On t-homogeneous permutation sets (Q798757)

Let \(S_ n\) be the symmetric group on \(X=\{1,...,n\}\). A subset Z of \(S_ n\) is t-transitive if the following condition holds: for any two t- subsets x,y in X, there exists some z in Z that moves x to y, and the number of such elements in Z is a constant that is independent of the choice of x and y. We prove that a t-homogeneous subset in \(S_ n\) is also (t-1)-homogeneous for 2\(\leq t\leq (n/2)\).