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 a₁,...,aₙ so that a₁,a₂²,...,aₙⁿ 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.