Rigidity of the second symmetric product of the pseudo-arc

Let P denote the pseudo-arc and let F2(P)={{p,q}:p,q∈P} denote the second symmetric product of P. The main result in this paper is the following: if E:F2(P)→F2(P) is an embedding, then there is an embedding e:P→P such that E({p,q})={e(p),e(q)}. We obtain that the autohomeomorphisms of F2(P) are induced, P has rigid hyperspace F2(P), and the degree of homogeneity of F2(P) is exactly 3.