Sets of Operations on Set of 3 Elements/Automorphism Group of D/Isomorphism Classes

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 $S$.

Let $\oplus \in \DD$.

Then the isomorphism class of $\oplus$ consists of $6$ elements, all of which are in $\DD$.

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 $8.14 \ \text{(b)}$