diff --git a/Advanced/Quotient Groups/main.tex b/Advanced/Quotient Groups/main.tex deleted file mode 100755 index 5c8b96a..0000000 --- a/Advanced/Quotient Groups/main.tex +++ /dev/null @@ -1,52 +0,0 @@ -% use [nosolutions] flag to hide solutions. -% use [solutions] flag to show solutions. -\documentclass[ - solutions, - singlenumbering, - shortwarning, - unfinished -]{../../resources/ormc_handout} -\usepackage{../../resources/macros} - -\usepackage{units} - -\uptitlel{Advanced 2} -\uptitler{Fall 2023} -\title{Quotient Groups} -\subtitle{Prepared by \githref{Mark} on \today{}} - - -\def\znz#1{\nicefrac{\mathbb{Z}}{#1\mathbb{Z}}} - -\begin{document} - - \maketitle - - \input{parts/0 mod} - \input{parts/1 groups} - \input{parts/2 subgroups} - - - % Rough outline: - % - % Part 1: (DONE) - % mod, eqrel, eqclass. - % - % Part 2: (IN PROGRESS) - % groups, Z/nZx, graphs, isomorphism. - % generators, generating sets. - % - % Part 3: (IN PROGRESS) - % subgroups, isomorphic subgroups, - - % TODO: - % - % cosets - % normal subgroups - % quotient groups - % Understand Z/nZ - % Functions as objects (groups of functions) - % Q/Z problems (mod generalization) - % isomorphism groups (which are iso to symmetric group) - -\end{document} diff --git a/Advanced/Quotient Groups/parts/0 mod.tex b/Advanced/Quotient Groups/parts/0 mod.tex deleted file mode 100644 index 67ff605..0000000 --- a/Advanced/Quotient Groups/parts/0 mod.tex +++ /dev/null @@ -1,157 +0,0 @@ -\section{Modular Arithmetic} - -I'm sure you're all familiar with modular arithmetic. -In this section, our goal is to define \textit{equivalence relations}, -\textit{equivalence classes}, and use them to formally define arithmetic in mod $n$. - - -\problem{} -Compute the following: - -\begin{itemize} - \item $5 + 3 \pmod{4}$ - \item $7 \times 4 \pmod{9}$ - \item $-4 \pmod{5}$ - \item $3^{-1} \pmod{7}$ -\end{itemize} - -\vfill - - -\definition{} -An \textit{equivalence relation} on a set $A$ -is a symbol that makes a statement about two elements of $A$. -For example, $=$ is an equivalence relation on the set of integers. - -\vspace{2mm} - -An equivalence relation must satisfy the following properties: -\begin{itemize} - \item Reflexivity: $x \sim x$ for all $x \in A$ - \item Symmetry: if $x \sim y$, $y \sim x$ for any $x, y \in A$ - \item Transitivity: if $x \sim y$ and $y \sim z$, then $x \sim z$ -\end{itemize} - - -\problem{} -Which of the following are equivalence relations on $\mathbb{Z}$? -\begin{itemize} - \item $>$ - \item $\leq$ - \item $\Bumpeq$, where $a \Bumpeq b$ if $|a| = |b|$ - \item $\neq$ -\end{itemize} - - -\vfill -\pagebreak - - - - - - - - -\problem{} -Consider the relation $\equiv_n$ on $\mathbb{Z}$, where $a \equiv_n b$ holds iff $a \equiv b \pmod{n}$. \par -Show that $\equiv_n$ is an equivalence relation. - -\vfill - - -\definition{} -Say we have an equivalence relation $\sim$ on a set $A$. \par -The \textit{equivalence class} of $x$ is the set of all elements that are $\sim$ to $x$. \par -Here are a few examples: \par -\begin{itemize}[itemsep=2mm] - \item - The equivalence class of $2$ in $\mathbb{Z}$ under the relation $=$ is $\{2\}$, \par - since the only $x$ that satisfies $x = 2$ is $2$. - - \item - The equivalence class of $9$ in $\mathbb{Z}$ under the relation $\Bumpeq$ - from \ref{abseq} is $\{-9, 9\}$. -\end{itemize} - - -\problem{} -What is the equivalence class of $3$ in $\mathbb{Z}$ under $\equiv_5$? \par -\hint{Remember that $\mathbb{Z}$ contains both positive and negative numbers.} - -\begin{solution} - $\{..., -7, -2, 3, 8, 12, ... \}$ -\end{solution} - - -\vfill - -\problem{} -Let $A$ be a set and $\sim$ an equivalence relation. \par -Show that every element of $A$ is in \textit{exactly one} equivalence class. \par -\hint{What properties does an equivalence relation satisfy?} - - -\vfill - - - -We now have a proper definition of \say{mod $n$:} \par -it is the equivalence relation $a \equiv_n b$, which is usually written as $a \equiv b \pmod{n}$. \par -We will use this definition throughout this handout. - -\note[Note]{ - This is different than the \say{mod} operator $a ~\%~ b $, - which is defined as the remainder of $a \div b$. -} - - -\pagebreak - - -\definition{} -Given any $x \in \mathbb{Z}$, $[x]_n$ is the equivalence class of $x$ under $\equiv_n$. - - -\problem{} -Compute the following: -\begin{itemize}[itemsep = 1mm] - \item $[5]_3 + [4]_3$ - \item $[-2]_7 + [9]_7$ -\end{itemize} - -\vfill - -\problem{} -Does $[4]_3 + [7]_5$ make sense? - -\vfill - -\problem{} -Find all $n$ that satisfy -$[5]_n \times [17]_n = [3]_n + [2]_n$ \par -\hint{$[a]_n = [b]_n$ iff $n$ divides $a - b$, by definition of mod.} - -\begin{solution} - $[85] = [12] ~\implies~ n ~|~ 85 - 12 ~\implies~ n ~|~ 73 ~\implies~ n \in \{1, 73\}$ -\end{solution} - - -\vfill - -\definition{} -$\znz{n}$ (pronounced \say{$\mathbb{Z}$ mod $n \mathbb{Z}$}) is the set of equivalence classes of $\equiv_n$ on $\mathbb{Z}$. \par -For example, $\znz{5} = \{~ [0]_5,~ [1]_5,~ [2]_5,~ [3]_5,~ [4]_5 ~\}$. \par - -\vspace{2mm} - -This notation may seem a bit odd, but don't let it confuse you. \par -One of our goals today is to understand what exactly $\znz{n}$ means. - -\problem{} -What is $\znz{6}$? - -\vfill - - -\pagebreak \ No newline at end of file diff --git a/Advanced/Quotient Groups/parts/1 groups.tex b/Advanced/Quotient Groups/parts/1 groups.tex deleted file mode 100644 index 3df0aaa..0000000 --- a/Advanced/Quotient Groups/parts/1 groups.tex +++ /dev/null @@ -1,158 +0,0 @@ -\section{Groups} - -\definition{} -A \textit{group} $G = (S, \ast)$ consists of a set $S$ and a binary operator $\ast$. \par -By definition, a group always has 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 There is an \textit{identity} $e \in G$, so that $a \ast e = a \ast e = a$ for all $a \in G$. - \item Any $a \in G$ has an \textit{inverse} $a^{-1} \in G$ that satisfies $a \ast a^{-1} = a^{-1} \ast a = e$. \par -\end{enumerate} - -\note[Note]{ - Commutativity is \textit{not} a required property of a group! \\ - In most cases, $a \ast b \neq b \ast a$. -} - - - -\problem{} -Is $(\znz{5}, +)$ a group? \par -How about $(\znz{5}, -)$? \par -\hint{In this problem, $+$ and $-$ work just as you'd expect.} - -\vfill - -\problem{} -What is the smallest possible group? - -\begin{solution} - Let $(G, \ast)$ be our group, where $G = \{e\}$ and $\ast$ is defined by the identity $e \ast e = e$ - - Verifying that the trivial group is a group is trivial. -\end{solution} - -\vfill - -\problem{} -How many distinct groups have two elements? \par -\hint{ - Two groups are \say{the same} if the elements of one can be renamed to get the other. \\ - A group is fully defined by its multiplication table. -} - - -\vfill -\pagebreak - - - -%\problem{} -%Is $(\znz{17}, \times)$ a group? \par -%How should we modify $\znz{17}$ to make it one? - -%\problem{} -%Is $(\znz{6}, \times)$ a group? \par -%How should we modify $\znz{6}$ to make it one? \par -%\hint{ -% Be careful, this isn't as easy as \ref{firstcross}. \\ -% Which elements aren't invertible? -%} - - -%\definition{} -%Building on problems \ref{num:firstcross} and \ref{num:secondcross}, we'll define $(\znz{n})^\times$ as the multiplicative -%group of integers mod $n$. \par - -%Specifically, $(\znz{n})^\times$ is the set of all integers coprime to $n$. \par - -%\vspace{2mm} - -%For example, $(\znz{6})^\times = \{1, 5\}$ \par -%and $(\znz{15})^\times = \{1, 2, 4, 7, 8, 11, 13, 14\}$ \par - -%\vspace{2mm} - -%Note that $0$ is the identity in $\znz{n}$ and $1$ is the identity in $(\znz{n})^\times$\hspace{-1.5ex}. \par -%\note[Note]{ -% Also, notice that we've omitted the operations $+$ and $\times$ in the two groups above. \\ -% These operations are implicitly \say{attached} to $\znz{n}$ and $(\znz{n})^\times$\hspace{-1.5ex}, \\ -% and we rarely write them for the sake of cleaner notation. -%} - - -\vfill - - -\definition{} -Let $G$ be a group, $a$ an element of $G$, and $n \in \mathbb{Z}^+$. \par -$a^n$ is the defined as $a \ast a \ast ... \ast a$, repeated $n$ times. - -\vspace{1mm} - -Note that this is \textbf{not} \say{normal} exponentiation! \par -If our group's operator is $+$ (for example, $\znz{5}$), $a^n = a + ... + a$, \par -which you'll recognize as multipication. - -\vspace{1mm} - -Beware of this odd notation. By convention, we use \say{multiplicative} notation -when working with groups---so, $a \ast b$ may also be written as $ab$, -and $a \ast a \ast a$ may be written as $a^3$. - -\vspace{1mm} - -Again, remember that $a^n$ simply means \say{$\ast$ $a$ with itself $n$ times,} \par -regardless of the specific operator our group uses. - -\problem{} -Let $a$ be an element of a finite group. \par -Show that there is a positive integer $n$ so that $a^n = e$. \par - -\vspace{2mm} - -The smallest such $n$ defines the \textit{order} of $g$. - -\vfill - -\problem{} -Find the order of 5 in $(\znz{25}, +)$. \par -%Find the order of 2 in $((\znz{17})^\times, \times)$. \par -Find the order of 2 in $(\znz{7}, +)$. \par - -\vfill -\pagebreak - - -\definition{} -Let $G$ be a group. \par -We say a $g \in G$ is a \textit{generator} of $G$ -if every element in $G$ can be written as some power of $g$. - -\vspace{2mm} - -If $G$ has a generator, we say $G$ is \textit{cyclic.} - -\problem{} -Find a generator of $\znz{7}$. Then, find a generator of $(\znz{7})^\times$ - -\vfill - -\definition{} -Let $G$ be a group. \par -The \textit{order} of $G$ is the number of elements in $G$. \par -We'll write this as $|G|$, using the same notation we use with sets. \par -\note[Note]{ - Don't confuse the order of an \textbf{element} - with the order of a \textbf{group}! -} - -\problem{} -Let $G$ be a cyclic group, and let $g$ be any generator in $G$. \par -Show that $\text{ord}(g) = |G|$. \par -\hint{Contradiction.} - -\vfill -\pagebreak \ No newline at end of file diff --git a/Advanced/Quotient Groups/parts/2 subgroups.tex b/Advanced/Quotient Groups/parts/2 subgroups.tex deleted file mode 100644 index ff6d5e2..0000000 --- a/Advanced/Quotient Groups/parts/2 subgroups.tex +++ /dev/null @@ -1,23 +0,0 @@ -\section{Subgroups} - -\definition{} -Let $G$ be a group, and let $H$ be a subset of $G$. \par -We say $H$ is a \textit{subgroup} of $G$ if $H$ is also a group -(with the operation $\ast$). - -\definition{} -Let $S$ be a subset of $G$. \par -The \textit{group generated by $S$} consists of all elements of $G$ \par -that may be written as a combination of elements in $S$ - -\vspace{2mm} - -We will denote this group as $\langle S \rangle$. \par -Convince yourself that $\langle g \rangle = G$ if $g$ generates $G$. - -\problem{} -What is the subgroup generated by $\{7, 8\}$ in $(\znz{15})^\times$? \par -Is this the whole group? - -\problem{} -Show that the group generated by $S$ is indeed a group. \ No newline at end of file