\documentclass{amsart} \usepackage{tikz} \usepackage{ulem} \usepackage{amsthm} \usepackage{stmaryrd}%\fatsemi \newcommand{\dom}{\mathrm{Dom}} \newcommand{\ran}{\mathrm{Ran}} \newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\defiff}{\stackrel{def}{\Longleftrightarrow}} \theoremstyle{definition}\newtheorem{df}{\bf Definition} \newtheorem{ex}{\bf Example} \newtheorem{e}{\bf Exercise} \newtheorem{prop}{\bf Proposition} \newtheorem{thm}{\bf Theorem} \newtheorem{lem}{\bf Lemma} \newtheorem{cor}{\bf Corollary} \begin{document} \title{Introduction to algebra --- handout 7} \maketitle \section{Structures with one binary operation} \begin{df} Algebraic structure $\langle S, \cdot\rangle$ is a {\bf semigroup} if $\cdot$ is an associative binary operation on $S$, i.e. \begin{equation*} \forall abc\enskip a\cdot(b\cdot c)=(a\cdot b)\cdot c. \end{equation*} \end{df} \begin{df} A semigroup is called {\bf monoid} if there is an identity element in it, i.e., \begin{equation*} \exists e \forall a \enskip a\cdot e=e\cdot a =a. \end{equation*} \end{df} \begin{df} Algebraic structure $\langle G, \cdot\rangle$ is a {\bf group} if it is a monoid in which every element has an inverse, i.e., \begin{equation*} \forall a \exists a^{-1}\enskip a \cdot a^{-1}=a^{-1}a=e. \end{equation*} \end{df} \begin{df} Commutative groups are often called {\bf abelian} groups. \end{df} \begin{ex}${}$ \begin{enumerate} \item Finite text over an alphabet form a semigroup (monoid) with respect to concatenation. \item Odd numbers with multiplication $\langle 2\Z+1,\cdot\rangle$ is a monoid. \item Even numbers with addition $\langle 2\Z,+\rangle$ is a group. \item $\langle \Z,\cdot \rangle$ is a monoid. \item $\langle \Z,+\rangle$ is an abelian group ($e=0$ and $a^{-1}=-a$). \item $\langle \{0\},+\rangle\cong \langle \{1\},\cdot\rangle$ is an abelian group (up to isomorphism the only one element group). \item $\langle \{-1,1\},\cdot\rangle$ is an abelian group (up to isomorphism the only two element group). \item $\langle \{-1,1,i,-i\},\cdot\rangle$ is an abelian group (where $i=\sqrt{-1}$). \item Positive real (or rational) numbers with multiplication is an abelian group. \item Nonzero real (or rational or complex) numbers with multiplication is an abelian group. %\item Complex numbers $\C=\{a+bi: a\in\R,\;b\in\R,\;i^2=-1\}$ with absolute value $1$ (i.e., $a^2+b^2=1$) is a group with multiplication. \item Permutations (bijections) of any set with respect of composition is a group. The group of permutations of an $n$ element set is denoted by $S_n$. \item The isometries of the plane with respect to composition is a group. \item Symmetries of a geometric shape with composition form a group. \item Let $a\sim b$ iff $m|b-a$ and let $\Z_m=\Z/\sim$. Then $\langle \Z_m,+\rangle$ is an abelian group. \end{enumerate} \end{ex} \begin{prop} Let $G$ be a group. \begin{enumerate} \item If $f\cdot x=x\cdot f=x$ for all $x\in G$, then $f=e$. \item If $a\cdot b=e$, then $b=a^{-1}$ and $a=b^{-1}$. \item If $ax=bx$, then $a=b$ (right cancelation). \item If $ya=yb$, then $a=b$ (cancelation). \item $(a^{-1})^{-1}=a$ and $(ab)^{-1}=b^{-1}a^{-1}$. \end{enumerate} \end{prop} \begin{df} A nonempty subset $H$ of group $G$ is a {\bf subgroup} of $G$, in symbols $H\le G$ if \begin{itemize} \item If $a,b\in H$, then $a\cdot b\in H$. \item $e\in H$. \item If $a\in H$, then $a^{-1}\in H$. \end{itemize} \end{df} \begin{ex} ${}$ \begin{enumerate} \item $\langle Z,+\rangle\le \langle\Q,+\rangle \le \langle \R,+\rangle \le \langle \C,+\rangle$ \item $\langle \{1\},\cdot\rangle\le \langle\{-1,1\},\cdot \rangle\le \langle\{-1,1,i,-i\},\cdot \rangle\le \langle \C\{0\},\cdot\rangle$ \item \raisebox{-18pt}{\begin{tikzpicture} \node at (-1.6,0) {$\langle 6\Z,+\rangle$}; \node[rotate=15] at (-0.8,0.2) {$\le$}; \node[rotate=15] at (0.8,-0.2) {$\le$}; \node at (0,0.4) {$\langle 2\Z,+\rangle$}; \node[rotate=-15] at (0.8,0.2) {$\le$}; \node[rotate=-15] at (-0.8,-0.2) {$\le$}; \node at (0,-0.4) {$\langle 3\Z,+\rangle$}; \node at (1.4,0) {$\langle \Z,+\rangle$}; \end{tikzpicture}} %\item $\langle 6\Z+\rangle\le\langle 2\Z+\rangle\le\langle \Z+\rangle$ \item The translations of the plane is a subgroup of the isometries of the plane. \item The isometries of the plane fixing the origin is a subgroup of the isometries of the plane. \end{enumerate} \end{ex} \begin{e} Prove that $\emptyset\neq H\subseteq G$ is a subgroup of group $G$ if and only if $a,b\in H$ implies $ab^{-1}\in H$. \end{e} Let us note that the intersection of subgroups is a subgroup. \begin{df} Let $X$ be a subset of group $G$. The intersection of subgroups containing $X$ is called the {\bf subgroup generated by} $X$ and it is denoted by $\langle X\rangle$. \end{df} \begin{thm} Let $L(G)$ be the set of subgroups of a group $G$. $L(G)$ is a lattice with respect to inclusion. \end{thm} \begin{proof} $H\land K=H\cap K$ and $H\lor K=\langle H\cup K\rangle$. \end{proof} \begin{df} Let $A$ and $B$ two nonempty subsets of group $G$. \begin{equation*} AB:=\{ab:a\in A \text{ and } b\in B\}\qquad A^{-1}:=\{a^{-1}: a\in A\} \end{equation*} \end{df} \noindent {\it Notation}: we write $aB$ in place of $\{a\}B$. \smallskip Let us note $HH=H$ and $H^{-1}=H$ if $H$ is a subgroup. \begin{prop} Let $G$ be a group and let $H$ be an arbitrary subset of $G$. Let relation $a\sim b$ hold iff $a^{-1}b\in H$. Then \begin{itemize} \item $\sim$ is an equivalence relation. \item The equivalence class of $a\in G$ is $aH$. The equivalence classes are called {\bf left cosets}. \item $|aH|=|H|$ for all $a\in G$. \end{itemize} \end{prop} The notion of {\bf right cosets} can be introduced analogously. \begin{df} The number of left cosets is called the {\bf index} of $H$ in $G$, denoted by $[G:H]$. \end{df} \begin{thm}[Lagrange] Let $G$ be a finite group and let $H\le G$ be an arbitrary subgroup of $G$. Then \begin{equation*} |G|=|H|[G:H]. \end{equation*} \end{thm} \begin{cor} If $|G|=n$, then $a^n=e$ for all $a\in G$. \end{cor} \begin{df} Let $a\equiv b\; (mod\; m)$ denote the equivalence relation on positive integers defined as $m|b-a$. \end{df} \begin{prop} If $a\equiv b\;(mod\; m)$ and $c\equiv d\;(mod\; m)$, then \begin{itemize} \item $a+c \equiv b+d \;(mod\; m)$ and \item $ac \equiv bd \;(mod\; m)$. \end{itemize} \end{prop} \begin{df} Positive integers $m$ and $n$ are called {\bf coprime} if $1$ is the only common divisor of $m$ and $n$. {\bf Euler $\phi$-function} is the function that maps positive integer $n$ to the number of coprimes primes less than $n$. \end{df} The residue classes coprime to $m$ is a group having $\phi(m)$ elements. Consequently: \begin{thm}[Euler] Let $a$ and $m$ be coprime, then $a^{\phi(m)}\equiv 1\;(mod\; m)$. \end{thm} Let us note that $\phi(p)=p-1$ if $p$ is a prime. Consequently: \begin{thm}[Fermat] $a^p\equiv a\;(mod\; p)$. \end{thm} \begin{df} Two elements $a$ and $b$ of a group $G$ are called {\bf conjugate} if there is an element $g$ of $G$ such that $a=gbg^{-1}$. \end{df} \begin{e} Prove that being conjugate is an equivalence realtion on $G$. \end{e} \begin{df} A subgroup $N$ of $G$ is called a {\bf normal subgroup}, in symbols $N\triangleleft G$, $gng^{-1}\in N$ for all $g\in G$ and $n\in N$. \end{df} \end{document}