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?
Alternating and special orthogonal groups which are simple
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$\begingroup$
group-theory
lie-groups
simple-groups
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4Seems like the strong law of small numbers to me: http://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). – 2011-04-19
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8I love "It is well-known by those who know it well" :-) – 2011-05-19
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5Well, there is an [informal principle](http://en.wikipedia.org/wiki/Field_with_one_element) saying that $A_d$ is $SO(d;\mathbb F_1)$... – 2011-06-18
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0How are $A_1$ and $A_2$ not simple? Aren't they trivial? – 2011-07-18