Timeline for The mysterious significance of local subgroups in finite group theory
Current License: CC BY-SA 4.0
16 events
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| Dec 25, 2023 at 17:20 | comment | added | Richard Lyons | By the Frattini argument, every nontrivial subquotient $M/N$ of a finite group $G$ (by which I mean $1<N\le M\le G$) is isomorphic to a subquotient of a $p$-local subgroup of $G$ for any prime divisor $p$ of $|N|$. | |
| Dec 24, 2023 at 12:38 | answer | added | Geoff Robinson | timeline score: 13 | |
| Dec 22, 2023 at 10:40 | history | edited | semisimpleton | CC BY-SA 4.0 |
Mentioned the Thompson order formula, suggested by @Geoff Robinson in comments
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| S Dec 22, 2023 at 6:26 | history | bounty ended | semisimpleton | ||
| S Dec 22, 2023 at 6:26 | history | notice removed | semisimpleton | ||
| Dec 22, 2023 at 0:05 | history | edited | Sam Hopkins |
edited tags
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| Dec 21, 2023 at 12:17 | comment | added | Geoff Robinson | There are unresolved mysteries around this question. You might also mention the so-called Thompson order formula, which proves that if the finite group $G$ has more than one conjugacy class of involutions, then $|G|$ is precisely determined by $2$-local information | |
| Dec 21, 2023 at 11:27 | answer | added | Tom De Medts | timeline score: 15 | |
| Dec 21, 2023 at 7:20 | comment | added | semisimpleton | What I'm looking for is an "interpretation" of local subgroups. For example, prime ideals in a commutative ring can be interpreted as points in a space, and then elements of the ring can be interpreted as functions on that space. Is there some interpretation (not necessarily geometric) that relates local subgroups to some other familiar concepts? For example (as I mentioned in the main post), I remember reading somewhere (but I can't remember where!) that there is some analogy between local analysis and Tits' theory of buildings. Do you know of any references for this? | |
| Dec 21, 2023 at 7:12 | comment | added | semisimpleton | I assume in that last sentence you meant finite simple group | |
| Dec 21, 2023 at 5:53 | comment | added | Milo Moses | (2/2) As for the quote in Ronald Solomon's article, I'd say that I agree. The most important (and sometimes largest) subgroups of a given group $G$ are its (Sylow) p-subgroups. If you take the normalizers of these subgroups, you have a chance of getting something even larger. Additionally, when you're studying finite groups you don't need to worry about the normalizer of a proper subgroup not being proper. | |
| Dec 21, 2023 at 5:49 | comment | added | Milo Moses | (1/2) I assume that you're already well-convinced of the power and utility of p-subgroups, and the local-to-global results they provide. My intuition for why p-local subgroups would be interesting is that, generically, normalizers of interesting subgroups will also be interesting subgroups. I don't see an a-priori reason there to be a deeper reason than that. | |
| Dec 21, 2023 at 4:53 | history | edited | semisimpleton | CC BY-SA 4.0 |
added 284 characters in body; edited title
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| S Dec 21, 2023 at 4:50 | history | bounty started | semisimpleton | ||
| S Dec 21, 2023 at 4:50 | history | notice added | semisimpleton | Draw attention | |
| Dec 18, 2023 at 11:28 | history | asked | semisimpleton | CC BY-SA 4.0 |