“Then I will do it myself”, said the little red hen. And she did.
Since nobody took up the challenge, I had to do it myself.
Let λ be a partition of n. We say that a permutation group G of degree n is λ-transitive if given any two ordered partitions of {1,2,…,n} with shape λ, there is an element of G carrying each part of the first to the corresponding part of the second; and G is λ-transitive if given any two partitions of shape λ, there is an element of G carrying the first to the second.
Theorem If G is λ-homogeneous but not λ-transitive, where λ is not the partition with all parts of size 1, then one of the following happens:
- λ = (n−t,1,1,…,1), and G is t-homogeneous but not t-transitive;
- n = tk, λ = (k,k,…k), and G fixes a point and acts as the symmetric or alternating group on the remaining points;
- λ = (3,3), and G = PSL(2,5).
Also, the lovely arguments using character theory of the symmetric group, regular 2-graphs, and so on, all had to go in the dustbin, because everything became so much simpler! When this happens, there is a tendency to feel that something is lost, but the gains outweigh the losses.
I still use the Classification of Finite Simple Groups at one point (just to say there are no 6-transitive groups except symmetric and alternating groups), but I suspect that even this is not really necessary.