TITLE: Highly transitive groups (without assuming the classification of finite simple groups)
QUESTION [25 upvotes]: 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).

REPLY [15 votes]: There is a classical result of Wielandt that if you assume the Schreier conjecture (that the outer automorphism group of an finite nonabelian simple groups is solvable), then a group of degree n other that A_n or S_n is at most 7-transitive. Unfortunately, the only known proof of the Schreier conjecture uses the classification of finite simple groups.