Sets of Operations on Set of 3 Elements/Automorphism Group of B/Isomorphism Classes
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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 $S$.
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 $\BB$.
Proof
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Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.14 \ \text{(b)}$