Symmetric Group on 3 Letters

From ProofWiki
Jump to navigation Jump to search

Group Example

Let $S_3$ denote the set of permutations on $3$ letters.


The symmetric group on $3$ letters is the algebraic structure:

$\struct {S_3, \circ}$

where $\circ$ denotes composition of mappings.


It is usually denoted, when the context is clear, without the operator: $S_3$.


Cycle Notation

It can be expressed in the form of permutations given in cycle notation as follows:

\(\ds e\) \(:=\) \(\ds \text { the identity mapping}\)
\(\ds p\) \(:=\) \(\ds \tuple {1 2 3}\)
\(\ds q\) \(:=\) \(\ds \tuple {1 3 2}\)


\(\ds r\) \(:=\) \(\ds \tuple {2 3}\)
\(\ds s\) \(:=\) \(\ds \tuple {1 3}\)
\(\ds t\) \(:=\) \(\ds \tuple {1 2}\)


Cayley Table

$\begin{array}{c|cccccc}

\circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$


Group Presentation

Its group presentation is:

$S_3 := \gen {a, b: a^3 = b^2 = \paren {a b}^2 = e}$

Hence:

$\begin{array}{c|cccccc}
     & e     & a     & a^2   & b     & a b   & a^2 b \\

\hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$


Order of Elements

The orders of the various elements of $S_3$ are:

\(\ds e: \, \) \(\ds \) \(\) \(\ds \) Order $1$
\(\ds \tuple {123}: \, \) \(\ds \tuple {123}^2\) \(=\) \(\ds \tuple {132}\)
\(\ds \tuple {123} \tuple {132}\) \(=\) \(\ds e\) hence Order $3$
\(\ds \tuple {132}: \, \) \(\ds \tuple {132}^2\) \(=\) \(\ds \tuple {123}\)
\(\ds \tuple {132} \tuple {123}\) \(=\) \(\ds e\) hence Order $3$
\(\ds \tuple {12}: \, \) \(\ds \tuple {12}^2\) \(=\) \(\ds e\) hence Order $2$
\(\ds \tuple {13}: \, \) \(\ds \tuple {13}^2\) \(=\) \(\ds e\) hence Order $2$
\(\ds \tuple {23}: \, \) \(\ds \tuple {23}^2\) \(=\) \(\ds e\) hence Order $2$


Subgroups

The subsets of $S_3$ which form subgroups of $S_3$ are:

\(\ds \) \(\) \(\ds S_3\)
\(\ds \) \(\) \(\ds \set e\)
\(\ds \) \(\) \(\ds \set {e, \tuple {123}, \tuple {132} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {12} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {13} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {23} }\)


Normal Subgroups

Consider the subgroups of $S_3$:

The subsets of $S_3$ which form subgroups of $S_3$ are:

\(\ds \) \(\) \(\ds S_3\)
\(\ds \) \(\) \(\ds \set e\)
\(\ds \) \(\) \(\ds \set {e, \tuple {123}, \tuple {132} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {12} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {13} }\)
\(\ds \) \(\) \(\ds \set {e, \tuple {23} }\)


Of those, the normal subgroups in $S_3$ are:

$S_3, \set e, \set {e, \tuple {123}, \tuple {132} }$


Generators

Let:

\(\ds G_1\) \(=\) \(\ds \set {\tuple {123}, \tuple {12} }\)
\(\ds G_2\) \(=\) \(\ds \set {\tuple {13}, \tuple {23} }\)


Then:

\(\ds S_3\) \(=\) \(\ds \gen {G_1}\)
\(\ds \) \(=\) \(\ds \gen {G_2}\)

where $\gen G$ denotes the group generated by a subset $G$ of $S_3$.


Centralizers

The centralizers of each element of $S_3$ are given by:

\(\ds \map {C_{S_3} } e\) \(=\) \(\ds S_3\)
\(\ds \map {C_{S_3} } {123}\) \(=\) \(\ds \set {e, \tuple {123}, \tuple {132} }\)
\(\ds \map {C_{S_3} } {132}\) \(=\) \(\ds \set {e, \tuple {123}, \tuple {132} }\)
\(\ds \map {C_{S_3} } {12}\) \(=\) \(\ds \set {e, \tuple {12} }\)
\(\ds \map {C_{S_3} } {23}\) \(=\) \(\ds \set {e, \tuple {23} }\)
\(\ds \map {C_{S_3} } {13}\) \(=\) \(\ds \set {e, \tuple {13} }\)


Normalizers of Subgroups

The normalizers of each subgroup of $S_3$ are given by:

\(\ds \map {N_{S_3} } {\set e}\) \(=\) \(\ds S_3\)
\(\ds \map {N_{S_3} } {\set {e, \tuple {123}, \tuple {132} } }\) \(=\) \(\ds S_3\)
\(\ds \map {N_{S_3} } {\set {e, \tuple {12} } }\) \(=\) \(\ds \set {e, \tuple {12} }\)
\(\ds \map {N_{S_3} } {\set {e, \tuple {13} } }\) \(=\) \(\ds \set {e, \tuple {13} }\)
\(\ds \map {N_{S_3} } {\set {e, \tuple {23} } }\) \(=\) \(\ds \set {e, \tuple {23} }\)
\(\ds \map {N_{S_3} } {S_3}\) \(=\) \(\ds S_3\)


Center

The center of $S_3$ is given by:

$\map Z {S_3} = \set e$


Conjugacy Classes

The conjugacy classes of $S_3$ are:

\(\ds \) \(\) \(\ds \set e\)
\(\ds \) \(\) \(\ds \set {\tuple {123}, \tuple {132} }\)
\(\ds \) \(\) \(\ds \set {\tuple {12}, \tuple {13}, \tuple {23} }\)


Also see

Retrieved from "https://proofwiki.org/w/index.php?title=Symmetric_Group_on_3_Letters&oldid=522204"