From 7bad77dcd94e96536f9249c1415b6071846d914c Mon Sep 17 00:00:00 2001 From: Mark Date: Thu, 19 Jan 2023 11:58:53 -0800 Subject: [PATCH] Added group theory parts --- Advanced/Group Theory/main.tex | 3 +- Advanced/Group Theory/parts/01 groups.tex | 98 ++++++++++++------- .../Group Theory/parts/02 isomorphism.tex | 56 +++++++++++ Advanced/Group Theory/parts/03 bonus.tex | 47 +++++++++ 4 files changed, 166 insertions(+), 38 deletions(-) create mode 100644 Advanced/Group Theory/parts/02 isomorphism.tex create mode 100644 Advanced/Group Theory/parts/03 bonus.tex diff --git a/Advanced/Group Theory/main.tex b/Advanced/Group Theory/main.tex index 566da71..cc3b7e4 100755 --- a/Advanced/Group Theory/main.tex +++ b/Advanced/Group Theory/main.tex @@ -13,12 +13,13 @@ {Group Theory} { - Based on a lesson by Janet Chen \\ Prepared by Mark on \today } \input{parts/00 review} \input{parts/01 groups} + \input{parts/02 isomorphism} + \input{parts/03 bonus} \end{document} \ No newline at end of file diff --git a/Advanced/Group Theory/parts/01 groups.tex b/Advanced/Group Theory/parts/01 groups.tex index 121a7f2..0ae08f3 100755 --- a/Advanced/Group Theory/parts/01 groups.tex +++ b/Advanced/Group Theory/parts/01 groups.tex @@ -7,8 +7,8 @@ A group must 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: $(a \ast b) \ast c = a \ast (b \ast c)$ - \item There is an \textit{identity} $\overline{0} \in G$, so that $a \ast \overline{0} = a$ for all $a \in G$. - \item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = \overline{0}$. $b$ is called the \textit{inverse} of $a$. \\ + \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 $a$. \\ This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise. \end{enumerate} @@ -59,6 +59,11 @@ Is $(G, \circ)$ a group? \vfill \pagebreak +\problem{} +Show that if $G$ has four elements, $(G, \ast)$ is abelian. + +\vfill + \problem{} Show that a group has exactly one identity element. \vfill @@ -67,6 +72,18 @@ Show that a group has exactly one identity element. Show that each element in a group has exactly one inverse. \vfill +\problem{} +Show that... +\begin{itemize} + \item $e^{-1} = 1$ + \item $(a^{-1})^{-1} = a$ +\end{itemize} +\vfill + +\problem{} +Show that $(a^m)^{-1} = (a^{-1})^m$ for all $a \in G$ and $m \in \mathbb{Z}$. +\vfill + \problem{} Let $(G, \ast)$ be a group and $a, b, c \in G$. Show that... \begin{itemize} @@ -75,15 +92,16 @@ Let $(G, \ast)$ be a group and $a, b, c \in G$. Show that... \end{itemize} What does this mean intuitively? \vfill +\pagebreak \problem{} Let $(G, \ast)$ be a finite group (i.e, $G$ has finitely many elements), and let $g \in G$. \\ -Show that $\exists~n \in Z^+$ so that $g^n = \overline{0}$ \\ +Show that $\exists~n \in Z^+$ so that $g^n = e$ \\ \hint{$g^n = g \ast g \ast ... \ast g$ $n$ times.} \vspace{2mm} -The smallest such $n$ defines the \textit{order} of $(G, \ast)$. +The smallest such $n$ defines the \textit{order} of $g$. \vfill @@ -92,42 +110,48 @@ What is the order of 5 in $(\mathbb{Z}/25, +)$? \\ What is the order of 2 in $((\mathbb{Z}/17)^\times, \times)$? \\ \vfill +\pagebreak \problem{} -Show that if $G$ has four elements, $(G, \ast)$ is abelian. +Let $e, a, b, c$ be counterclockwise rotations of a square by $0, \frac{\pi}{2}, \pi,$ and $\frac{3\pi}{2}$. \\ +Create a multiplication table for this group. +\vfill + +\problem{} +Let $d, f, g, h$ correspond to reflections of the square along the following axis. \\ +Create a multiplication table for this group. + +\begin{center} +\begin{tikzpicture}[scale=2] + \draw (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0); + + \draw[gray] (1.25,1.25) -- (-0.25,-0.25) node[below left]{$d$}; + \draw[gray] (1.25,-0.25) -- (-0.25,1.25) node[above left]{$f$}; + \draw[gray] (0.5,-0.25) -- (0.5,1.25) node[above]{$g$}; + \draw[gray] (-0.25, 0.5) -- (1.25,0.5) node[right]{$h$}; + +\end{tikzpicture} +\end{center} +\vfill + +\problem{} +Create a multiplication table for all symmetries of a square. +\vfill +\pagebreak + +\problem{} +Create a multiplication table for all symmetries of a rhombus. +\vfill +\pagebreak + +\problem{} +Find the order of each element in... +\begin{itemize} + \item The group of symmetries of a square + \item The group of symmetries of a rhombus +\end{itemize} + \vfill \pagebreak -\definition{} -Recall your tables from \ref{modtables}: \\ - -\begin{center} -\begin{tabular}{c | c c c c} - + & 0 & 1 & 2 & 3 \\ - \hline - 0 & 0 & 1 & 2 & 3 \\ - 1 & 1 & 2 & 3 & 0 \\ - 2 & 2 & 3 & 0 & 1 \\ - 3 & 3 & 0 & 1 & 2 \\ -\end{tabular} -\hspace{1cm} -\begin{tabular}{c | c c c c} - \times & 1 & 2 & 3 & 4 \\ - \hline - 1 & 1 & 2 & 4 & 3 \\ - 2 & 2 & 4 & 3 & 1 \\ - 3 & 4 & 3 & 1 & 2 \\ - 4 & 3 & 1 & 2 & 4 \\ -\end{tabular} -\end{center} - -Look at these tables and convince yourself that $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times )$ are the same group. \\ -We say that two such groups are \textit{isomorphic}. - -\vspace{2mm} - -Intuitively, this means that these two groups have the same algebraic structure. We can translate statements about addition in $\mathbb{Z}/4$ into statements about multiplication in $(\mathbb{Z}/5)^\times$ \\ - -\pagebreak - diff --git a/Advanced/Group Theory/parts/02 isomorphism.tex b/Advanced/Group Theory/parts/02 isomorphism.tex new file mode 100644 index 0000000..2975af6 --- /dev/null +++ b/Advanced/Group Theory/parts/02 isomorphism.tex @@ -0,0 +1,56 @@ +\section{Isomorphism} + +\definition{} +We say two groups are \textit{isomorphic} if we can create a bijective mapping between them. + +\problem{} +Recall your tables from \ref{modtables}: \\ +\begin{center} +\begin{tabular}{c | c c c c} + + & 0 & 1 & 2 & 3 \\ + \hline + 0 & 0 & 1 & 2 & 3 \\ + 1 & 1 & 2 & 3 & 0 \\ + 2 & 2 & 3 & 0 & 1 \\ + 3 & 3 & 0 & 1 & 2 \\ +\end{tabular} +\hspace{1cm} +\begin{tabular}{c | c c c c} + \times & 1 & 2 & 3 & 4 \\ + \hline + 1 & 1 & 2 & 4 & 3 \\ + 2 & 2 & 4 & 3 & 1 \\ + 3 & 4 & 3 & 1 & 2 \\ + 4 & 3 & 1 & 2 & 4 \\ +\end{tabular} +\end{center} +Are $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times)$ isomorphic? If they are, find a bijection that maps one to the other. +\vfill + +\problem{} +Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. Show that $f(e_A) = e_B$, where $e_A$ and $e_B$ are the identities of $A$ and $B$. +\vfill + +\problem{} +Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. \\ +Show that $f(a^{-1}) = f(a)^{-1}$ for all $a \in A$. +\vfill + +\problem{} +Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$. Show that $f(a)$ and $a$ have the same order. + +\vfill +\pagebreak + +\problem{} +Find all distinct groups of two elements. \\ +Find all distinct groups of three elements. \\ +Groups that are isomorphic are not distinct. + +\vfill + +\problem{} +Show that the groups $(\mathbb{R}, +)$ and $(\mathbb{Z}^+, \times)$ are isomorphic. +\vfill + +\pagebreak \ No newline at end of file diff --git a/Advanced/Group Theory/parts/03 bonus.tex b/Advanced/Group Theory/parts/03 bonus.tex new file mode 100644 index 0000000..e2c15a3 --- /dev/null +++ b/Advanced/Group Theory/parts/03 bonus.tex @@ -0,0 +1,47 @@ +\section{Bonus} + +\problem{} +Find the inverse of 19 in $\mathbb{Z}/23$ \\ +\hint{Recall the Euclidian Algorithm} + + +\begin{solution} + 17 +\end{solution} +\vfill + +\problem{} +Prove Lagrange's theorem: + +$$ + a^p = a \text{ (mod p)} +$$ + +\vfill + +\problem{} +Show that $a$ has an inverse mod $m$ iff $\gcd(a, m) = 1$ \\ + +\begin{solution} + Assume $a^\star$ is the inverse of $a \pmod{m}$. \\ + Then $a^\star \times a \equiv 1 \pmod{m}$ \\ + + Therefore, $aa^\star - 1 = km$, and $aa^\star - km = 1$ \\ + We know that $\gcd(a, m)$ divides $a$ and $m$, therefore $\gcd(a, m)$ must divide $1$. \\ + $\gcd(a, m) = 1$ \\ + + Now, assume $\gcd(a, m) = 1$. \\ + By the Extended Euclidean Algorithm, we can find $(u, v)$ that satisfy $au+mv=1$ \\ + So, $au-1 = mv$. \\ + $m$ divides $au-1$, so $au \equiv 1 \pmod{m}$ \\ + $u$ is $a^\star$. +\end{solution} + +\vfill + + +\problem{} +Show that for any integers $a, b, c$, \\ +$\gcd(ac + b, a) = \gcd(a, b)$\\ + +\vfill \ No newline at end of file