## Group Theory and Linear Algebra

Last updated: 23 September 2014

## Lecture 20: Symmetric groups and subgroups generated by a subset

Let $G$ be a group and $g\in G\text{.}$ The order of $G$ is $\text{Card}\left(G\right)$ and the order of $g$ is the smallest $k\in {ℤ}_{>0}$ such that ${g}^{k}=1\text{.}$

### The symmetric group ${S}_{n}$

$S1 = { } with Card(S1)=1 S2 = { , } with Card(S2)=2 S3 = { , , , , , } with$ $since · = = · = = · = = · = =$ Then $\text{Card}\left({S}_{3}\right)=6\text{.}$

The order of $\begin{array}{c} \end{array}$ is 3 and the order of $\begin{array}{c} \end{array}$ is 2.

The symmetric group ${S}_{n}$ is the set $Sn = { 12⋯n ···· ···· 12⋯n | each top dot is connected to a unique bottom dot, each bottom dot is connected to some top dot, no two top dots are connected to the same bottom dot } = { bijective function from 12n ··⋯· to 12n ··⋯· }$ with product given by ${w}_{1} · {w}_{2} = {w}_{1} {w}_{2}$

### Different representations of the same permutation

A permutation is an element of ${S}_{n}\text{.}$ Say $w = (diagram notation) = ( 12345678 45731268 ) (two line notation) = (1437625)(8)(cycle notation) = ( 00001000 00000100 00010000 10000000 01000000 00000010 00100000 00000001 ) (matrix notation)$

Let $G$ be a group. Let $S$ be a subset of $G\text{.}$ The subgroup generated by $S$ is the subgroup $⟨S⟩\subseteq G$ such that

 (a) $S\subseteq ⟨S⟩,$ (b) If $H$ is a subgroup of $G$ and $S\subseteq H$ then $⟨S⟩\subseteq H\text{.}$
So $⟨S⟩$ is the smallest subgroup of $G$ containing $S\text{.}$

$G={S}_{3},$ $S=\left\{\begin{array}{c} \end{array}\right\}\text{.}$

Then $⟨S⟩= { , , } with .$

$G={S}_{3},$ $S=\left\{\begin{array}{c} \end{array}\right\}\text{.}$

Then $⟨S⟩= { , with } .$

Note: $ℤ2ℤ = {0,1}with +01 001 110 ℤ3ℤ = {0,1,2}with +012 0012 1120 2201$ The function $f: ℤ3ℤ ⟶ { , , } 0 ⟼ 0 ⟼ 2 ⟼$ is an isomorphism.

The subgroups of ${S}_{3}$ are ${S}_{3} \left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\right\} \left\{\begin{array}{c}\end{array}\right\}$

## Notes and References

These are a typed copy of Lecture 20 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on September 7, 2011.