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

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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:

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


Proof

Let $n$ denote the number of operations of $\DD$ which have an identity element.


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 operations on $S$ which have an identity element.

Let:

$A$ denote the number of operations in $\AA$ which have an identity element
$B$ denote the number of operations in $\BB$ which have an identity element
$C$ denote the total number of operations in $\CC_1$, $\CC_2$ and $\CC_3$ which have an identity element.


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

$N = A + B + C + D$

From Count of Binary Operations with Identity:

$N = 3^5 = 243$

Then we have:

From Automorphism Group of $\AA$: Operations with Identity: $A = 0$
From Automorphism Group of $\BB$: Operations with Identity: $B = 0$
From Automorphism Group of $\CC_n$: Operations with Identity: $C = 3 \times 9$


Hence we have:

\(\ds n\) \(=\) \(\ds N - A - B - C\)
\(\ds \) \(=\) \(\ds 243 - 0 - 0 - 27\)
\(\ds \) \(=\) \(\ds 216\)

$\blacksquare$


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