Created the Abstract Algebra theorems and definitions cheat sheet
This commit is contained in:
@@ -0,0 +1,5 @@
|
||||
\section{Applications of Sylow Theorems}
|
||||
|
||||
\begin{theorem}[Cyclic Groups of Order $\mathbf{pq}$]
|
||||
If $G$ is a group of order $pq$, where $p$ and $q$ are primes, $p < q$, and $p$ does not divide $q - 1$, then $G$ is cyclic. In particular, $G$ is isomorphic to $\Z_{pq}$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\chapter{Sylow Theorems}
|
||||
\subimport{./}{conjugacy-classes.tex}
|
||||
\subimport{./}{the-class-equation.tex}
|
||||
\subimport{./}{the-sylow-theorems.tex}
|
||||
\subimport{./}{applications-of-sylow-theorems.tex}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Conjugacy Classes}
|
||||
|
||||
\begin{definition}[Conjugacy Class of $\mathbf{a}$]
|
||||
Let $a$ and $b$ be elements of a group $G$. We say that $a$ and $b$ are \textit{conjugate} in $G$ (and call $b$ the \textit{conjugate} of $a$) if $xax^{-1}=b$ for some $x$ in $G$. The \textit{conjugacy class of $a$} is the set $\cl(a) = \{xax^{-1}\ \vert\ x \in G\}$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Number of Conjugates of $\mathbf{a}$]
|
||||
Let $G$ be a finite group and let $a$ be an element of $G$. Then, $\abs{\cl(a)} = \abs{G:C(a)}$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[$\mathbf{\abs{\cl(a)}}$ Divides $\mathbf{\abs{G}}$]
|
||||
In a finite group, $\abs{\cl(a)}$ divides $\abs{G}$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,15 @@
|
||||
\section{The Class Equation}
|
||||
|
||||
\begin{corollary}[Class Equation]
|
||||
For any finite group $G$,
|
||||
\[ \abs{G} = \sum \abs{G:C(a)} \]
|
||||
where the sum runs over one element of $a$ from each conjugacy class of $G$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[$\mathbf{p}$-Groups Have Nontrivial Centers]
|
||||
Let $G$ be a nontrivial finite group whose order is a power of a prime $p$. Then $\Z(G)$ has more than one element.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Groups of Order $\mathbf{p^2}$ Are Abelian]
|
||||
If $\abs{G}=p^2$, where $p$ is prime, then $G$ is Abelian.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,29 @@
|
||||
\section{The Sylow Theorems}
|
||||
|
||||
\begin{theorem}[Existence of Subgroups of Prime-Power Order (Sylow's First Theorem, 1872)]
|
||||
Let $G$ be a finite group and let $p$ be a prime. If $p^k$ divides $\abs{G}$, then $G$ has at least one subgroup of order $p^k$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Sylow $\mathbf{p}$-Subgroup]
|
||||
Let $G$ be a finite group and let $p$ be a prime. If $p^k$ divides $\abs{G}$ and $p^{k+1}$ does not divide $\abs{G}$, then any subgroup of $G$ of order $p^k$ is called a \textit{Sylow $p$-subgroup of $G$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{corollary}[Cauchy's Theorem]
|
||||
Let $G$ be a finite group and let $p$ be a prime that divides the order of $G$. Then $G$ has an element of order $p$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{definition}[Conjugate Subgroups]
|
||||
Let $H$ and $K$ be subgroups of a group $G$. We say that $H$ and $K$ are \textit{conjugate} in $G$ if there is an element in $G$ such that $H = gKg^{-1}$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Sylow's Second Theorem]
|
||||
If $H$ is a subgroup of a finite group $G$ and $\abs{H}$ is a power of a prime $p$, then $H$ is contained in some Sylow $p$-subgroup of $G$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Sylow's Third Theorem]
|
||||
Let $p$ be a prime and let $G$ be a group of order $p^km$, where $p$ does not divide $m$. Then the number $n$ of Sylow $p$-subgroups of $G$ is equal to 1 modulo $p$ and divides $m$. Furthermore, any two Sylow $p$-subgroups of $G$ are conjugate.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[A Unique Sylow $\mathbf{p}$-Subgroup Is Normal]
|
||||
A Sylow $p$-subgroup of a finite group $G$ is a normal subgroup of $G$ if and only if it is the only Sylow $p$-subgroup of $G$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Finite Simple Groups}
|
||||
\subimport{./}{historical-background.tex}
|
||||
\subimport{./}{nonsimplicity-tests.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Historical Background}
|
||||
|
||||
\begin{definition}[Simple Group]
|
||||
A group is \textit{simple} if its only normal subgroups are the identity subgroup and the group itself.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,21 @@
|
||||
\section{Nonsimplicity Tests}
|
||||
|
||||
\begin{theorem}[Sylow Test for Nonsimplicity]
|
||||
Let $n$ be a positive integer that is not prime, and let $p$ be a prime divisor of $n$. If 1 is the only divisor of $n$ that is equal to 1 modulo $p$, then there does not exist a simple group of order $n$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[$\mathbf{2\cdot}$Odd Test]
|
||||
An integer of the form $2 \cdot n$, where $n$ is an odd number greater than 1, is not the order of a simple group.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Generalized Cayley Theorem]
|
||||
Let $G$ be a group and let $H$ be a subgroup of $G$. Let $S$ be the group of all permutations of the left cosets of $H$ in $G$. Then there is a homomorphism from $G$ into $S$ whose kernel lies in $H$ and contains every normal subgroup of $G$ that is contained in $H$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Index Theorem]
|
||||
If $G$ is a finite group and $H$ is a proper subgroup of $G$ such that $\abs{G}$ does not divide $\abs{G:H}!$, then $H$ contains a nontrivial normal subgroup of $G$. In particular, $G$ is not simple.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Embedding Theorem]
|
||||
If a finite non-Abelian simple group $G$ has a subgroup of index $n$, then $G$ is isomorphic to a subgroup of $A_n$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,7 @@
|
||||
\chapter{Generators and Relations}
|
||||
\subimport{./}{motivation.tex}
|
||||
\subimport{./}{definitions-and-notation.tex}
|
||||
\subimport{./}{free-group.tex}
|
||||
\subimport{./}{generators-and-relations.tex}
|
||||
\subimport{./}{classification-of-groups-of-order-up-to-15.tex}
|
||||
\subimport{./}{characterization-of-dihedral-groups.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Characterization of Dihedral Groups}
|
||||
|
||||
\begin{theorem}[Characterization of Dihedral Groups]
|
||||
Any group generated by a pair of elements of order 2 is dihedral.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Classification of Groups of Order Up to 15}
|
||||
|
||||
\begin{theorem}[Classification of Groups of Order 8 (Cayley, 1859)]
|
||||
Up to isomorphism, there are only five groups of order 8: $\Z_8$, $\Z_4 \oplus \Z_2$, $\Z_2 \oplus \Z_2 \oplus \Z_2$, $D_4$, and the quaternions.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,11 @@
|
||||
\section{Definitions and Notation}
|
||||
|
||||
\begin{remark}
|
||||
For any set $S=\{a,b,c,\dots\}$ of distinct symbols, we create a new set $S^{-1} = \{a^{-1},b^{-1},c^{-1},\dots\}$ by replacing each $x$ in $S$ by $x^{-1}$. Define the set $W(S)$ to be the collection of all formal finite strings of the form $x_1x_2\dots x_k$, where each $x_i \in S \cup S^{-1}$. The elements of $W(S)$ are called \textit{words from $S$}. We also permit the string with no elements to be in $W(S)$. this word is called the \textit{empty word} and is denoted by $e$.
|
||||
|
||||
We may define a binary operation on the set $W(S)$ by juxtaposition; that is, if $x_1x_2\dots x_k$ and $y_1y_2\dots y_t$ belong to $W(S)$, then so does $x_1x_2\dots x_ky_1y_2\dots y_t$. Observe that this operation is associative and the empty word is the identity. Also, notice that a word such as $aa^{-1}$ is not the identity, because we are treating the elements of $W(S)$ as formal symbols with no implied meaning.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Equivalence Classes of Words]
|
||||
For any pair of elements $u$ and $v$ of $W(S)$, we say that $u$ is related to $v$ if $v$ can be obtained from $u$ by a finite sequence of insertions or deletions of words of the form $xx^{-1}$ or $x^{-1}x$, where $x \in S$.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Free Group}
|
||||
|
||||
\begin{theorem}[Equivalence Classes Form a Group]
|
||||
Let $S$ be a set of distinct symbols. For any word $u$ in $W(S)$, let $\overline{u}$ denote the set of all words in $W(S)$ equivalent to $u$ (that is, $\overline{u}$ is the equivalence class containing $u$). Then the set of all equivalence classes of elements of $W(S)$ is a group under the operation $\overline{u}\cdot\overline{v} = \overline{uv}$. This group is called a \textit{free group on $S$}.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Universal Mapping Property]
|
||||
Every group is a homomorphic image of a free group.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Universal Factor Group Property]
|
||||
Every group is isomorphic to a factor group of a free group.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,20 @@
|
||||
\section{Generators and Relations}
|
||||
|
||||
\begin{definition}[Generators and Relations]
|
||||
Let $G$ be a group generated by some subset $A=\{a_1,a_2,\dots,a_n\}$ and let $F$ be the free group on $A$. Let $W = \{w_1,w_2,\dots,w_t\}$ be a subset of $F$ and let $N$ be the smallest normal subgruop of $F$ containing $W$. We say that $G$ is \textit{given by the generators $a_1,a_2,\dots,a_n$ and the relations $w_1=w_2=\dots=w_t=e$} if there is an isomorphism from $F/N$ onto $G$ that carries $a_iN$ to $a_i$.
|
||||
|
||||
\noindent The notation for this situation is
|
||||
\[ G = \lr{a_1,a_2,\dots,a_n\ \vert\ w_1=w_2=\dots=w_t=e} \]
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Dyck's Theorem (1882)]
|
||||
Let
|
||||
\[ G = \lr{a_1,a_2,\dots,a_n\ \vert\ w_1=w_2=\dots=w_t=e} \]
|
||||
and let
|
||||
\[ \overline{G}=\lr{a_1,a_2,\dots,a_n\ \vert\ w_1=w_2=\dots=w_t=w_{t+1}=\dots=w_{t+k}=e} \]
|
||||
Then $\overline{G}$ is a homomorphic image of $G$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Largest Group Satisfying Defining Relations]
|
||||
If $K$ is a group satisfying the defining relations of a finite group $G$ and $\abs{K} \geq \abs{G}$, then $K$ is isomorphic to $G$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Motivation}
|
||||
|
||||
\begin{remark}
|
||||
In this chapter, we present a convenient way to define a group with certain prescribed properties. Simply put, we begin with a set of elements that we want to generate the group, and a set of equations (called \textit{relations}) that specify the conditions that these generators are to satisfy. Among all such possible groups, we will select one that is as large as possible. This will uniquely determine the group up to isomorphism.
|
||||
\end{remark}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Symmetry Groups}
|
||||
\subimport{./}{isometries.tex}
|
||||
\subimport{./}{classification-of-finite-plane-symmetry-gruops.tex}
|
||||
\subimport{./}{classification-of-finite-groups-of-rotations-in-R3.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Classification of Finite Groups of Rotations in $\mathbf{\R^3}$}
|
||||
|
||||
\begin{theorem}[Finite Groups of Rotations in $\mathbf{\R^3}$]
|
||||
Up to isomorphism, the finite groups of rotations in $\R^3$ are $\Z_n$, $D_n$, $A_r$, $S_4$, and $A_5$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Classification of Finite Plane Symmetry Groups}
|
||||
|
||||
\begin{theorem}[Finite Symmetry Groups in the Plane]
|
||||
The only finite plane symmetry groups are $\Z_n$ and $D_n$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Isometries}
|
||||
|
||||
\begin{remark}
|
||||
It is convenient to begin our discussion with the definition of an isometry (from the Greek \textit{isometros}, meaning "equal measure") in $\R^n$.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Isometry]
|
||||
An \textit{isometry} of $n$-dimensional space $\R^n$ is a function from $\R^n$ onto $\R^n$ that preserves distance.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Symmetry Group of a Figure in $\mathbf{\R^n}$]
|
||||
Let $F$ be a set of points in $\R^n$. the \textit{symmetry group of $F$} in $\R^n$ is the set of all isometries of $\R^n$ that carry $F$ onto itself. The group operation is function composition.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Frieze Groups and Crystallographic Groups}
|
||||
\subimport{./}{the-frieze-groups.tex}
|
||||
\subimport{./}{the-crystallographic-groups.tex}
|
||||
\subimport{./}{identification-of-plane-periodic-patterns.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Identification of Plane Periodic Patterns}
|
||||
|
||||
\begin{remark}
|
||||
A \textit{lattice of points} of a pattern is a set of images of any particular point acted on by the translation group of the pattern. A \textit{lattice unit} of a pattern whose translation subgroup is generated by $u$ and $v$ is a parallelogram formed by a point of the pattern and its image under $u,v$, and $u + v$. A \textit{generating region} (or \textit{fundamental region}) of a periodic pattern is the smallest portion of the lattice unit whose images under the full symmetry of the group of the pattern cover the plane.
|
||||
\end{remark}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{The Crystallographic Groups}
|
||||
|
||||
\begin{remark}
|
||||
The seven frieze groups catalog all symmetry groups that leave a design invariant under all multiples of just one translation. However, there are 17 additional kinds of discrete plane symmetry groups that arise from infinitely repeating designs in a plane. these groups are the symmetry groups of plane patterns whose subgroups of translations are isomorphic to $\Z \oplus \Z$. Consequently, the patterns are invariant under linear combinations of two linearly independent translations. These 16 groups were first studied by the 19th-century crystallographers and often called the \textit{plane crystallographic groups}. Another term occasionally used for these groups is \textit{wallpaper groups}.
|
||||
\end{remark}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{The Frieze Groups}
|
||||
|
||||
\begin{remark}
|
||||
In this chapter, we discuss an interesting collection of infinite symmetry groups that arise from periodic designs in a plane. There are two types of such groups. The \textit{discrete frieze groups} are the plane symmetry groups of patterns whose subgroups of translations are isomorphic to $\Z$. These kinds of designs are the ones used for decorative strips and for patterns on jewelry. In mathematics, familiar examples include the graphs of $y=\sin(x)$, $y=\tan(x)$, $y=\abs{\sin(x)}$, and $\abs{y} = \sin(x)$. After we analyze the discrete frieze groups, we examine the discrete symmetry groups of plane patterns whose subgroups of translations are isomorphic to $\Z \oplus \Z$.
|
||||
\end{remark}
|
||||
@@ -0,0 +1,10 @@
|
||||
\section{Burnside's Theorem}
|
||||
|
||||
\begin{definition}[Elements Fixed by $\mathbf{\phi}$]
|
||||
For any group $G$ of permutations on a set $S$ and any $\phi$ in $G$, we let $\fix(\phi) = \{i \in S\ \vert\ \phi(i)=i\}$. This set is called the \textit{elements fixed by $\phi$} (or more simply, "fix of $\phi$").
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Burnside's Theorem]
|
||||
If $G$ is a finite group of permutations on a set $S$, then the number of orbits of elements of $S$ under $G$ is
|
||||
\[ \frac{1}{\abs{G}}\sum_{\phi \in G}\abs{\fix(\phi)} \]
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Symmetry and Counting}
|
||||
\subimport{./}{motivation.tex}
|
||||
\subimport{./}{burnsides-theorem.tex}
|
||||
\subimport{./}{group-action.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Group Action}
|
||||
|
||||
\begin{remark}
|
||||
Our informal approach to counting the number of objects that are considered nonequivalent can be made formal as follows. If $G$ is a group and $S$ is a set of objects, we say that $G$ \textit{acts on} $S$ if there is a homomorphism $\gamma$ from $G$ to sym$(S)$, the group of all permutations on $S$. (The hommomorphism is sometimes called the \textit{group action}.) For convenience, we denote the image of $g$ under $\gamma$ as $\gamma_g$. Then two objects $x$ and $y$ in $S$ are viewed as equivalent under the action of $G$ if and only if $\gamma_g(x) = y$ for some $g$ in $G$. Notice that when $\gamma$ is one-to-one, the elements of $G$ may be regarded as permutations on $S$. On the other hand, when $\gamma$ is not one-to-one, the elements of $G$ may still be regarded as permutations on $S$, but there are distinct elements $g$ and $h$ in $G$ such that $\gamma_g$ and $\gamma_h$ induce the same permutation on $S$ [that is, $\gamma_g(x) = \gamma_h(x)$ for all $x$ in $S$]. Thus, a group acting on a set is a natural generalization of the permutation group concept.
|
||||
\end{remark}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Motivation}
|
||||
|
||||
\begin{remark}
|
||||
In general, we say that two designs (arrangements of beads) $A$ and $B$ are \textit{equivalent under a group $G$} of permutations of the arrangements if there is an element $\phi$ in $G$ such that $\phi(A) = B$. That is, two designs are equivalent under $G$ if they are in the same orbit of $G$. It follows, then, that the number of nonequivalent designs under $G$ is simply the number of orbits of designs under $G$. (The set being permuted is the set of all possible designs or arrangements.)
|
||||
\end{remark}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Cayley Digraphs of Groups}
|
||||
\subimport{./}{the-cayley-digraph-of-a-group.tex}
|
||||
\subimport{./}{hamiltonian-circuits-and-paths.tex}
|
||||
@@ -0,0 +1,20 @@
|
||||
\section{Hamiltonian Circuits and Paths}
|
||||
|
||||
\begin{remark}
|
||||
Obviously, this idea can be applied to any digraph; that is, one starts at some vertex and attempts to traverse the digraph by moving along arcs in such a way that each vertex is visited exactly once before returning to the starting vertex. (To go from $x$ to $y$, there must be an arc from $x$ to $y$.) Such a sequence of arcs is called a \textit{Hamiltonian circuit} in the digraph. A sequence of arcs that passes through each vertex exactly once without returning to the starting point is called a \textit{Hamiltonian path}. In the rest of this chapter, we concern ourselves with the existence of Hamiltonian circuits and paths in Cayley digraphs.
|
||||
\end{remark}
|
||||
|
||||
\begin{theorem}[A Necessary Condition]
|
||||
Cay$(\{(1,0),(0,1)\}:\Z_m \oplus \Z_n)$ does not have a Hamiltonian circuit when $m$ and $n$ are relatively prime and greater than 1.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[A Sufficient Condition]
|
||||
Cay$(\{(1,0),(0,1)\}:\Z_m \oplus \Z_n)$ has a Hamiltonian circuit when $n$ divides $m$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Abelian Groups Have Hamiltonian Paths]
|
||||
Let $G$ be a finite Abelian group, and let $S$ be any (nonempty*) generating set for $G$. Then Cay$(S:G)$ has a Hamiltonian path.\\
|
||||
|
||||
|
||||
\noindent *If $S$ is the empty set, it is customary to define $\lr{S}$ as the identity group. We prefer to ignore this trivial case.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{The Cayley Digraph of a Group}
|
||||
|
||||
\begin{definition}[Cayley Digraph of a Group]
|
||||
Let $G$ be a finite group and let $S$ be a set of generators for $G$. We define a digraph Cay$(S:G)$, called the \textit{Cayley digraph of $G$ with generating set $S$}, as follows.
|
||||
\begin{enumerate}
|
||||
\item Each element of $G$ is a vertex of Cay$(S:G)$.
|
||||
\item For $x$ and $y$ in $G$, there is an arc from $x$ to $y$ if and only if $xs=y$ for some $s \in S$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Introduction to Algebraic Coding Theory}
|
||||
\subimport{./}{linear-codes.tex}
|
||||
\subimport{./}{parity-check-matrix-decoding.tex}
|
||||
\subimport{./}{coset-decoding.tex}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Coset Decoding}
|
||||
|
||||
\begin{theorem}[Coset Decoding Is Nearest-Neighbor Decoding]
|
||||
In coset decoding, a received word $w$ is decoded as a code word $c$ such that $d(w,c)$ is a minimum.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Syndrome]
|
||||
If an $(n,k)$ linear code over $\F$ has parity-check matrix $H$, then, for any vector $u$ in $\F^n$, the vector $uH$ is called the \textit{syndrome} of $u$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Same Coset-Same Syndrome]
|
||||
Let $C$ be an $(n,k)$ linear code over $\F$ with a parity-check matrix $H$. Then, two vectors of $\F^n$ are in the same coset of $C$ if and only if they have the same syndrome.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,19 @@
|
||||
\section{Linear Codes}
|
||||
|
||||
\begin{definition}[Linear Code]
|
||||
An $(n,k)$ \textit{linear code} of a finite field $\F$ is a $k$-dimensional subspace $V$ of the vector space
|
||||
\[ \F^n = \underbrace{\F \oplus \F \oplus \dots \oplus \F}_\text{$n$ copies} \]
|
||||
over $\F$. The members of $V$ are called the \textit{code words}. When $\F$ is $\Z_2$, the code is called \textit{binary}.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Hamming Distance, Hamming Weight]
|
||||
The \textit{Hamming distance} between two vectors in $\F^n$ is the number of components in which they differ. The \textit{Hamming weight} of a vector is the number of nonzero components of the vector. The \textit{Hamming weight} of a linear code is the minimum weight of any nonzero vector in the code.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Properties of Hamming Distance and Hamming Weight]
|
||||
For any vectors $u$, $v$ and $w$, $d(u,v) \leq d(u,w) + d(w,v)$ and $d(u,v) = \text{wt}(u-v)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Correcting Capability of a Linear Code]
|
||||
If the Hamming weight of a linear code is at least $2t + 1$, then the code can correct any $t$ or fewer errors. Alternatively, the same code can detect any $2t$ or fewer errors.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{Parity-Check Matrix Decoding}
|
||||
|
||||
\begin{lemma}[Orthogonality Relation]
|
||||
Let $C$ be a systematic $(n,k)$ linear code over $\F$ with a standard generator matrix $G$ and parity-check matrix $H$. Then, for any vector $v$ in $\F^n$, we have $vH=0$ (the zero vector) if and only if $v$ belongs to $C$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[Parity-Check Matrix Decoding]
|
||||
Parity-check matrix decoding will correct any single error if and only if the rows of the parity-check matrix are nonzero and no one row is a scalar multiple of any other row.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{An Introduction to Galois Theory}
|
||||
\subimport{./}{fundamental-theorem-of-galois-theory.tex}
|
||||
\subimport{./}{solvability-of-polynomials-by-radicals.tex}
|
||||
@@ -0,0 +1,17 @@
|
||||
\section{Fundamental Theorem of Galois Theory}
|
||||
|
||||
\begin{definition}[Automorphism, Galois Group, Fixed Field of $\mathbf{H}$]
|
||||
Let $\E$ be an extension field of the field $\F$. An \textit{automorphism of $\E$} is a ring isomorphism from $\E$ onto $\E$. The \textit{Galois group} of $\E$ over $\F$, $\gal(\E/\F)$, is the set of all automorphisms of $\E$ that take every element of $\F$ to itself. If $H$ is a subgroup of $\gal(\E/\F)$, then set
|
||||
\[ \E_H = \{x \in \E\ \vert\ \phi(x) = x,\ \forall\ \phi \in H\} \]
|
||||
is called the \textit{fixed field of $H$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Fundamental Theorem of Galois Theory]
|
||||
Let $\F$ be a field of characteristic 0 or a finite field. If $\E$ is the splitting field over $\F$ for some polynomial in $\F[x]$, then the mapping from the set of subfields of $\E$ containing $\F$ to the set of subgroups of $\gal(\E/\F)$ given by $\K \to \gal(\E/\F)$ is a one-to-one correspondence. Furthermore, for any subfield $\K$ of $\E$ containing $\F$,
|
||||
\begin{enumerate}
|
||||
\item $[\E:\K] = \abs{\gal(\E/\K)}$ and $[\K:\F] = \abs{\gal(\E/\F)} / \abs{\gal(\E/\K)}$. [The index of $\gal(\E/\K)$ in $\gal(\E/\F)$ equals the degree of $\K$ over $\F$.]
|
||||
\item If $\K$ is the splitting field of some polynomial in $\F[x]$, then $\gal(\E/\K)$ is a normal subgroup of $\gal(\E/\F)$ and $\gal(\K/\F)$ is isomorphic to $\gal(\E/\F)/\gal(\E/\K)$.
|
||||
\item $\K = \E_{\gal(\E/\K)}$. [The fixed field of $\gal(\E/\K)$ is $\K$.]
|
||||
\item If $H$ is a subgroup of $\gal(\E/\F)$, then $H=\gal(\E/\E_H)$. [The automorphism group of $\E$ fixing $\E_H$ is $H$.]
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,27 @@
|
||||
\section{Solvability of Polynomials by Radicals}
|
||||
|
||||
\begin{definition}[Solvable by Radicals]
|
||||
Let $\F$ be a field, and let $f(x) \in \F[x]$. We say that $f(x)$ is \textit{solvable by radicals over $\F$} if $f(x)$ splits in some extension $\F(a_1,a_2,\dots,a_n)$ of $\F$ and there exist positive integers $k_1,\dots,k_n$ such that $a_1^{k_1} \in \F$ and $a_i^{k_i} \in \F(a_1,\dots,a_{i-1})$ for $i=2,\dots,n$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Solvable Group]
|
||||
We say that a group $G$ is \textit{solvable} if $G$ has a series of subgroups
|
||||
\[ \{e\} = H_0 \subset H_1 \subset H_2 \subset \dots \subset H_k = G \]
|
||||
where, for each $0 \leq i <k$, $H_i$ is normal in $H_{i + 1}$ and $H_{i + 1}/H_i$ is Abelian.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Splitting Field of $\mathbf{x^n - a}$]
|
||||
Let $\F$ be a field of characteristic 0 and let $a \in \F$. If $\E$ is the splitting field of $x^n-a$ over $\F$, then the Galois group $\gal(\E/\F)$ is solvable.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Factor Group of a Solvable Group is Solvable]
|
||||
A factor group of a solvable group is solvable.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[$\mathbf{N}$ and $\mathbf{G/N}$ Implies $\mathbf{G}$ Is Solvable]
|
||||
Let $N$ be a normal subgroup of a group $G$. If both $N$ and $G/N$ are solvable, then $G$ is solvable.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[(Galois) Solvable by Radicals Implies Solvable Group]
|
||||
Let $\F$ be a field of characteristic 0 and let $f(x) \in \F[x]$. Suppose the $f(x)$ splits in $\F(a_1,a_2,\dots,a_t)$, where $a_1^{n_1} \in \F$ and $a_i^{n_i} \in \F(a_1,\dots,a_{i-1})$ for $i=2,\dots,t$. Let $\E$ be the splitting field for $f(x)$ over $\F$ in $\F(a_1,a_2,\dots,a_t)$. Then the Galois group $\gal(\E/\F)$ is solvable.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Cyclotomic Extensions}
|
||||
\subimport{./}{cyclotomic-polynomials.tex}
|
||||
\subimport{./}{the-constructible-regular-n-gons.tex}
|
||||
@@ -0,0 +1,27 @@
|
||||
\section{Cyclotomic Polynomials}
|
||||
|
||||
\begin{remark}
|
||||
Recall from Example 2 in Chapter 16 that the complex zeros of $x^n-1$ are 1, $\omega = \cos(2\pi/n) = i\sin(2\pi/n)$, $\omega^2, \omega^3,\dots,\omega^{n-1}$. Thus, the splitting field of $x^n-1$ over $\Q$ is $\Q(\omega)$. This field is called the \textit{$n$th cyclotomic extension of $\Q$}, and the irreducible factors of $x^n-1$ over $\Q$ are called the \textit{cyclotomic polynomials}.
|
||||
|
||||
Since $\omega=\cos(2\pi/n) + i\sin(2\pi/n)$ generates a cyclic group of order $n$ under multiplication, we know from Corollary 3 of Theorem 4.2 that the generators of $\lr{\omega}$ are the elements of the form $\omega^k$, where $1 \leq k \leq n$ and $\gcd(n,k) = 1$. These generators are called the \textit{primitive $n$th roots of unity}. Recalling that we use $\phi(n)$ to denote the number of positive integers less than or equal to $n$ and relatively prime to $n$, we see that for each positive integer $n$ there are precisely $\phi(n)$ primitive $n$th roots of unity. The polynomials whose zeros are the $\phi(n)$ primitive $n$th roots of unity have a special name.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}
|
||||
For any positive integer $n$, let $\omega_1,\omega_2,\dots,\omega_{\phi(n)}$ denote the primitive $n$th roots of unity. the \textit{$n$th cyclotomic polynomial over $\Q$} is the polynomial $\Phi_n(x) = (x-\omega_1)(x-\omega_2)\dots(x-\omega_{\phi(n)})$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
For every positive integer $n$, $x^n-1 = \Pi_{d\vert n}\Phi_d(x)$, where the product runs over all positive divisors $d$ of $n$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
For every positive integer $n$, $\Phi_n(x)$ has integer coefficients.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[(Gauss)]
|
||||
The cyclotomic polynomials $\Phi_n(s)$ are irreducible over $\Z$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
Let $\omega$ be a primitive $n$th root of unity. Then $\gal(\Q(\omega)/\Q) \approx U(n)$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{The Constructible Regular $\mathbf{n}$-gons}
|
||||
|
||||
\begin{lemma}
|
||||
Let $n$ be a positive integer and let $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$. Then $\Q(\cos(2\pi/n)) \subseteq \Q(\omega)$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[(Gauss, 1796)]
|
||||
It is possible to construct the regular $n$-gon with a straightedge and compass if and only if $n$ has the form $2^kp_1p_2\dots p_t$, where $k \geq 0$ and the $p_i$'s are distinct primes of the form $2^m + 1$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,11 @@
|
||||
\part{Special Topics}
|
||||
\subimport{chapters/chapter-24/}{chapter-24.tex}
|
||||
\subimport{chapters/chapter-25/}{chapter-25.tex}
|
||||
\subimport{chapters/chapter-26/}{chapter-26.tex}
|
||||
\subimport{chapters/chapter-27/}{chapter-27.tex}
|
||||
\subimport{chapters/chapter-28/}{chapter-28.tex}
|
||||
\subimport{chapters/chapter-29/}{chapter-29.tex}
|
||||
\subimport{chapters/chapter-30/}{chapter-30.tex}
|
||||
\subimport{chapters/chapter-31/}{chapter-31.tex}
|
||||
\subimport{chapters/chapter-32/}{chapter-32.tex}
|
||||
\subimport{chapters/chapter-33/}{chapter-33.tex}
|
||||
Reference in New Issue
Block a user