Oct 2, 2021 at 12:34 | comment | added | John Coleman | This might be a good question for Mathematics Educators. | |
| Oct 2, 2021 at 6:43 | comment | added | Kapil | It allows one to construct a covering of $\mathbb{P}^1$ which is (a) ramified at only 3 points (b) has covering group $S_n$. This covering is defined over a number field. (A 2-generated group is a quotient of the free group with 2 generators which is the fundamental group of $\mathbb{P}^1-3 \mathrm{points}$. Is this example too advanced? | |
| Oct 1, 2021 at 23:07 | history | became hot network question | |||
| Oct 1, 2021 at 22:01 | comment | added | Derek Holt | @LSpice All finite simple groups are $2$-generated. In fact the probability of any two randomly chosen elements generating a nonabelian simple group $G$ tends to $1$ as $|G| \to \infty$. | |
| Oct 1, 2021 at 21:20 | comment | added | LSpice | It's not my question, but I'd be interested, even more generally than the significance of $\operatorname S_n$ being $2$-generated, in the significance of any group being $2$-generated. For example, I seem to remember that most simple groups are $2$-generated. Is that just a fact, or does it have any further implications? | |
| Oct 1, 2021 at 19:49 | answer | added | Pace Nielsen | timeline score: 11 | |
| Oct 1, 2021 at 18:59 | answer | added | Jukka Kohonen | timeline score: 14 | |
| Oct 1, 2021 at 17:08 | answer | added | Derek Holt | timeline score: 16 | |
| Oct 1, 2021 at 17:00 | comment | added | Matthieu Romagny | @Sam Hopkins you're right. In fact I am not particularly interested in this 2-set, and I'd be pleased to bo told concrete applications of having a pair of generating elements in $S_n$. | |
| Oct 1, 2021 at 16:59 | comment | added | Matthieu Romagny | @YCor I called the set standard because it is the one that is found in all algebra textbooks for students. But I agree there are many 2-element generating sets: a theorem of Isaacs and Zieschang says that any nonidentity permutation can be completed to such a set. | |
| Oct 1, 2021 at 15:36 | comment | added | Jules Lamers | I don't know what it's particularly useful for, but find this presentation interesting as a reminder that the Coxeter presentation by simple transpositions (which I like and use) is by no means minimal: to me the interesting part is that there's a generating set with just two elements | |
| Oct 1, 2021 at 15:31 | comment | added | Sam Hopkins | It's a little unclear from the question whether you're most interested in the fact that these particular two elements generate $S_n$, or whether you care more about the observation that $S_n$ is $2$-generated in general. | |
| Oct 1, 2021 at 15:25 | comment | added | YCor | I never heard or thought of this as "the" standard generating subset, it's one among many. It's maybe just the simplest one on 2 generators we can describe in mathematical terms. (By the way it also works in the symmetric group over $\mathbf{Z}$, using the shift $+1$ as infinite cycle, generating the whole finite support symmetric group.) A little farther from teaching level: It's also known to be highly inefficient compared to "generic pairs", if one wants to get the size of the Cayley graph as small as possible, or if one wants to generate random elements by multiplying random generators. | |
| Oct 1, 2021 at 15:15 | answer | added | Geoff Robinson | timeline score: 24 | |
| Oct 1, 2021 at 15:05 | answer | added | Thomas Browning | timeline score: 40 | |
| Oct 1, 2021 at 15:04 | history | asked | Matthieu Romagny | CC BY-SA 4.0 |