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4
Advanced/Symmetric Group/main.tex
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@ -35,14 +35,14 @@
\section{Bonus problems}
\problem{}
Show that $x$ has a multiplicative inverse mod $n$ iff $\text{gcd}(x, n) = 1$
Show that $x \in \mathbb{Z}^+$ has a multiplicative inverse mod $n$ iff $\text{gcd}(x, n) = 1$
\vfill
\problem{}
Let $\sigma = (\sigma_1 \sigma_2 ... \sigma_k)$ be a $k$-cycle in $S_n$, and let $\tau$ be an arbitrary element of $S_n$. \par
Show that $\tau \sigma \tau^{-1}$ = $\bigl(\tau(\sigma_1), \tau(\sigma_2), ..., \tau(\sigma_k)\bigr)$ \par
\hint{As usual, $\sigma$ is a permutation. Thus, $\sigma(x)$ is the value at position $x$ after applying $\sigma$.}
\hint{As usual, $\tau$ is a permutation. Thus, $\tau(x)$ is the value at position $x$ after applying $\tau$.}
\vfill

42
Advanced/Symmetric Group/parts/0 intro.tex
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@ -1,18 +1,7 @@
\section{Introduction}
\definition{Intuitive permutations}
Intuitively, a \textit{permutation} is an ordered arrangement of a set of objects. \par
For example, $123$, $312$, and $231$ are all permutations of 1, 2, and 3.
\problem{}
List all permutations on three objects. \par
How many permutations of $n$ objects are there?
\vfill
\definition{Formal permutations}<permadef>
\definition{}
Let $\Omega$ be an arbitrary set of $n$ objects. \par
A \textit{permutation} on $\Omega$ is a bijective map $f: \Omega \to \Omega$.
@ -26,24 +15,31 @@ The permutation $[312]$ is given by a map $f$ defined by the following table:
\item $f(3) = 2$
\end{itemize}
Similarly, the \textit{trivial permutation} $[123]$ is given by the identity map $f(x) = x$.
\problem{}
What map corresponds to the permutation $[321]$?
List all permutations on three objects. \par
How many permutations of $n$ objects are there?
\vfill
\problem{}
What map corresponds to the permutation $[321]$?
\vfill
\problem{}
Why do we define permutations as a \textit{bijective} map?
What map corresponds to the \say{do-nothing} permutation? \par
Write it as a function and in square-bracket notation. \par
\note[Note]{We usually call this the \textit{trivial permutation}}
\vfill
\pagebreak
We can visualize permutations with a diagram we'll call the \say{braid.}
We can visualize permutations with a \textit{string diagram}, shown below. \par
The arrows in this diagram denote the image of $f$ for each possible input.
Two examples are below:
@ -161,8 +157,8 @@ The rightmost diagram uses arbitrary, meaningless labels.
It shouldn't be hard to see that despite the different \say{output} order (2134 and 1432), \par
the same permutation is depicted in all three diagrams. This example demonstrates two things:
\begin{itemize}[itemsep=2mm]
\item First, the items of our set do not have any meaning. \par
$\Omega$ is just a set of arbitrary \textit{things}, which we may label however we like.
\item First, the names of the items in our set do not have any meaning. \par
$\Omega$ is just a set of $n$ arbitrary things, which we may label however we like.
\item Second, permutations are verbs. We do not care about the \say{output} of a certain permutation,
we care about what it \textit{does}. We could, for example, describe the permutation above as
@ -176,16 +172,10 @@ Why, then, do we order our elements when we talk about permutations? As noted be
If we assign a natural order to the elements of $\Omega$ (say, 1234), we can identify permutations by simply listing
their output:
Clearly, $[1234]$ represents the trivial permutation, $[2134]$ represents \say{swap first two,}
and $[4123]$ represents \say{cycle left.}
and $[4123]$ represents \say{cycle right.}
\problem{}
Draw braids for $[4123]$ and $[2341]$.
Draw string diagrams for $[4123]$ and $[2341]$.
\vfill
Finally, note that permutations (as defined in \ref{permadef}) are \textit{not} \say{orderings of a certain set.} \par
They are defined as \textit{bijective maps}, which can be written as orderings of a given array. \par
Remember: permutations are verbs!
\pagebreak

14
Advanced/Symmetric Group/parts/1 cycle.tex
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@ -2,8 +2,10 @@
\section{Cycle Notation}
\definition{Order}
The \textit{order} of a permutation $f$ is the smallest $n$ so that $f^n(x) = x$ for all $x$. \par
In other words, if we repeat this permutation $n$ times, we get back to where we started.
The \textit{order} of a permutation $f$ is the smallest positive $n$ so that $f^n(x) = x$ for all $x$. \par
In other words: if we repeat this permutation $n$ times, we get back to where we started. \par
Note that the order is given by the \textit{smallest} positive integer $n$. There may be more than one!
\vspace{2mm}
@ -44,7 +46,7 @@ Naturally, the identity permutation has order one.
\problem{}
What is the order of $[2314]$? \par
How about $[4321]$? \par
\note[Note]{You shouldn't need to draw any braids to solve this problem.}
\note[Note]{You shouldn't need to draw any strings to solve this problem.}
\vfill
@ -168,6 +170,9 @@ The permutation $[431265]$ is a bit more interesting---it contains of two cycles
\end{center}
Another name we'll often use for two-cycles is \textit{transposition}. \par
Any permutation that swaps two adjacent elements is called a transposition. \par
\problem{}
Find all cycles in $[5342761]$.
@ -417,7 +422,8 @@ Be careful.
\problem{}
Look at the last two permutations in \ref{insquare}, $(1234)$ and $(3412)$. \par
These are \textit{identical}---they are the same cycle written in two different ways. \par
List all other ways to write this cycle. \hint{There are two more.}
List all other ways to write this cycle. \hint{There are two more.} \par
\note{Also, note that the last two permutations in \ref{insquare} are the same.}
\pagebreak

59
Advanced/Symmetric Group/parts/2 groups.tex
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@ -6,11 +6,11 @@ Before we continue, we must introduce a bit of notation:
\item $S_n$ is the set of permutations on $n$ objects.
\item $\mathbb{Z}_n$ is the set of integers mod $n$.
\item $\mathbb{Z}_n^\times$ is the set of integers mod $n$ with multiplicative inverses, which is \par
the set of integers smaller than $n$ and coprime to $n$\footnotemark{}\hspace{-1ex}. \par
\item $\mathbb{Z}_n^\times$ is the set of integers mod $n$ with multiplicative inverses. \par
In other words, it is the set of integers smaller than $n$ and coprime to $n$.\footnotemark{} \par
For example, $\mathbb{Z}_{12}^\times = \{1, 5, 7, 11\}$.
\footnotetext{We proved this in another handout, but you make take it as fact here.}
\footnotetext{We proved this in another handout, but you may take it as fact here.}
\end{itemize}
\problem{}
@ -26,7 +26,7 @@ Groups always have the following properties:
\begin{enumerate}
\item $G$ is closed under $\ast$. In other words, $a, b \in G \implies a \ast b \in G$.
\item $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
\item $\ast$ is \textit{associative}: $(a \ast b) \ast c = a \ast (b \ast c)$ for all $a,b,c \in G$
\item There is an \textit{identity} $e \in G$, so that $a \ast e = a \ast e = a$ for all $a \in G$.
\item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = b \ast a = e$. $b$ is called the \textit{inverse} of $a$. \par
This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise.
@ -42,7 +42,7 @@ Is $(\mathbb{Z}_5, -)$ a group? \par
\problem{}
What is the smallest group?
What is the group with the fewest elements?
\begin{solution}
Let $(G, \star)$ be our group, where $G = \{x\}$ and $\star$ is defined by $x \star x = x$
@ -63,7 +63,10 @@ What is the smallest group?
\problem{}
Show that function composition is associative
\vfill
\problem{}
@ -82,7 +85,7 @@ The smallest such $n$ defines the \textit{order} of $g$.
\begin{examplesolution}
We've already done a special case of this problem! \par
Look back through the handout and find it, then rewrite your proof for an arbitrary group.
Find it in this handout, then rewrite your proof for an arbitrary (finite) group.
\end{examplesolution}
@ -116,34 +119,25 @@ We say $g$ is a \textit{generator} if every other element of $G$ may be written
Say the size of a group $G$ is $n$. \par
If $g$ is a generator, what is its order? \par
Provide a proof.
\vfill
\problem{}
Find the two generators in $(\mathbb{Z}, +)$ \par
Then, find all generators of $(\mathbb{Z}_5, +)$
\vfill
\problem{}
How many groups have only one generator?
\begin{solution}
The order of a generator must equal the order of its group.
Only one: the trivial group. The inverse of a generator is also a generator!
\end{solution}
\vfill
\problem{}
Find the only generator of $(\mathbb{Z}^+, +)$ \par
Then, find all generators of $(\mathbb{Z}_5, +)$
\vfill
\pagebreak
\definition{}
Let $S$ be a subset of the elements in $G$. \par
@ -168,13 +162,4 @@ We've already found a few generating sets of $S_n$. What are they?
\end{solution}
\vfill
\problem{}
Find the smallest set that generates $(\mathbb{Z}^+, +)$. \par
\vfill
\problem{}
Find the smallest set that generates $(\mathbb{Z}, +)$. \par
\vfill
\pagebreak

37
Advanced/Symmetric Group/parts/3 subgroup.tex
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@ -6,19 +6,18 @@ What elements do $S_2$ and $S_3$ share?
Consider the sets $\{1, 2\}$ and $Omega_3 = \{1,2,3\}$. Clearly, $\{1, 2\} \subset \{1, 2, 3\}$. \par
Consider the sets $\{1, 2\}$ and $\{1,2,3\}$. 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.
This however, isn't as interesting as it could be. Remember that $S_2$ and $S_3$ are \textit{groups}, not \textit{sets}: \par
their elements come with structure, which the \say{subset} relation does not capture.
\vspace{2mm}
Therefore, the \say{subset} relation isn't particularly useful when applied to groups. \par
We instead use a similar relation: subgroups.
To account for this, we'll define 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
@ -73,19 +72,21 @@ Show that $S_3$ is a subgroup of $S_4$.
\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}
How many subgroups of $S_4$ are behave like to $S_3$? \par
\note{
Of course, \say{behaves like} is a very hand-wavy relationship. \\
Formally, this is called \textit{isomorphism}, but we'll formally define that
in a later lesson.
}
\vfill
\problem{}
What is the order of $S_3$ and $S_4$? \par
What are the orders of $S_3$ and $S_4$? \par
How is this related to \ref{firstindex}?
\begin{solution}
@ -93,8 +94,8 @@ How is this related to \ref{firstindex}?
\vspace{2mm}
This solution is written using index notation, but the class
doesn't yet need to know what it means.
This solution is written using index notation, \par
but the class doesn't need to know what it means yet.
\end{solution}
\vfill
@ -108,16 +109,14 @@ How many instances of each does $S_4$ contain?
\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 $S_4$).
How many subgroups of $\mathbb{Z}_4$ are isomorphic to $S_4$?.
\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
There are four instances of $\mathbb{Z}_4$ in $S_4$, 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}