Timeline for Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Aug 16, 2021 at 15:43 | comment | added | mme | I feel that this suggests more than anything else that when we enumerate groups by cardinality, we are counting them incorrectly. Should they be perhaps weighted relative to the number of prime factors of |G|, and their size? 2-groups are so prevalent because they have lots of room for nontrivial extensions, so perhaps one should weight a group based on whether its cardinality plausibly supports lots of extensions. | |
| Jul 29, 2021 at 16:45 | comment | added | Emil Jeřábek | To be honest, I don’t; I learned these things from scattered remarks here on MO by more knowledgeable people. The one I could find now is mathoverflow.net/a/164218 (which only explicitly mentions 2-groups of class 2, not the more refined structure). AFAIUI, the supporting evidence, beyond some computation, are the classical (roughly matching) upper and lower bounds on the number of groups by Higman (doi.org/10.1112/plms/s3-10.1.24) and Sims (doi.org/10.1112/plms/s3-15.1.151), where the lower bound comes from groups of the form I mentioned in the previous comment. | |
| Jul 29, 2021 at 16:17 | comment | added | JoshuaZ | @EmilJeřábek Huh. I did not know that and that's a bit surprising! Do you have a reference for that conjecture? | |
| Jul 29, 2021 at 15:55 | comment | added | Emil Jeřábek | (My previous comment is confusing. It is, of course, not true in general that a class-2 nilpotent 2-group is an extension of an elementary abelian group by another one. However, it is conjectured that most finite groups are of this form, i.e., class-2 nilpotent 2-groups $G$ such that both $G/Z(G)$ and $Z(G)=G'$ are elementary abelian.) | |
| Jul 29, 2021 at 15:15 | comment | added | Emil Jeřábek | Well, most groups are conjectured to be not just $2$-groups, but also class-$2$ nilpotent. I’d say that that’s indeed a fairly strong “well-behavedness” condition, just short of being abelian, and the fact that they are formed as an extension of $(C_2)^n$ by $(C_2)^m$ does make their structure rather more boring than general finite groups (even though it still leaves room for them to be messy enough). | |
| Jul 29, 2021 at 13:05 | comment | added | JoshuaZ | @IlmariKaronen I'm not sure if that's what is going on here. A lot of those other 2-groups are complicated, hard to describe and aren't very well-behaved at all. They aren't boring, they just don't show up. | |
| Jul 29, 2021 at 8:25 | comment | added | Ilmari Karonen | … I've ran into something quite similar in applied math before: whereas a random family of $k$ permutations of an $n$-element set is quite useful as an ideal block cipher in cryptography, a random commutative family of $k$ permutations of an $n$-element set turns out, with probability tending to 1 as $k\to\infty$, to be extremely boring and useless. | |
| Jul 29, 2021 at 8:25 | comment | added | Ilmari Karonen | This seems like an instance of a generic phenomenon where 1) we have a parametric family of objects we're interested in (e.g. groups with at most $n$ elements), and 2) it contains a subfamily (e.g. 2-groups) whose size grows faster with $n$ than the size of any other subfamily, such that for large $n$ almost all objects in the family belong to that particular subfamily, but 3) objects in the subfamily have properties that make their behavior very simple and boring, so that few if any of them are of any interest to anyone. | |
| Jul 28, 2021 at 11:45 | comment | added | Kvothe | @JoshuaZ, thanks, that is probably fair. It is however not covered in undergraduate Physics ;) | |
| Jul 28, 2021 at 11:39 | comment | added | JoshuaZ | @Kvothe I've added a link to p-group. (It would not have occurred to me though that p-group needed to be defined since it is a pretty standard abstract algebra term which I would think would be standard in the undergrad math curriculum.) | |
| Jul 28, 2021 at 11:38 | history | edited | JoshuaZ | CC BY-SA 4.0 |
Adding link to p-group
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| Jul 28, 2021 at 10:24 | comment | added | David Zhang | @Kvothe I understand your frustration, but "p-group" is standard terminology in abstract algebra which is freely used on this site. It is a group in which each element has order $p^k$, or equivalently for finite groups, a group of order $p^n$. | |
| Jul 28, 2021 at 8:47 | comment | added | Kvothe | For someone who does not know what a 2-group or a p-group is, the link is extremely confusing. Is it linking to a definition that is NOT relevant to this answer?) How about including a link that is relevant to this answer in that case. | |
| Jul 27, 2021 at 2:00 | comment | added | Benjamin Steinberg | Or most semigroups have a zero and the product of any three elements are zero but nobody ever considers these | |
| Jul 27, 2021 at 0:32 | comment | added | Richard Stanley | In the same vein, most finite partially ordered sets have height two (i.e., longest chain of cardinality three) by ams.org/journals/tran/1975-205-00/S0002-9947-1975-0369090-9, not at all like the ones occurring "naturally." | |
| Jul 26, 2021 at 21:46 | history | edited | JoshuaZ | CC BY-SA 4.0 |
clarify definition of 2-group
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| Jul 26, 2021 at 21:26 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Jul 26, 2021 at 19:20 | history | edited | JoshuaZ | CC BY-SA 4.0 |
grammar, spelling
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| Jul 26, 2021 at 19:07 | history | answered | JoshuaZ | CC BY-SA 4.0 |