Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $k\in {\mathbb{Z}}_{>0}$ such that $g^k=1\text{.}$

The symmetric group $S_n$

$$\begin{array}{ccc}{\displaystyle S_1}& =& {\displaystyle \left\{\begin{array}{c} \end{array}\right\}\text{with}\begin{array}{cc}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\end{array}\text{Card}\left(S_1\right)=1}\\ {\displaystyle S_2}& =& {\displaystyle \{\begin{array}{c} \end{array},\begin{array}{c} \end{array}\}\text{with}\begin{array}{ccc}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\end{array}\text{Card}\left(S_2\right)=2}\\ {\displaystyle S_3}& =& {\displaystyle \{\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\}\text{with}}\end{array}$$ $$\begin{array}{ccccccc}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\end{array}\text{since}\begin{array}{ccccc}{\displaystyle \begin{array}{c} \end{array}·\begin{array}{c} \end{array}}& =& {\displaystyle \genfrac{}{}{0ex}{}{\begin{array}{c} \end{array}}{\begin{array}{c} \end{array}}}& =& {\displaystyle \begin{array}{c} \end{array}}\\ {\displaystyle \begin{array}{c} \end{array}·\begin{array}{c} \end{array}}& =& {\displaystyle \genfrac{}{}{0ex}{}{\begin{array}{c} \end{array}}{\begin{array}{c} \end{array}}}& =& {\displaystyle \begin{array}{c} \end{array}}\\ {\displaystyle \begin{array}{c} \end{array}·\begin{array}{c} \end{array}}& =& {\displaystyle \genfrac{}{}{0ex}{}{\begin{array}{c} \end{array}}{\begin{array}{c} \end{array}}}& =& {\displaystyle \begin{array}{c} \end{array}}\\ {\displaystyle \begin{array}{c} \end{array}·\begin{array}{c} \end{array}}& =& {\displaystyle \genfrac{}{}{0ex}{}{\begin{array}{c} \end{array}}{\begin{array}{c} \end{array}}}& =& {\displaystyle \begin{array}{c} \end{array}}\end{array}$$ 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 $$\begin{array}{ccc}{\displaystyle S_n}& =& {\displaystyle \left\{\begin{array}{cccc}1& 2& \cdots & n\\ ·& ·& ·& ·\\ & \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \\ ·& ·& ·& ·\\ 1& 2& \cdots & n\end{array}\hspace{0.17em}\right|\hspace{0.17em}\genfrac{}{}{0ex}{}{{\displaystyle \genfrac{}{}{0ex}{}{\text{each top dot is connected to a unique}}{\text{bottom dot, each bottom dot is connected}}}}{{\displaystyle \genfrac{}{}{0ex}{}{\text{to some top dot, no two top dots are}}{\text{connected to the same bottom dot}}}}\}}\\ & =& {\displaystyle \left\{\text{bijective function from}\begin{array}{cccc}1& 2& & n\\ ·& ·& \cdots & ·\end{array}\text{to}\begin{array}{cccc}1& 2& & n\\ ·& ·& \cdots & ·\end{array}\right\}}\end{array}$$ with product given by $$\begin{array}{c} w1 \end{array}·\begin{array}{c} w2 \end{array}=\begin{array}{c} w1 w2 \end{array}$$

Different representations of the same permutation

A permutation is an element of $S_n\text{.}$ Say $$\begin{array}{ccc}{\displaystyle w}& =& {\displaystyle \begin{array}{c} \end{array}\text{(diagram notation)}}\\ & =& {\displaystyle \left(\begin{array}{cccccccc}1& 2& 3& 4& 5& 6& 7& 8\\ 4& 5& 7& 3& 1& 2& 6& 8\end{array}\right)\text{(two line notation)}}\\ & =& {\displaystyle \left(1437625\right)\left(8\right)\text{(cycle notation)}}\\ & =& {\displaystyle \left(\begin{array}{cccccccc}0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1\end{array}\right)\text{(matrix notation)}}\end{array}$$

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 $$⟨S⟩=\{\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\}\text{with}\begin{array}{cccc}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\end{array}\text{.}$$

$G=S_3,$ $S=\left\{\begin{array}{c} \end{array}\right\}\text{.}$

Then $$⟨S⟩=\{\begin{array}{c} \end{array},\begin{array}{c} \end{array}\text{with}\begin{array}{ccc}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\\ \begin{array}{c} \end{array}& \begin{array}{c} \end{array}& \begin{array}{c} \end{array}\end{array}\}\text{.}$$

Note: $$\begin{array}{ccc}{\displaystyle \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$2\mathbb{Z}$}\right.}& =& {\displaystyle \{0,1\}\text{with}\begin{array}{ccc}+& 0& 1\\ 0& 0& 1\\ 1& 1& 0\end{array}}\\ {\displaystyle \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$3\mathbb{Z}$}\right.}& =& {\displaystyle \{0,1,2\}\text{with}\begin{array}{cccc}+& 0& 1& 2\\ 0& 0& 1& 2\\ 1& 1& 2& 0\\ 2& 2& 0& 1\end{array}}\end{array}$$ The function $$\begin{array}{cccc}f:& \raisebox{1ex}{$\mathbb{Z}$}\!\left/ \!\raisebox{-1ex}{$3\mathbb{Z}$}\right.& ⟶& \{\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\}\\ & 0& ⟼& \begin{array}{c} \end{array}\\ & 0& ⟼& \begin{array}{c} \end{array}\\ & 2& ⟼& \begin{array}{c} \end{array}\end{array}$$ is an isomorphism.

The subgroups of $S_3$ are $$\begin{array}{c} S3 {,} {,} {,} {,,} {} \end{array}$$

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.

page history