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

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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:

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


Proof

Let $n$ denote the number of commutative operations of $\DD$.


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 $S$
\(\ds \CC_1\) \(:\) \(\ds \set {I_S, \tuple {a, b} }\)
\(\ds \CC_2\) \(:\) \(\ds \set {I_S, \tuple {a, c} }\)
\(\ds \CC_3\) \(:\) \(\ds \set {I_S, \tuple {b, c} }\)


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

Let:

$A$ denote the number of commutative operations in $\AA$
$B$ denote the number of commutative operations in $\BB$
$C$ denote the total number of commutative operations in $\CC_1$, $\CC_2$ and $\CC_3$.


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

$N = A + B + C + D$

From Count of Commutative Binary Operations on Set:

$N = 3^6 = 729$

Then we have:

From Automorphism Group of $\AA$: Operations with Identity: $A = 1$
From Automorphism Group of $\BB$: Operations with Identity: $B = 8$
From Automorphism Group of $\CC_n$: Operations with Identity: $C = 3 \times 8$


Hence we have:

\(\ds n\) \(=\) \(\ds N - A - B - C\)
\(\ds \) \(=\) \(\ds 729 - 1 - 8 - 24\)
\(\ds \) \(=\) \(\ds 696\)

$\blacksquare$


Sources

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