I have an idle curiosity about a funny coincidence between two sequences of groups. It is well-known by those who know it well that the alternating group $A_d$ of degree $d$ is simple if and only if $d\not\in\{1,2,4\}$. By what seems like an astonishing coincidence to me, the special orthogonal group $SO(d,\mathbb{R})$ is simple if and only if $d\not\in\{1,2,4\}$. Is this merely a coincidence?
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- 4$\begingroup$ Seems like the strong law of small numbers to me: en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers . The fact that $A_3$ is simple is essentially an accident (it's too small not to be simple). $\endgroup$Qiaochu Yuan– Qiaochu Yuan2011-04-19 17:24:10 +00:00Commented Apr 19, 2011 at 17:24
- 8$\begingroup$ I love "It is well-known by those who know it well" :-) $\endgroup$joriki– joriki2011-05-19 09:14:39 +00:00Commented May 19, 2011 at 9:14
- 5$\begingroup$ Well, there is an informal principle saying that $A_d$ is $SO(d;\mathbb F_1)$... $\endgroup$Grigory M– Grigory M2011-06-18 12:33:28 +00:00Commented Jun 18, 2011 at 12:33
- $\begingroup$ How are $A_1$ and $A_2$ not simple? Aren't they trivial? $\endgroup$MartianInvader– MartianInvader2011-07-18 16:51:06 +00:00Commented Jul 18, 2011 at 16:51
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