127 lines
3.1 KiB
TeX
127 lines
3.1 KiB
TeX
\section{Subgroups}
|
|
|
|
\problem{}<s2s3share>
|
|
What elements do $S_2$ and $S_3$ share?
|
|
\vspace{2cm}
|
|
|
|
|
|
|
|
Consider the sets $\{1, 2\}$ and $Omega_3 = \{1,2,3\}$. Clearly, $\{1, 2\} \subset \{1, 2, 3\}$. \par
|
|
Can we say something similar about $S_2$ and $S_3$?
|
|
|
|
\vspace{2mm}
|
|
|
|
Looking at \ref{s2s3share}, we may want to say that $S_2 \subset S_3$ since every element of $S_2$ is in $S_3$. \par
|
|
This reasoning, however, is not correct. Remember that $S_2$ and $S_3$ are \textit{groups}, not \textit{sets}: \par
|
|
their elements come with structure.
|
|
|
|
\vspace{2mm}
|
|
|
|
Therefore, the \say{subset} relation isn't particularly useful when applied to groups. \par
|
|
We instead use a similar relation: subgroups.
|
|
|
|
\definition{}
|
|
Let $G$ and $G'$ be groups. We say $G'$ is a \textit{subgroup} of $G$ (and write $G' \subset G$) if the following are true:\par
|
|
(Note that $x, y$ are elements of $G$, and $xy$ is multiplication in $G$)
|
|
\begin{itemize}
|
|
\item the set of elements in $G'$ is a subset of the set of elements in $G$.
|
|
\item the identity of $G$ is in $G'$
|
|
\item $x,y \in G' \implies xy \in G'$
|
|
\item $x \in G' \implies x^{-1} \in G'$
|
|
\end{itemize}
|
|
|
|
The above definition may look faily scary, but the idea behind a subgroup is simple. \par
|
|
Consider $S_3$ and $S_4$, the groups of permutations of $3$ and $4$ elements. \par
|
|
|
|
\vspace{2mm}
|
|
|
|
Say we have a set of four elements and only look at the first three. \par
|
|
$S_3$ fully describes all the ways we can arrange those three elements:
|
|
|
|
\begin{center}
|
|
\begin{tikzpicture}[scale=0.5]
|
|
\node (1a) at (0, 0.5) {1};
|
|
\node (2a) at (1, 0.5) {2};
|
|
\node (3a) at (2, 0.5) {3};
|
|
\node (4a) at (3, 0.5) {4};
|
|
|
|
\node (2b) at (0, -2) {2};
|
|
\node (3b) at (1, -2) {3};
|
|
\node (1b) at (2, -2) {1};
|
|
\node (4b) at (3, -2) {4};
|
|
|
|
\draw[line width = 0.3mm, ->, ogreen]
|
|
(4a)
|
|
-- ($(4a) + (0, -1)$)
|
|
-- ($(4b) + (0,1)$)
|
|
-- (4b);
|
|
|
|
\line{1a}{1b}
|
|
\line{2a}{2b}
|
|
\line{3a}{3b}
|
|
|
|
\node[fill=white,draw=oblue,line width=0.3mm] at (1, -0.75) {$S_3$};
|
|
|
|
\end{tikzpicture}
|
|
\end{center}
|
|
|
|
|
|
\problem{}
|
|
Show that $S_3$ is a subgroup of $S_4$.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
|
|
|
|
|
|
\problem{}<firstindex>
|
|
How many subgroups of $S_4$ are equal to $S_3$?
|
|
|
|
\begin{solution}
|
|
Four, since there are four ways to pick three things from $S_4$.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
What is the order of $S_3$ and $S_4$? \par
|
|
How is this related to \ref{firstindex}?
|
|
|
|
\begin{solution}
|
|
$|S_4| = |S_3| \times [S_4 : S_3]$
|
|
|
|
\vspace{2mm}
|
|
|
|
This solution is written using index notation, but the class
|
|
doesn't yet need to know what it means.
|
|
\end{solution}
|
|
|
|
\vfill
|
|
|
|
\problem{}
|
|
$S_4$ also has $S_2$ and the trivial group as subgroups. \par
|
|
How many instances of each does $S_4$ contain?
|
|
|
|
\vfill
|
|
|
|
|
|
\problem{}
|
|
$(\mathbb{Z}_4, +)$ is also a subgroup of $S_4$. Find it! \par
|
|
How many copies of $Z_4$ are in $S_4$? \par
|
|
(You'll need to re-label elements, since we usually use different notation for $\mathbb{Z}_4$ and $S_4$).
|
|
|
|
\begin{solution}
|
|
A good hint is \say{look at generators.}
|
|
|
|
\vspace{4mm}
|
|
|
|
There are four instances of $\mathbb{Z}_4$ in $S_4$, \par
|
|
each of which is generated by a 4-cycle of $S_n$. \par
|
|
(i.e, the group generated by $(1234)$ is isomorphic to $\mathbb{Z}_4$)
|
|
\end{solution}
|
|
|
|
|
|
\vfill
|
|
\pagebreak
|