Hello, apparently finite groups which are n-transitive with n>5 are only the permutation groups Sn or the alternating groups An+2, see e.g. page 226 this book by Isaacs <http://books.google.fr/books?id=pCLhYaMUg8IC&pg=PA226>


Is this characterization useful at all? For instance, are there famous proofs (maybe in a geometric context), which use it to, say, show that a certain group is in fact some An ?



Use of n-transitivity in finite group theory
This fact is used in a nice way by [Dunfield and Thurston](http://arxiv.org/abs/math/0502567) to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution. (The multiple transitivity appears in Thm 7.4.)


Also: I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.


**EDIT:** In response to Noah's query, I looked up a reference; the relevant theorem is due to Michael Larsen and is often called "Larsen's Alternative." You can read about it in section 1 of [this paper of Katz](http://muse.jhu.edu/journals/american_journal_of_mathematics/info/docs/hida_pdfs/hida18.pdf). 


For the specific problem of distinguishing a subgroup from a group by means of moments, [the 2005 paper of Guralnick and Tiep](http://www.ams.org/ert/2005-009-05/S1088-4165-05-00192-5/home.html) is relevant.


