Let $p \ge 11$ be a prime number, $k,n$ be positive integers such that $n|gcd(p-1,k-1)$ and $p > k > n \ge 5$. Let $s \in \mathbb Z_p$ such that $ord_p(s) = n$. Is it possible that the sets $A = \{1,2,3,\dots,k-1\}$ and $B = \{k,k+1,k+2,\dots,p-1\}$ of classes modulo $p$ satisfy $sA = A$ and $sB = B$, simultaneously? I mean, are $A$ and $B$ invariants by multiplication by $s$?
Thanks!