# Alternating and special orthogonal groups which are simple

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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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). –  Qiaochu Yuan Apr 19 '11 at 17:24
I love "It is well-known by those who know it well" :-) –  joriki May 19 '11 at 9:14
Well, there is an informal principle saying that $A_d$ is $SO(d;\mathbb F_1)$... –  Grigory M Jun 18 '11 at 12:33
How are $A_1$ and $A_2$ not simple? Aren't they trivial? –  MartianInvader Jul 18 '11 at 16:51