Show that there is a group which can act on X transitively.

Question

Assume X be a finite set of order n≥1 which has been partitioned into t non-empty, pairwise disjoint, subsets, X=P1∪ P2 ∪⋯∪ Pt, some t≥1.

Consider the case when t=1. Show that there is a group which can act on X transitively.

Answer 1

In this case, X is just equal to P_1, because there is only one part. The symmetric group of X, defined to be the group of all bijections from X to X, acts transitively on X. For any elements x,y in X the transposition (xy) implements the transitivity.