On Distinct Powers in Finite Groups

Sun conjectured that if a group $G$ has no elements of orders $2,3,...,n+1$, then every $n$-element subset $A$ can be ordered as $a_1,...,a_n$ so that $a_1,a_2^2,...,a_n^n$ are pairwise distinct. We prove the conjecture for finite groups when $n=4,5,6$. The proofs combine collision-counting arguments with reductions to finite-field assignment problems; the remaining finite cases for $n=6$ are verified exhaustively. We also prove the conjecture for every $n$ when no four elements of $A$ generate the same cyclic subgroup.