On symmetric products of continua

We show that the nth symmetric product of a continuum is unicoherent if n ⩾ 3. We prove that the arc is the only finite dimensional continuum for which its second symmetric product is homeomorphic to its hyperspace of subcontinua, and that for any finite dimensional continuum X its hyperspace of subcontinua is not homeomorphic to its nth symmetric product for any n ⩾ 3.