2024-01-03 11:44:34 -08:00
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\section{Subgroups}
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\problem{}<s2s3share>
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What elements do $S_2$ and $S_3$ share?
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\vspace{2cm}
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2024-01-03 22:25:15 -08:00
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Consider the sets $\{1, 2\}$ and $\{1,2,3\}$. Clearly, $\{1, 2\} \subset \{1, 2, 3\}$. \par
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2024-01-03 11:44:34 -08:00
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Can we say something similar about $S_2$ and $S_3$?
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\vspace{2mm}
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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
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2024-01-03 22:25:15 -08:00
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This however, isn't as interesting as it could be. Remember that $S_2$ and $S_3$ are \textit{groups}, not \textit{sets}: \par
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their elements come with structure, which the \say{subset} relation does not capture.
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2024-01-03 11:44:34 -08:00
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\vspace{2mm}
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2024-01-03 22:25:15 -08:00
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To account for this, we'll define a similar relation: subgroups.
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2024-01-03 11:44:34 -08:00
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\definition{}
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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
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(Note that $x, y$ are elements of $G$, and $xy$ is multiplication in $G$)
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\begin{itemize}
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\item the set of elements in $G'$ is a subset of the set of elements in $G$.
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\item the identity of $G$ is in $G'$
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\item $x,y \in G' \implies xy \in G'$
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\item $x \in G' \implies x^{-1} \in G'$
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\end{itemize}
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The above definition may look faily scary, but the idea behind a subgroup is simple. \par
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Consider $S_3$ and $S_4$, the groups of permutations of $3$ and $4$ elements. \par
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\vspace{2mm}
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Say we have a set of four elements and only look at the first three. \par
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$S_3$ fully describes all the ways we can arrange those three elements:
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\begin{center}
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\begin{tikzpicture}[scale=0.5]
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\node (1a) at (0, 0.5) {1};
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\node (2a) at (1, 0.5) {2};
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\node (3a) at (2, 0.5) {3};
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\node (4a) at (3, 0.5) {4};
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\node (2b) at (0, -2) {2};
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\node (3b) at (1, -2) {3};
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\node (1b) at (2, -2) {1};
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\node (4b) at (3, -2) {4};
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\draw[line width = 0.3mm, ->, ogreen]
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(4a)
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-- ($(4a) + (0, -1)$)
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-- ($(4b) + (0,1)$)
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-- (4b);
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\line{1a}{1b}
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\line{2a}{2b}
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\line{3a}{3b}
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\node[fill=white,draw=oblue,line width=0.3mm] at (1, -0.75) {$S_3$};
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\end{tikzpicture}
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\end{center}
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\problem{}
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Show that $S_3$ is a subgroup of $S_4$.
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\vfill
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\pagebreak
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2024-01-04 11:34:55 -08:00
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\definition{}
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Let $G$ and $H$ be groups. We say that $G$ and $H$ are \textit{isomorphic} (and write $A \simeq B$) \par
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if there is a bijection $f: G \to H$ with the following properties:
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\begin{itemize}
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\item $f(e_G) = e_H$, where $e_G$ is the identity in $G$
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\item $f(x^{-1}) = f(x)^{-1}$ for all $x$ in $G$
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\item $f(xy) = f(x)f(y)$ for all $x, y$ in $G$
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\end{itemize}
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Intuitively, you can think of isomorphism as a form of equivalence. \par
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If two groups are isomorphic, they only differ by the names of their elements. \par
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The function $f$ above tells us how to map one set of labels to the other.
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\problem{}
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Show that $\mathbb{Z}_7^\times$ and $\mathbb{Z}_9^\times$ are isomorphic.
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\hint{
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Build a bijection with the above properties. \\
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Remember that a group is fully defined by its multiplication table.
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}
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\vfill
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\problem{}
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2024-01-08 10:56:22 -08:00
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Show that $\mathbb{Z}_{10}^\times$ and $\mathbb{Z}_5^\times$, and $\mathbb{Z}_4$ are isomorphic.
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2024-01-04 11:34:55 -08:00
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\hint{
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Build a bijection with the above properties. \\
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Remember that a group is fully defined by its multiplication table.
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2024-01-03 22:25:15 -08:00
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}
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2024-01-03 11:44:34 -08:00
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2024-01-04 11:34:55 -08:00
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\vfill
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\problem{}
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Show that isomorphism is transitive. \par
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That is, if $A \simeq B$ and $B \simeq C$, then $A \simeq C$.
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\vfill
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\pagebreak
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\problem{}<firstindex>
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How many subgroups of $S_4$ are isomorphic to $S_3$? \par
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2024-01-03 22:25:15 -08:00
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\vfill
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2024-01-03 11:44:34 -08:00
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\problem{}
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2024-01-03 22:25:15 -08:00
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What are the orders of $S_3$ and $S_4$? \par
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2024-01-03 11:44:34 -08:00
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How is this related to \ref{firstindex}?
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\begin{solution}
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$|S_4| = |S_3| \times [S_4 : S_3]$
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\vspace{2mm}
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2024-01-03 22:25:15 -08:00
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This solution is written using index notation, \par
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but the class doesn't need to know what it means yet.
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2024-01-03 11:44:34 -08:00
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\end{solution}
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\vfill
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\problem{}
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$S_4$ also has $S_2$ and the trivial group as subgroups. \par
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How many instances of each does $S_4$ contain?
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\vfill
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\problem{}
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$(\mathbb{Z}_4, +)$ is also a subgroup of $S_4$. Find it! \par
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2024-01-03 22:25:15 -08:00
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How many subgroups of $\mathbb{Z}_4$ are isomorphic to $S_4$?.
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2024-01-03 11:44:34 -08:00
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\begin{solution}
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A good hint is \say{look at generators.}
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\vspace{4mm}
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2024-01-03 22:25:15 -08:00
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There are four instances of $\mathbb{Z}_4$ in $S_4$, each of which is generated by a 4-cycle of $S_n$. \par
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2024-01-03 11:44:34 -08:00
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(i.e, the group generated by $(1234)$ is isomorphic to $\mathbb{Z}_4$)
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\end{solution}
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\vfill
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\pagebreak
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