... ... @@ -2,7 +2,7 @@ \section{Definition of the Group} \begin{definition} \begin{definition}[Group] A \textbf{group} is a set $G$ equipped with a binary operation $*$ such that % \begin{enumerate} ... ... @@ -24,7 +24,7 @@ for all $x \in G$. The other equations follow from these axioms. \end{remark} \begin{definition} \begin{definition}[Abelian, Commutative] A group $G$ is called \textbf{abelian} if it satisfies the \textbf{commutative law}. \end{definition} ... ... @@ -50,13 +50,13 @@ \end{enumerate} \end{proposition} \begin{definition} \begin{definition}[Semigroup] A \textbf{semigroup} is a set having an associative operation; a \textbf{monoid} is a semigroup $S$ having a (two-sided) identity element 1; that is, $1s = s= s1$ for all $s \in S$. \end{definition} \begin{definition} \begin{definition}[Order] Let $G$ be a group and let $a \in G$. If $a^k = 1$ for some $k \geq 1$, then the smallest such exponent $k \geq 1$ is called the \textbf{order} of $a$; if no such power exists, then we say that $a$ has \textbf{infinite order}. ... ... @@ -71,12 +71,12 @@ If $G$ is a finite group, then every $x \in G$ has finite order. \end{proposition} \begin{definition} \begin{definition}[Group order] If $G$ is a finite group, then the number of elements in $G$, denoted by $|G|$, is called the \textbf{order} of $G$. \end{definition} \begin{definition} \begin{definition}[Dihedral group] If $\pi_{n}$ is a regular polygon with $n \geq 3$ vertices $v_1, \ldots, v_n$ and center $O$, then the symmetry group $\Sigma(\pi_n)$ is called the \textbf{dihedral group} of order $2n$, denoted by $D_{2n}$ or $D_{n}$. The ... ... @@ -107,7 +107,7 @@ \subsection{Subgroups} \begin{definition} \begin{definition}[Subgroup, Proper Subgroup] A subset $H$ of a group $G$ is a \textbf{subgroup} if % \begin{enumerate} ... ... @@ -135,7 +135,7 @@ group and $\N$ is closed under addition, but $\N$ is not a subgroup of $\Z$. \end{example} \begin{definition} \begin{definition}[Cyclic Group] If $G$ is a group and $a \in G$, then the \textbf{cyclic subgroup} of $G$ \textbf{generated} by $a$, denoted by $\langle a \rangle$, is % ... ... @@ -162,7 +162,7 @@ \cdot)$for some$m \geq 1$. \end{lemma} \begin{definition} \begin{definition}[Generator] If$X$is a subset of a group$G$, then$\langle X \rangle$is called the \textbf{subgroup generated by} X. \end{definition} ... ... @@ -170,7 +170,7 @@ \subsection{Cosets and Lagrange's Theorem} \begin{definition} \begin{definition}[Coset] If$H$is a subgroup of a group$G$and$a \in G$, then the \textbf{(left) coset}$aH$is % ... ... @@ -199,7 +199,7 @@ \end{enumerate} \end{lemma} \begin{definition} \begin{definition}[Index] The \textbf{index} of a subgroup$H$in$G$, denoted by$[G:H]$, is the number of \emph{left} cosets of$H$in$G$. \end{definition} ... ... @@ -229,7 +229,7 @@ \section{Homomorphisms} \begin{definition} \begin{definition}[Homomorphism] Let$(G,*)$and$(H,\circ)$be groups. A \textbf{homomorphism} is a function satisfying % ... ... @@ -248,7 +248,7 @@ \end{enumerate} \end{lemma} \begin{definition} \begin{definition}[Kernel, Image] If$f \colon G \to H$is a homomorphism, define the \textbf{kernel} of$fby % \begin{align*} ... ... @@ -262,12 +262,12 @@ \end{align*} \end{definition} \begin{definition} \begin{definition}[Conjugate] LetG$be a group. A \textbf{conjugate} of$a \in G$is an element in$G$of the form$gag^{-1}$for some$g \in G$. \end{definition} \begin{definition} \begin{definition}[Cojugation] If$G$is a group and$g \in G$, then \textbf{conjugation by}$g$is the function defined by for all$a \in G$% ... ... @@ -285,7 +285,7 @@ \end{enumerate} \end{proposition} \begin{definition} \begin{definition}[Normal Subgroup] A subgroup$K$of a group$G$is called a \textbf{normal subgroup} it is invariant under conjugation from elements in G. That is,$\forall g \in G: \gamma_g (K) = K$. If$K$is a normal subgroup of$G$, we write$K ... ... @@ -313,7 +313,7 @@ \end{enumerate} \end{proposition} \begin{definition} \begin{definition}[Center] The \textbf{center} of a group $G$, $Z(G)$, is defined by % \begin{align*} ... ... @@ -331,7 +331,7 @@ is normal. \end{example} \begin{definition} \begin{definition}[Inner automorphism, Outer automorphism] The conjugations are called \textbf{inner automorphisms}. An automorphism that is not inner is called \textbf{outer automorphism}. The set $\mathrm{Aut}(G)$ of all the automorphisms of $G$ is itself a group. Furthermore, ... ... @@ -532,7 +532,7 @@ contains a subgroup of order $d$. \end{proposition*} \begin{definition*} \begin{definition*}[Direct Product] If $H$ and $K$ are groups, then their \textbf{direct product}, denoted by $H \times K$, is the set of all ordered pairs $(h,k), h \in H, k \in K$, equipped with the operation ... ... @@ -567,7 +567,7 @@ \section{Group Actions and the Sylow Theorems} \begin{definition} \begin{definition}[Group action] Let $G$ be a group and $T$ a set. An \textbf{action} of $G$ on $T$ is a map % \begin{align*} ... ... @@ -583,7 +583,7 @@ \end{enumerate} \end{definition} \begin{definition} \begin{definition}[Group of symmetries] Let $T$ be a set. Then we define \begin{align*} \Sym(T) := \set{f \colon T \to T, f \text{~bijective}} ... ... @@ -618,20 +618,20 @@ functions. \end{remark} \begin{definition} \begin{definition}[Invariant subset, Fixed point] If $G$ acts on $T$ and $S \subset T$ is such that $g \cdot s \in S, \forall s \in S$, then $G$ also acts on $S$. We call $S$ a $\mathbf{G}$\textbf{-invariant subset}. If $S$ consists of a single element and is invariant, we call that point a \textbf{fixed point} of the action. \end{definition} \begin{definition} \begin{definition}[Faithful group action] Let $G$ act on $T$. If the homomorphism $\alpha \colon G \to \Sym(T)$ is injective, then the action is called \textbf{faithful}, i.e. $e_{G} \in G$ is the only element of $G$ such that $g \cdot t = t, \forall t \in T$. \end{definition} \begin{definition} \begin{definition}[Orbit, Orbit map] Let $G$ act on $T$. Let $t \in T$. The map $\gamma_t \colon G \to T$ such that $\gamma_t(g) = g \cdot t$ is called the \textbf{orbit map} associated to $t$. Its image defines a set called the $\mathbf{G}$\textbf{-orbit} of $t$ in $T$ ... ... @@ -647,7 +647,7 @@ relation. Its equivalence classes are the orbits of $G$ in $T$. \end{lemma} \begin{definition} \begin{definition}[Transitive group action] Let $G$ act on $T$. % \begin{enumerate} ... ... @@ -657,7 +657,7 @@ \end{enumerate} \end{definition} \begin{definition} \begin{definition}[Stabilizer] Let $G$ be a group and $T$ a non-empty set with a $G$-action. Let $t \in T$. We define the \textbf{stabilizer} of $t$ in $G$ to be % ... ... @@ -723,13 +723,13 @@ \section{The Symmetric Group} \begin{definition} \begin{definition}[Permutation] A \textbf{permuation} of a set $X$ is a bijection of $X$ to itself. We denote the family of all permutations of a set $X$ by $S_X$. If $X = \set{1, 2, \ldots, n}$ we write $S_n$. \end{definition} \begin{definition} \begin{definition}[Cycle] Let $i_1, i_2, \ldots, i_r$ be distinct integers in $X = \{1, 2, \ldots, n\}$. If $\alpha \in S_n$ fixes the other integers in $X$ (if any) and if % ... ... @@ -750,7 +750,7 @@ A 2-cycle is called \textbf{transposition}. \end{definition} \begin{definition} \begin{definition}[Disjoint permutations] Two permutations $\alpha, \beta \in S_n$ are \textbf{disjoint} if every $i$ moved by one is fixed by the other. \end{definition} ... ... @@ -783,7 +783,7 @@ \end{enumerate} \end{proposition} \begin{definition} \begin{definition}[Complete factorization] A \textbf{complete factorization} of a permutation $\alpha$ is a factorization of $\alpha$ into disjoint cycles that contains exactly one 1-cycle $(i)$ for every $i$ fixed by $\alpha$. ... ... @@ -795,7 +795,7 @@ in which the cycles occur. \end{theorem} \begin{definition} \begin{definition}[Cycle Structure] Two permutations have the \textbf{same cycle structure} if, for each $r \geq 1$, their complete factorizations have the same number of r-cycles. \end{definition} ... ... @@ -837,13 +837,13 @@ transpositions. \end{proposition} \begin{definition} \begin{definition}[Even permutation, odd permutation] A permutation $\alpha \in S_n$ is \textbf{even} if it is a product of an even number of transpositions; $\alpha$ is \textbf{odd} if it is not even. The \textbf{parity} of a permutations is whether is is even or odd. \end{definition} \begin{definition} \begin{definition}[Sign of permutation] If $\alpha \in S_n$ and $\alpha = \beta_1 \hdots \beta_t$ is a complete factorization into disjoint cycles, then \textbf{signum} $\alpha$ is defined by ... ... @@ -876,7 +876,7 @@ where $n = |G|$. \end{lemma} \begin{definition} \begin{definition}[Alternating group] The subset % \begin{align*} ... ... @@ -889,7 +889,7 @@ \section{Simple Groups} \begin{definition} \begin{definition}[Simple group] A group $G$ is called \textbf{simple} if $G \neq \set{1}$ and $G$ has no normal subgroups other than $\set{1}$ and $G$ itself. \end{definition} ... ...
 ... ... @@ -35,7 +35,7 @@ \end{enumerate} \end{proposition} \begin{definition} \begin{definition}[Extension Field, finite extension, degree of extension] If $K$ is a field containing $k$ as a subfield, then $K$ is called an \textbf{extension field} of $k$, denoted by $K/k$. An extension field $K/k$ is a \textbf{finite extension} if $K$ is a finite-dimensional vector space over ... ... @@ -43,7 +43,7 @@ of $K/k$. \end{definition} \begin{definition} \begin{definition}[Algebraic and transcendental elements] Let $K/k$ be an extension field. An element $\alpha \in K$ is \textbf{algebraic} over $k$ if there is some nonzero polynomial $f(x) \in k[x]$ having $\alpha$ as a root; otherwise, $\alpha$ is ... ... @@ -60,7 +60,7 @@ If $K/k$ is a finite extension field, then $K/k$ is an algebraic extension. \end{proposition} \begin{definition} \begin{definition}[Adjoint] If $K/k$ is an extension field and $\alpha \in K$, then $k(\alpha)$ is the intersection of all those subfields of $K$ containing $k$ and $\alpha$; we call $k(\alpha)$ the subfield of $K$ obtained by \textbf{adjoining} $\alpha$ ... ... @@ -91,7 +91,7 @@ \end{enumerate} \end{theorem} \begin{definition} \begin{definition}[Minimal Polynomial] If $K/k$ is an extension field and $\alpha \in K$ is algebraic over $k$, then the unique monic irreducible polynomial $p(x) \in k[x]$ having $\alpha$ as a root is called the \textbf{minimal polynomial} of $\alpha$ over $k$; it is ... ... @@ -115,7 +115,7 @@ with $f$ a product of linear polynomials in $K[x]$. \end{theorem} \begin{definition} \begin{definition}[Splitting field] If $K/k$ is an extension field and $f(X) \in k[x]$ is nonconstant, then $f$ \textbf{splits over} $K$ if $f(x) = a(x-z_1) \hdots (x-z_n)$, where $z_1, \ldots, z_n$ are in $K$ and $a \in k$. An extension field $E/k$ is called a ... ... @@ -140,7 +140,7 @@ elements. \end{theorem} \begin{definition} \begin{definition}[Primitive element] If $k$ is a finite field, a generator of the cyclic group $k^{\times}$ is called a \textbf{primitive element} of $k$. \end{definition} ... ... @@ -202,13 +202,13 @@ \section{Algebraic Closure} \begin{definition} \begin{definition}[Algebraially closed] A field $K$ is \textbf{algebraically closed} if any non-constant $f(x) \in K[x]$ splits in linear factors over $K$, or equivalently, any non-constant $f(x) \in K[x]$ has a root in $K$. \end{definition} \begin{definition} \begin{definition}[Algebraic cloure] Let $K$ be a field. An \textbf{algebraic closure} of $K$ is an algebraic extension $L/K$ such that $L$ is algebraically closed. \end{definition} ... ... @@ -222,8 +222,8 @@ $K$. \end{theorem} \begin{definition} If $F/k$ and $K/k$ are extension fields, then a k-\textbf{map} is a ring \begin{definition}[$k$-map] If $F/k$ and $K/k$ are extension fields, then a $k$-\textbf{map} is a ring homomorphism $\varphi: F \to K$ that fixes $k$ pointwise. \end{definition} ... ...
 ... ... @@ -9,7 +9,7 @@ \section{The Definition of the Module} \begin{definition} \begin{definition}[Left module, Right module] Let $R$ be a ring. A \textbf{left} $R$\textbf{-module} is an additive abelian group $M$ equipped with a \textbf{scalar multiplication} $R \times M \to M$, denoted by ... ... @@ -56,7 +56,7 @@ \end{remark} \end{definition} \begin{definition} \begin{definition}[$R$-homomorphism] If $R$ is a ring and $M$ and $N$ are both left $R$-modules (or both right $R$-modules), then a function $f \colon M \to N$ is an $R$-\textbf{homomorphism} (or $R$\textbf{-map}) if ... ... @@ -72,23 +72,23 @@ \section{Fundamental Concepts and Results} \begin{definition} Let $M$ be an $R$-module. We define a submodule $N$ of $M$ as a subgroup of the additive group $(M, +, 0)$ which is closed under the action of the elements of $R$; that is, if $a \in R$ and $y \in N$, then $ay \in N$. \begin{definition}[Submodule] Let $M$ be an $R$-module. We define a \textbf{submodule} $N$ of $M$ as a subgroup of the additive group $(M, +, 0)$ which is closed under the action of the elements of $R$; that is, if $a \in R$ and $y \in N$, then $ay \in N$. \end{definition} \begin{theorem} Let $M$ and $M'$ be two $R$-modules and $\phi \colon M \to M'$ an $R$-homomorphism. Then $\ker \phi \subset M$ and $\im \phi \subset M'$ are submodules of $M$ and $M'$, respectively. Furthermore, the first isomorphism theorem also holds for modules, i.e.\ theorem also holds for modules, i.e~ \begin{align*} M / \ker \phi \cong \im \phi \end{align*} \end{theorem} \begin{definition} \begin{definition}[Cyclic module] Let $M$ be an $R$-module. Then $M$ is said to be a \textbf{cyclic module} if $M = Rx = \{ ax \mid a \in R\}$ where $x \in M$. This is in direct analogy to a cyclic group, namely they are both generated by one element. ... ... @@ -99,7 +99,7 @@ we have the homomorphism $\mu_x \colon R \to Rx$ with $r \mapsto rx$. Evidently, this homomorphism is surjective and we have $M = Rx \cong R/\ker \mu_x$. This motivates the following definition. \begin{definition} \begin{definition}[Annihilator] Let $M = Rx$ be a cyclic $R$-module. Let $\mu_x \colon R \to Rx$ be defined by $r \mapsto rx$. Then we define the \textbf{annihilator} of $\mu_x$ % ... ... @@ -130,7 +130,7 @@ Evidently, this homomorphism is surjective and we have $M = Rx \cong R/\ker \section{Direct Sums of Modules} \begin{definition} \begin{definition}[Direct sum of modules] Let$M_1, \dotsc, M_n$be modules over the same ring$R$. Let$M$be the product set$M_ 1 \times \dotsm \times M_n$. In the special case of the free module we define addition, the zero element and multiplication by elements in ... ... @@ -193,7 +193,7 @@$K$. Then we have$M \cong D^{(n)}/K$. The core idea is now to examine$K$. elements. \end{theorem} \begin{definition} \begin{definition}[Equivalence of Matrices] Let$D$be a PID\@. Two matrices$A, B \in M_{m \times n}(D)$are said to be \textbf{equivalent} if there exist invertible matrices$P \in M_{n}(D), Q \in M_{n}(D)$, such that$B = PAQ$. ... ... @@ -228,7 +228,7 @@$K$. Then we have$M \cong D^{(n)}/K$. The core idea is now to examine$K$. This decomposition is in general not unique! \end{remark} \begin{definition} \begin{definition}[Torsion module] Let$M$be a finitely generated module over a PID and define the \textbf{torsion module} of$M$,$\tor M$, by % ... ... @@ -245,7 +245,7 @@$K$. Then we have$M \cong D^{(n)}/K$. The core idea is now to examine$K$. and a free module. \end{theorem} \begin{definition} \begin{definition}[$p$-component] If$p$is a prime, we define the \textbf{p-component}$M_p$of$Mby \begin{align*} M_p = \set{y \in M \mid p^k y = 0 \text{~for some~} k \in \N}. ... ... @@ -265,7 +265,7 @@K$. Then we have$M \cong D^{(n)}/K$. The core idea is now to examine$K$. = t$ and $\ann z_i = \ann w_i$ for all $1 \leq i \leq s$. \end{theorem} \begin{definition} \begin{definition}[Finitely generated] An abelian Group $G$ is called \textbf{finitely generated} if there exist finitely many elements $x_1, \dotsc, x_s \in G$ such that every $x$ in $G$ can be written in the form ... ...