Sets of Operations on Set of 3 Elements/Automorphism Group of D

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Theorem

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

Let $\DD$ be the set of all operations $\circ$ on $S$ such that the group of automorphisms of $\struct {S, \circ}$ forms the set $\set {I_S}$, where $I_S$ is the identity mapping on $S$.


Then:

$\DD$ has $19 \, 422$ elements.


Isomorphism Classes

Let $\oplus \in \DD$.

Then the isomorphism class of $\oplus$ consists of $6$ elements, all of which are in $\DD$.


Operations with Identity

$216$ of the operations of $\DD$ has an identity element.


Commutative Operations

$696$ of the operations of $\DD$ is commutative.


Proof

Let $n$ denote the cardinality of $\DD$.

Equivalently, $n$ equals the number of operations $\circ$ on $S$ on which the only automorphism is $I_S$.


Recall these definitions:

Let $\AA$, $\BB$, $\CC_1$, $\CC_2$ and $\CC_3$ be respectively the set of all operations $\circ$ on $S$ such that the groups of automorphisms of $\struct {S, \circ}$ are as follows:

\(\ds \AA\) \(:\) \(\ds \map \Gamma S\) where $\map \Gamma S$ is the symmetric group on $S$
\(\ds \BB\) \(:\) \(\ds \set {I_S, \tuple {a, b, c}, \tuple {a, c, b} }\) where $I_S$ is the identity mapping on $S$
\(\ds \CC_1\) \(:\) \(\ds \set {I_S, \tuple {a, b} }\)
\(\ds \CC_2\) \(:\) \(\ds \set {I_S, \tuple {a, c} }\)
\(\ds \CC_3\) \(:\) \(\ds \set {I_S, \tuple {b, c} }\)


Lemma

$\set {\AA, \BB, \CC_1, \CC_2, \CC_3, \DD}$ forms a partition of the set of all operations on $S$.

$\Box$


Let $N$ be the total number of operations on $S$.

From the lemma, and from the Fundamental Principle of Counting:

$N = \card \AA + \card \BB + \card {\CC_1} + \card {\CC_2} + \card {\CC_3} + \card {\DD}$

From Count of Binary Operations on Set:

$N = 3^{3^2} = 3^9 = 19 \, 683$

Then we have:

From Automorphism Group of $\AA$: $\card \AA = 3$
From Automorphism Group of $\BB$: $\card \BB = 3^3 - 3 = 24$
From Automorphism Group of $\CC_n$: for $n = 1, 2, 3: \card {\CC_n} = 3^4 - 3 = 78$

Hence we have:

\(\ds \DD\) \(=\) \(\ds N - \card \AA - \card \BB - 3 \card {\CC_1}\)
\(\ds \) \(=\) \(\ds 19 \, 683 - 3 - 24 - 3 \times 78\)
\(\ds \) \(=\) \(\ds 19 \, 422\)

$\blacksquare$


Sources

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