algebra2 notes.md -> latex

README.md

@@ -31,7 +31,7 @@ I hereby present a treasure trove of resources covering various topics in mathem
     - [Euclidean Algorithm](https://github.com/Rafisto/uni/blob/master/algebra/programy/zadanie39.py) - $gcd(a, b)$
     - [Extended Euclidean Algorithm](https://github.com/Rafisto/uni/blob/master/algebra/programy/zadanie40.py) - $ax + by = \gcd(a, b)$
     - [An extravagend function analysis](https://github.com/Rafisto/uni/blob/master/algebra/programy/zadanie49.py) - analysis of $f(n)=\left|\{(a,b) \mid 1 \leq a,b \leq n, \gcd(a,b)=1\}\right| \cdot n^{-2}$
-    - [RSA Algorithm Exponent Brute-Forcer](https://github.com/Rafisto/uni/blob/master/algebra2/rsa34.py) - Crack RSA exponent
+    - [RSA Algorithm Exponent Brute-Forcer](https://github.com/Rafisto/uni/blob/master/algebra2/programs/rsa34.py) - Crack RSA exponent
     - [XOR Cipher](https://github.com/Rafisto/uni/blob/master/logika/programy/xorcipher.py) - One time pad encryption, decryption, key generation and key swap.
 
 Copyright 2024 © Rafał Włodarczyk

algebra2/algebra2.tex

@@ -0,0 +1,234 @@
+\documentclass{article}
+
+\usepackage[english]{babel}
+\usepackage[utf8]{inputenc}
+\usepackage{polski}
+\usepackage[T1]{fontenc}
+
+\usepackage[margin=1.5in]{geometry} 
+
+\usepackage{color} 
+\usepackage{amsmath}
+\usepackage{amsfonts}                                                                   
+\usepackage{graphicx}                                                             
+\usepackage{booktabs}
+\usepackage{amsthm}
+\usepackage{pdfpages}
+\usepackage{wrapfig}
+\usepackage{hyperref}
+\usepackage{etoolbox}
+
+\makeatletter
+\newenvironment{definition}[1]{%
+    \trivlist
+    \item[\hskip\labelsep\textbf{Definition. #1.}]
+    \ignorespaces
+}{%
+    \endtrivlist
+}
+\newenvironment{fact}[1]{%
+    \trivlist
+    \item[\hskip\labelsep\textbf{Fact. #1.}]
+    \ignorespaces
+}{%
+    \endtrivlist
+}
+\newenvironment{theorem}[1]{%
+    \trivlist
+    \item[\hskip\labelsep\textbf{Theorem. #1.}]
+    \ignorespaces
+}{%
+    \endtrivlist
+}
+\newenvironment{information}[1]{%
+    \trivlist
+    \item[\hskip\labelsep\textbf{Information. #1.}]
+    \ignorespaces
+}{%
+    \endtrivlist
+}
+\newenvironment{identities}[1]{%
+    \trivlist
+    \item[\hskip\labelsep\textbf{Identities. #1.}]
+    \ignorespaces
+}{%
+    \endtrivlist
+}
+\makeatother
+
+\title{Abstract algebra and coding}  
+\author{Rafał Włodarczyk}
+\date{INA 2, 2024}
+
+\begin{document}
+
+\maketitle
+
+\tableofcontents
+
+\section{Definitions}
+
+\subsection{Group}
+
+A group is a set \( G \) along with an operation \( \cdot \) satisfying the following axioms:
+\begin{enumerate}
+    \item \textbf{Operation is defined}: \( \forall a, b \in G: a \cdot b \in G \)
+    \item \textbf{Operation is associative}: \( \forall a, b, c \in G: a \cdot (b \cdot c) = (a \cdot b) \cdot c \)
+    \item \textbf{Identity element exists}: \( \exists e \in G: \forall a \in G: a \cdot e = e \cdot a = a \)
+    \item \textbf{Inverse element exists}: \( \forall a \in G: \exists a^{-1} \in G: a \cdot a^{-1} = a^{-1} \cdot a = e \)
+\end{enumerate}
+
+\subsection{Subgroup}
+
+A subset \( H \) of a group \( G \) is a subgroup if:
+\begin{enumerate}
+    \item \( H \) is closed under the operation: \( \forall a, b \in H: a \cdot b \in H \)
+    \item \( H \) is closed under inverses: \( \forall a \in H: a^{-1} \in H \)
+    \item \( H \) contains the identity element: \( e \in H \)
+    \item \( H \) is closed under associativity: \( \forall a, b \in H: a \cdot b \in H \)
+\end{enumerate}
+
+It suffices to check closure under operation and inverses for \( H \).
+
+\subsection{Normal Subgroup}
+
+A subgroup \( H \) of a group \( G \) is normal in \( G \) if:
+\begin{enumerate}
+    \item \( H \) is a subgroup of \( G \):
+    \begin{itemize}
+        \item \( H \) is closed under the operation: \( \forall a, b \in H: a \cdot b \in H \)
+        \item \( H \) has an inverse element: \( \forall a \in H: a^{-1} \in H \)
+    \end{itemize}
+    \item \( H \) is closed under conjugation: \( \forall a \in G: aHa^{-1} = H \)
+\end{enumerate}
+
+\subsection{Group Homomorphism}
+
+A group homomorphism is a function \( f: G \to H \) satisfying:
+\[ f(a \cdot b) = f(a) \cdot f(b) \]
+
+\subsection{Kernel of a Homomorphism}
+
+The kernel of a homomorphism \( f \) is the set of elements in \( G \) mapped to the identity element in \( H \):
+\[ \ker f = \{ a \in G : f(a) = e_H \} \]
+
+\subsection{Image of a Homomorphism}
+
+The image of a homomorphism is the set of elements in \( H \) obtained by applying \( f \) to elements in \( G \):
+\[ \text{Im} f = \{ f(a) \in H : a \in G \} \]
+
+\subsection{Order of an Element in a Group}
+
+The order of an element \( a \) in a group \( G \) is defined as:
+\[ \text{ord}(a) = \min\{ n \in \mathbb{N} : a^n = e \} \]
+
+If no such \( n \) exists, \( a \) has infinite order.
+
+\subsection{Generator of a Group}
+
+An element \( a \) in a group \( G \) is a generator if:
+\[ \forall b \in G: \exists n \in \mathbb{Z}: b = a^n \]
+
+\subsection{Coset of a Group}
+
+The coset of a subgroup \( H \) in a group \( G \) is defined as:
+\begin{itemize}
+    \item Left coset: \( aH = \{ a \cdot h : h \in H \} \)
+    \item Right coset: \( Ha = \{ h \cdot a : h \in H \} \)
+    \item Double coset: \( aH = Ha \)
+\end{itemize}
+
+\subsection{Cyclic Group}
+
+A group \( G \) is cyclic if there exists an element \( a \in G \) such that:
+\[ G = \{ a^n : n \in \mathbb{Z} \} \]
+
+Thus, \( G \) is generated by one element \( a \).
+
+\subsection{Dihedral Group}
+
+The dihedral group \( D_n \) is the group of symmetries of a regular \( n \)-gon.
+
+\subsection{Quotient Group}
+
+The quotient group \( G/H \) of a group \( G \) by a normal subgroup \( H \) is the set of cosets of \( H \) in \( G \) with the operation:
+\[ (aH) \cdot (bH) = (a \cdot b)H \]
+
+\subsection{Ring}
+
+A ring \( R \) is a set with two operations \( + \) and \( \cdot \) satisfying:
+\begin{enumerate}
+    \item \( (R, +) \) is an abelian group
+    \item \( \cdot \) is associative: \( \forall a, b, c \in R: a \cdot (b \cdot c) = (a \cdot b) \cdot c \)
+    \item Distributivity of multiplication over addition:
+    \[ \forall a, b, c \in R: a \cdot (b + c) = a \cdot b + a \cdot c \quad \text{and} \quad (a + b) \cdot c = a \cdot c + b \cdot c \]
+\end{enumerate}
+
+\subsection{Invertible Element in a Ring}
+
+An element \( a \) in a ring \( R \) is invertible if there exists an element \( b \in R \) such that:
+\[ a \cdot b = b \cdot a = 1 \]
+
+The set of invertible elements is denoted as \( R^* = \{ a \in R : a \text{ is invertible} \} \)
+
+\subsection{Subring}
+
+A subring of a ring \( R \) is a subset \( S \subseteq R \) with operations \( + \) and \( \cdot \) such that:
+\begin{enumerate}
+    \item \( S \) is closed under addition: \( \forall a, b \in S: a + b \in S \)
+    \item \( S \) is closed under multiplication: \( \forall a, b \in S: a \cdot b \in S \)
+\end{enumerate}
+
+\subsection{Ring Homomorphism}
+
+A ring homomorphism is a function \( f: R \to S \) satisfying:
+\begin{enumerate}
+    \item \( f \) is a group homomorphism: \( f(a + b) = f(a) + f(b) \)
+    \item \( f \) is a ring homomorphism: \( f(a \cdot b) = f(a) \cdot f(b) \)
+\end{enumerate}
+
+\subsection{Ideal}
+
+An ideal of a ring \( R \) is a subset \( I \subseteq R \) satisfying:
+\begin{enumerate}
+    \item \( (I, +) \) is a subgroup of the abelian group \( (R, +) \)
+    \item \( I \) is closed under multiplication: \( \forall a, b \in I: a \cdot b \in I \)
+    \item \( I \) is closed under addition: \( \forall a, b \in I: a + b \in I \)
+    \item \( I \) is closed under multiplication by ring elements: \( \forall a \in I, r \in R: a \cdot r \in I \) and \( r \cdot a \in I \)
+\end{enumerate}
+
+\subsection{Principal Ideal}
+
+A principal ideal generated by an element \( a \in R \) is the set:
+\[ \langle a \rangle = \{ a \cdot r : r \in R \} \]
+
+\subsection{Quotient Ring}
+
+The quotient ring \( R/I \) of a ring \( R \) by an ideal \( I \) is the set of cosets of \( I \) in \( R \) with operations:
+\[ (a + I) + (b + I) = (a + b) + I \]
+\[ (a + I) \cdot (b + I) = (a \cdot b) + I \]
+
+\section{Theorems}
+
+\subsection{Lagrange's Theorem}
+
+If \( G \) is a finite group and \( H \) is a subgroup of \( G \), then the order of \( H \) divides the order of \( G \):
+\[ |G| = |H| \cdot [G : H] \]
+Or equivalently:
+\[ |H| \mid |G| \]
+
+\subsection{Chinese Remainder Theorem}
+
+If \( m_1, m_2, \ldots, m_n \) are pairwise coprime integers, then the system of congruences:
+\[ \begin{cases} x \equiv a_1 \pmod{m_1} \\ x \equiv a_2 \pmod{m_2} \\ \vdots \\ x \equiv a_n \pmod{m_n} \end{cases} \]
+has exactly one solution modulo \( m_1 \cdot m_2 \cdot \ldots \cdot m_n \).
+
+\subsection{Euler's Theorem}
+
+For any integer \( a \) coprime to \( n \), it holds that:
+\[ a^{\varphi(n)} \equiv 1 \pmod{n} \]
+
+\end{document}
+
+
+\end{document}