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:

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


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


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:

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


Subgroups

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

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


Normal Subgroups

Consider the subgroups of $S_3$:

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

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


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

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


Generators

Let:

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


Then:

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

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:

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


Normalizers of Subgroups

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

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


Center

The center of $S_3$ is given by:

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


Conjugacy Classes

The conjugacy classes of $S_3$ are:

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


Also see