From a2caac7e95377505e0b1df55b72fc85b201df52a Mon Sep 17 00:00:00 2001 From: Mark Date: Wed, 3 Jan 2024 22:25:15 -0800 Subject: [PATCH] Symmetric edits --- Advanced/Symmetric Group/main.tex | 4 +- Advanced/Symmetric Group/parts/0 intro.tex | 42 +++++-------- Advanced/Symmetric Group/parts/1 cycle.tex | 14 +++-- Advanced/Symmetric Group/parts/2 groups.tex | 59 +++++++------------ Advanced/Symmetric Group/parts/3 subgroup.tex | 37 ++++++------ 5 files changed, 68 insertions(+), 88 deletions(-) diff --git a/Advanced/Symmetric Group/main.tex b/Advanced/Symmetric Group/main.tex index b7ab866..9a0ae66 100755 --- a/Advanced/Symmetric Group/main.tex +++ b/Advanced/Symmetric Group/main.tex @@ -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 diff --git a/Advanced/Symmetric Group/parts/0 intro.tex b/Advanced/Symmetric Group/parts/0 intro.tex index c70bc04..3fc2538 100644 --- a/Advanced/Symmetric Group/parts/0 intro.tex +++ b/Advanced/Symmetric Group/parts/0 intro.tex @@ -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} +\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 \ No newline at end of file diff --git a/Advanced/Symmetric Group/parts/1 cycle.tex b/Advanced/Symmetric Group/parts/1 cycle.tex index b80ccbe..1e438c3 100755 --- a/Advanced/Symmetric Group/parts/1 cycle.tex +++ b/Advanced/Symmetric Group/parts/1 cycle.tex @@ -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 diff --git a/Advanced/Symmetric Group/parts/2 groups.tex b/Advanced/Symmetric Group/parts/2 groups.tex index 6eb4216..abb0ded 100755 --- a/Advanced/Symmetric Group/parts/2 groups.tex +++ b/Advanced/Symmetric Group/parts/2 groups.tex @@ -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 diff --git a/Advanced/Symmetric Group/parts/3 subgroup.tex b/Advanced/Symmetric Group/parts/3 subgroup.tex index 1bbfc72..f694f0d 100644 --- a/Advanced/Symmetric Group/parts/3 subgroup.tex +++ b/Advanced/Symmetric Group/parts/3 subgroup.tex @@ -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{} -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}