Timeline for Finite groups with lots of conjugacy classes, but only small abelian normal subgroups?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Jun 25, 2016 at 16:50 | answer | added | yakov | timeline score: 0 | |
| May 10, 2016 at 6:18 | comment | added | YCor | @JSE no, there are trivial examples with an unbounded commutator group, for instance $C_3^n\rtimes C_2$ where $C_k$ is cyclic of order $k$, $C_2$ acts on $C_3^n$ by $x\mapsto -x$ (all elements of $C_3^n$ are commutators). Of course it has an abelian subgroup of bounded order, but you can fix this by taking the direct product with the group $G_n(q)$ of my example and both the derived subgroup and the smallest index of an abelian subgroup tend to infinity. | |
| May 9, 2016 at 21:06 | comment | added | Yemon Choi | @YCor has already answered your question, but you might be interested in this old question I asked mathoverflow.net/questions/21071/… and the answer given by "BugsBunny". The question concerned a "fourth" version of being close to abelian, namely that all irrreducible characters have small degree; I think this is equivalent to your 3., but the bounds aren't very good | |
| May 9, 2016 at 20:56 | comment | added | Alexander Bors | @JSE: This also looks like an interesting condition. I will think about it, thank you! | |
| May 9, 2016 at 20:24 | vote | accept | Alexander Bors | ||
| May 9, 2016 at 20:06 | comment | added | JSE | YCor answered your question, but I note that YCor's example has very few commutators. Since "probability that two elements commute" is "probability that a random commutator is the identity," it might be reasonable to ask what happens if you take "almost-abelian" to mean "bounded number of commutators." | |
| May 9, 2016 at 19:54 | answer | added | YCor | timeline score: 14 | |
| May 9, 2016 at 19:25 | history | edited | YCor |
edited tags
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| May 9, 2016 at 14:32 | history | edited | Alexander Bors | CC BY-SA 3.0 |
extended a reference: [1, Theorem 4(iii)] -> [1, Theorems 4(iii) and 8(ii)], to cover both solvable and non-solvable groups
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| May 9, 2016 at 14:04 | history | asked | Alexander Bors | CC BY-SA 3.0 |