Sets of Operations on Set of 3 Elements/Automorphism Group of D/Operations with Identity
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:
\(\ds n\) | \(=\) | \(\ds N - A - B - C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 243 - 0 - 0 - 27\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 216\) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.14 \ \text{(c)}$