Sets of Operations on Set of 3 Elements/Isomorphism Classes

Theorem

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

Let $\MM$ be the set of all operations $\circ$ on $S$.

Then the elements of $\MM$ are divided in $3330$ isomorphism classes.

That is, up to isomorphism, there are $3330$ operations on $S$.

Proof

From Automorphism Group of $\AA$: Isomorphism Classes:

each element of $\AA$ is in its own isomorphism class.

Hence $\AA$ contributes $3$ isomorphism classes.

From Automorphism Group of $\BB$: Isomorphism Classes:

the $24$ elements of $\BB$ form $12$ isomorphism classes in pairs.

From Automorphism Group of $\CC_n$: Isomorphism Classes:

the $3 \times 78$ elements of $\CC$ form $78$ isomorphism classes in threes.

From Automorphism Group of $\DD$: Isomorphism Classes:

the $19 \, 422$ elements of $\DD$ form $3237$ isomorphism classes in sixes.

Thus there are $3 + 12 + 78 + 3237 = 3330$ isomorphism classes.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.14 \ \text{(b)}$