What is known about the classification of n-transitive group actions for n large without using the classification of finite simple groups? With the classification of finite simple groups a complete list of all 2-transitive group actions is known, in particular there are no 6-transitive groups other than the symmetric groups and the alternating groups. I want something like "there are no interesting n-transitive group actions for n sufficiently large" but without the classification theorem (however, I'd be happy if the n in that statement was obscenely large). Even any partial (but unconditional) results would interest me (like any n-transitive group for n sufficiently large needs to have properties X, Y, and Z).



Highly transitive groups (without assuming the classification of finite simple groups)
Marshall Hall's *The Theory of finite groups* only cites an asymptotic bound: a permutation group of degree n that isn't Sn or An can be at most t-transitive for t less than 3 log n. I suppose that was the state of the art at the time (late 1960s). There is an earlier paper by G.A. Miller on JSTOR that you can find by searching for "multiply transitive group".


There is a classical theorem of Jordan that classified sharply quadruply transitive permutation groups, i.e., those for which only the identity stabilizes a given set of 4 elements (from Wikipedia).


