Structure of Cardinality 3+ where Every Permutation is Automorphism is Idempotent

Theorem
Let $S$ be a set whose cardinality is at least $3$.

Let $\struct {S, \circ}$ be an algebraic structure on $S$ such that every permutation on $S$ is an automorphism on $\struct {S, \circ}$.

Then $\circ$ is an idempotent operation.

Proof
$\circ$ is not idempotent.

Then there exists $x \in S$ such that:
 * $\exists y \in S: x \circ x = y$

where $x \ne y$.

Because there are at least $3$ distinct elements of $S$ :
 * $\exists z \in S: z \ne x, z \ne y$

Let $f: S \to S$ be a permutation on $S$ such that:
 * $\map f x = x$
 * $\map f y = z$

We have:

This contradicts our assertion that $x$ and $z$ are distinct.

From Proof by Contradiction it follows that our assumption that $\circ$ is not idempotent must have been false.

Hence $\circ$ is idempotent.