Sets of Operations on Set of 3 Elements/Automorphism Group of A/Isomorphism Classes

Theorem
Let $S = \set {a, b, c}$ be a set with $3$ elements.

Let $\AA$ be the set of all operations $\circ$ on $S$ such that the group of automorphisms of $\struct {S, \circ}$ is the symmetric group on $S$, that is, $\map \Gamma S$.

Then:
 * The elements of $\AA$ are each in its own isomorphism class.

Proof
Recall from Automorphism Group of $\AA$ the elements of $\AA$, expressed in Cayley table form:


 * $\begin{array}{c|ccc}

\to & a & b & c \\ \hline a & a & b & c \\ b & a & b & c \\ c & a & b & c \\ \end{array} \qquad \begin{array}{c|ccc} \gets & a & b & c \\ \hline a & a & a & a \\ b & b & b & b \\ c & c & c & c \\ \end{array} \qquad \begin{array}{c|ccc} \circ & a & b & c \\ \hline a & a & c & b \\ b & c & b & a \\ c & b & a & c \\ \end{array}$

We have from Algebraic Structures formed by Left and Right Operations are not Isomorphic for Cardinality Greater than 1 that $\struct {S, \to}$ and $\struct {S, \gets}$ are not isomorphic.

Observe from the Cayley table $\circ$ that:
 * $(1): \quad \forall x, y \in S: x \ne y \implies x \circ y \ne x \land x \circ y \ne y$

there exists an isomorphism $\phi$ from $\struct {S, \to}$ to $\struct {S, \circ}$.

We have:

Then:

Hence:
 * $\map \phi c = \map \phi b$

which means $\phi$ is not an injection.

That is, $\phi$ is not a bijection and hence not an isomorphism.

Hence by Proof by Contradiction there exists no isomorphism from $\struct {S, \to}$ to $\struct {S, \circ}$.

In a similar way:

there exists an isomorphism $\phi$ from $\struct {S, \gets}$ to $\struct {S, \circ}$.

We have:

Then:

Hence:
 * $\map \phi c = \map \phi a$

which means $\phi$ is not an injection.

That is, $\phi$ is not a bijection and hence not an isomorphism.

Hence by Proof by Contradiction there exists no isomorphism from $\struct {S, \gets}$ to $\struct {S, \circ}$.

It has been shown that none of the elements of $\AA$ is isomorphism to any of the others.

The result follows by definition of isomorphism class.