Sets of Operations on Set of 3 Elements

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Let $\AA$, $\BB$, $\CC_1$, $\CC_2$, $\CC_3$ and $\DD$ 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 $S$
\(\ds \CC_1\) \(:\) \(\ds \set {I_S, \tuple {a, b} }\)
\(\ds \CC_2\) \(:\) \(\ds \set {I_S, \tuple {a, c} }\)
\(\ds \CC_3\) \(:\) \(\ds \set {I_S, \tuple {b, c} }\)
\(\ds \DD\) \(:\) \(\ds \set {I_S}\)

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


This article needs proofreading.
In particular: Several of the results are derived from ad hoc exhaustion techniques and appeals to intuition. Some have not been completed because of inadequate application. Some of them gloss over the specific results which are needed to justify the steps. In all cases, invitations to improvement are tendered.
If you believe all issues are dealt with, please remove {{Proofread}} from the code.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Proofread}} from the code.


The following results can be deduced:


Automorphism Group of $\AA$

$\AA$ has $3$ elements.


Automorphism Group of $\BB$

$\BB$ has $3^3 - 3$ elements.


Automorphism Group of $\CC_n$

Each of $\CC_1$, $\CC_2$ and $\CC_3$ has $3^4 - 3$ elements.


Automorphism Group of $\DD$

$\DD$ has $19 \, 422$ elements.


Isomorphism Classes

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 $S$.


Operations with Identity

Let $\NN$ be the set of all operations $\circ$ on $S$ which have an identity element.

Then the elements of $\NN$ are divided in $45$ isomorphism classes.

That is, up to isomorphism, there are $45$ operations on $S$ which have an identity element.


Commutative Operations

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

Then the elements of $\PP$ are divided in $129$ isomorphism classes.

That is, up to isomorphism, there are $129$ commutative operations on $S$ which have an identity element.


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Sets_of_Operations_on_Set_of_3_Elements&oldid=546564"