Timeline for The probability that two elements of a finite nonabelian simple group commute
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 19, 2019 at 22:04 | comment | added | DavidLHarden | Clarifying: I meant that $c(G) \leq 4$ and $G$ finite imply $|G| \leq 12$. | |
| Jan 19, 2019 at 22:03 | comment | added | DavidLHarden | In fact, that can be proven without the use of any representation theory. Let $c(G)$ denote the number of conjugacy classes of the finite group $G$, so that $cp(G) \geq \epsilon$ means $\frac{c(G)}{|G|} \geq \epsilon$. Then if $k$ is the size of the smallest non-identity conjugacy class of $G$, comparing that with the average size of one of them gives us $k \leq \frac{|G|-1}{c(G)-1}$. Since $c(G) \geq 5$ for any nonabelian finite simple $G$ (in fact, $c(G) \leq 4$ implies $|G| \leq 12$), we get $k < \frac{5|G|}{4c(G)} \leq \frac{5}{4\epsilon}$. Then remember $G$ acts faithfully on $k$ points. | |
| Sep 19, 2018 at 5:20 | history | edited | Yemon Choi |
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| Sep 19, 2018 at 4:07 | comment | added | Ian Agol | In fact, for any $\epsilon$, there will be at most finitely many simple groups $G$ with $cp(G) \geq \epsilon$. This follows from Dixon's argument (linked below) and the fact that for any $n$ there are at most finitely many finite simple groups with an irreducible representation of dimension $n$. See e.g. mathoverflow.net/a/27365/1345 | |
| Sep 19, 2018 at 3:45 | answer | added | Keith Kearnes | timeline score: 8 | |
| S Sep 19, 2018 at 3:33 | history | suggested | Mec | CC BY-SA 4.0 |
the probability is at most 1/12
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| Sep 19, 2018 at 3:25 | review | First posts | |||
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| Sep 19, 2018 at 3:24 | review | Suggested edits | |||
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| Sep 19, 2018 at 3:21 | history | asked | user129021 | CC BY-SA 4.0 |