**Fact.** Suppose that is a one-orbit set and

is an equivariant function. Then is a bijection.

**Proof. **Let be the set of equivariant endomorphisms from to . This is a finite monoid, which contains . By a standard result on finite monoids, there must be some natural number such that is idempotent, i.e.

Let be any element in the range of . By idempotency of , this is a fixpoint, i.e.

By equivariance of , it follows that

holds for every atom automorphism , and therefore is the identity on by the assumption that has one orbit. Since is the identity, it follows that has to be a bijection.

**Comment. **The fact also follows from the representation theorem, but we do not know if the representation theorem always holds.

It should be clarified that although the statement of this result looks like it holds for arbitrary transitive G-sets, it does not. A key step in the proof is that there are only finitely many equivariant endofunctions on , and this requires the finite support assumption.