# Highly transitive groups (without assuming the classification of finite simple groups)

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).

• You're assuming they're finite? Nov 18, 2009 at 19:15
• Yes, finite. Of course if n is an actual number then G might as well be finite. I suppose you could ask questions like this for n a cardinal, but I'm not so interested in that. Nov 18, 2009 at 19:25

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).

• Ah well, that's what I was afraid of. Nov 18, 2009 at 19:25
• I'm speculating wildly here, but there may be intermediate structural results in the classification that give you bounds without proceeding through the case-elimination. Nov 18, 2009 at 19:31
• As far as I can tell, there have been no classification-free improvements of Wielandt's 1934 paper. He proved that a $t$-transitive subgroup of $S_n$ which doesn't contain $A_n$ must satisfy $3\log(n-t)>t$ and $n-t\ge\binom{t}{[4t/5]}$. The latter bound is approximately $2\log n>t$ when $t$ is large. And I think Hall's book is a strange reference for this, since it only states a weaker version of Wielandt's result, with no proof. Aug 28, 2021 at 21:16

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.

Well, the theorem that $M_{11}$ and $M_{12}$ are the only sharply $k$-transitive groups for $k> 3$ is about 100 years older than the classification theorem...

Also, if $k>3$ (without the sharply hypothesis), $G$ must be simple, unless it is in $S_4$. Of course, that's why the classification is useful.

• Yeah, I almost remarked that I already knew this in the question and for some reason decided against it. Nov 18, 2009 at 19:23
• You should have. Nov 18, 2009 at 19:26