Sets of Operations on Set of 3 Elements/Automorphism Group of B
Theorem
Let $S = \set {a, b, c}$ be a set with $3$ elements.
Let $\BB$ be the set of all operations $\circ$ on $S$ such that the group of automorphisms of $\struct {S, \circ}$ forms the set $\set {I_S, \tuple {a, b, c}, \tuple {a, c, b} }$, where $I_S$ is the identity mapping on $S$.
Then:
- $\BB$ has $3^3 - 3$ elements.
Isomorphism Classes
Let $\oplus \in \BB$.
Then the isomorphism class of $\oplus$ consists of $\oplus$ and exactly one other operation $\otimes$ on $S$ such that $\otimes \in \BB$.
That is, the elements of $\BB$ are in doubleton isomorphism classes each of which is a subset of $\BB$.
Operations with Identity
- None of the operations of $\BB$ has an identity element.
Commutative Operations
- Exactly $8$ of the operations of $\BB$ is commutative.
Proof
Recall the definition of (group) automorphism:
- $\phi$ is an automorphism on $\struct {S, \circ}$ if and only if:
- $\phi$ is a permutation of $S$
- $\phi$ is a homomorphism on $\struct {S, \circ}$: $\forall a, b \in S: \map \phi {a \circ b} = \map \phi a \circ \map \phi b$
From Identity Mapping is Group Automorphism, $I_S$ is always an automorphism on $\struct {S, \circ}$.
Hence it is not necessary to analyse the effect of $I_S$ on $S$.
Let us denote each of the remaining elements of $\set {I_S, \tuple {a, b, c}, \tuple {a, c, b} }$ as follows:
| \(\ds p\) | \(:\) | \(\ds \map p a = b, \map p b = c, \map p c = a\) | ||||||||||||
| \(\ds q\) | \(:\) | \(\ds \map q a = c, \map q b = a, \map q c = b\) |
We select various product elements $x \circ y \in S$ and determine how $p$ and $q$ constrain other product elements as follows:
Then by definition of the above mappings $p$ and $q$, and the definition of a homomorphism, we obtain as follows:
- $(1)
- \quad a \circ a$
|
\circ & a & b & c \\ \hline a & a & & \\ b & & b & \\ c & & & c \\ \end {array}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
\circ & a & b & c \\ \hline a & b & & \\ b & & c & \\ c & & & a \\ \end {array}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
\circ & a & b & c \\ \hline a & c & & \\ b & & a & \\ c & & & b \\ \end {array}$ |
So selecting $a \circ a$ fixes $b \circ b$ and $c \circ c$.
- $(2)
- \quad a \circ b$
|
\circ & a & b & c \\ \hline a & & a & \\ b & & & b \\ c & c & & \\ \end {array}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
\circ & a & b & c \\ \hline a & & b & \\ b & & & c \\ c & a & & \\ \end {array}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
\circ & a & b & c \\ \hline a & & c & \\ b & & & a \\ c & b & & \\ \end {array}$ |
So selecting $a \circ b$ fixes $b \circ c$ and $c \circ a$.
- $(3)
- \quad a \circ c$
|
\circ & a & b & c \\ \hline a & & & a \\ b & b & & \\ c & & c & \\ \end {array}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
\circ & a & b & c \\ \hline a & & & b \\ b & c & & \\ c & & a & \\ \end {array}$ | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
\circ & a & b & c \\ \hline a & & & c \\ b & a & & \\ c & & b & \\ \end {array}$ |
So selecting $a \circ c$ fixes $b \circ a$ and $c \circ b$.
There are $3$ elements $x$ of $S$ with which $a \circ x$ can be made.
For each of these $3$, there are $3$ different elements $y$ such that $a \circ x = y$.
These collectively fix all the possible values of $b \circ x$ and $c \circ x$.
Hence they exhaust all possible operations $\circ$ on $S$ for which $p$ and $q$ are automorphisms.
Thus there are independently:
- $3$ different options for $a \circ a$
- $3$ different options for $a \circ b$
- $3$ different options for $a \circ c$
and therefore $3 \times 3 \times 3 = 3^3$ operations $\circ$ on $S$ for which $p$ and $q$ are automorphisms.
Note that from Automorphism Group of $\AA$, $3$ of these are also such that the group of automorphisms of $\struct {S, \circ}$ is the symmetric group on $S$.
So these are excluded from our count.
The result follows.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.14 \ \text{(a)}$