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

Theorem

Let $S = \set {a, b, c}$ be a set with $3$ elements.

Let $\CC_1$, $\CC_2$ 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:

\(\ds \CC_1\)

\(:\)

\(\ds \set {I_S, \tuple {a, b} }\)

\(\ds \CC_2\)

\(:\)

\(\ds \set {I_S, \tuple {a, c} }\)

\(\ds \CC_3\)

\(:\)

\(\ds \set {I_S, \tuple {b, c} }\)

where $I_S$ is the identity mapping on $S$.

Let $\CC = \CC_1 \cup \CC_2 \cup \CC_3$.

Let $\oplus \in \CC$.

Then the isomorphism class of $\oplus$ consists of $3$ elements: exactly one operation from each of $\CC_1$, $\CC_2$ and $\CC_3$.

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 $8.14 \ \text{(b)}$