Subgroup transitive on the subset with same cardinality Maybe there is some very obvious insight that i miss here, but i've asked this question also to other people and nothing meaningful came out:
If you have a subgroup G of $S_n$(the symmetric group on n elements), you can consider the natural action of G on the subset of $\{1,...,n\}$; my question is this:what are the G that act transitively on the subsets of the same cardinality; that is, whenever $A,B \subseteq \{1,...,n\}$ with $|A|=|B|$, there is  $g \in G$ so that $gA=B$.

By the Livingstone Wagner Theorem, (Livingstone, D., Wagner, A., Transitivity of finite permutation groups on unordered sets. Math. Z. 90 (1965) 393–403),  if $n \ge 2k$ and $G$ is transitive on $k$-subsets with $k \ge 5$, then $G$ is $k$-transitive. Using the classification of finite simple groups, $A_n$ and $S_n$ are the only finite $k$-transitive groups for $k \ge 6$, and the only 4- and 5-transitive groups are the Mathieu groups, so (at least for $n \ge 10$), the only permutation groups that are $k$-homogeneous for all $k$ are $A_n$ and $S_n$.
See also the similar discussion in Group actions transitive on certain subsets
