Sharply $(n-2)$-transitive Sets of Permutations

Let $S_n$ be the symmetric group on the set $[n]={1, 2, ldots, n}$. For $gin S_n$ let $fix(g)$ denote the number of fixed points of $g$. A subset $S$ of $S_n$ is called $t$-emph{transitive} if for any two $t$-tuples $(x_1,x_2,ldots,x_t)$ and $(y_1,y_2,ldots ,y_t)$ of distinct elements of $[n]$, there exists $gin S$ such that $x_{i}^g=y_{i}$ for any $1leq ileq t$ and additionally $S$ is called emph{sharply $t$-transitive} if for any given pair of $t$-tuples, exactly one element of $S$ carries the first to the second. In addition, a subset $S$ of $S_n$ is called $t$-intersecting if $fix(h^{-1}g)geq t$ for any two distinct permutations $h$ and $g$ of $S$. In this paper, we prove that there are only two sharply $(n-2)$-transitive subsets of $S_n$ and finally we establish some relations between sharply $k$-transitive subsets and $t$-intersecting subsets of $S_n$ where $k,tin mathbb{Z}$ and $0leq tleq kleq n$.