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Jul 27, 2013 at 5:04 history edited Venkataramana CC BY-SA 3.0
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Jul 16, 2012 at 14:08 comment added Igor Rivin This is stated (complete with proof very similar to what you give) in the paper of Venkataramana and Soifer (without attribution to Rapinchuk. They also give estimates for principal congruence subgroups (for which the number IS bounded by some function of $n$ only)
Jul 16, 2012 at 11:49 history answered Venkataramana CC BY-SA 3.0