Abstract Algebra

$ % macros % MathJax \newcommand{\bigtimes}{\mathop{\vcenter{\hbox{$\Huge\times\normalsize$}}}} % prefix with \! and postfix with \!\! % sized grouping symbols \renewcommand {\aa} [1] {\left\langle {#1} \right\rangle} % <> angle brackets \newcommand {\bb} [1] {\left[ {#1} \right]} % [] brackets \newcommand {\cc} [1] {\left\{ {#1} \right\}} % {} curly braces \newcommand {\mm} [1] {\lVert {#1} \rVert} % || || double norm bars \newcommand {\nn} [1] {\lvert {#1} \rvert} % || norm bars \newcommand {\pp} [1] {\left( {#1} \right)} % () parentheses % unit \newcommand {\unit} [1] {\bb{\mathrm{#1}}} % measurement unit % math \newcommand {\fn} [1] {\mathrm{#1}} % function name % sets \newcommand {\setZ} {\mathbb{Z}} \newcommand {\setQ} {\mathbb{Q}} \newcommand {\setR} {\mathbb{R}} \newcommand {\setC} {\mathbb{C}} % arithmetic \newcommand {\q} [2] {\frac{#1}{#2}} % quotient, because fuck \frac % trig \newcommand {\acos} {\mathrm{acos}} % \mathrm{???}^{-1} is misleading \newcommand {\asin} {\mathrm{asin}} \newcommand {\atan} {\mathrm{atan}} \newcommand {\atantwo} {\mathrm{atan2}} % at angle = atan2(y, x) \newcommand {\asec} {\mathrm{asec}} \newcommand {\acsc} {\mathrm{acsc}} \newcommand {\acot} {\mathrm{acot}} % complex numbers \newcommand {\z} [1] {\tilde{#1}} \newcommand {\conj} [1] {{#1}^\ast} \renewcommand {\Re} {\mathfrak{Re}} % real part \renewcommand {\Im} {\mathrm{I}\mathfrak{m}} % imaginary part % quaternions \newcommand {\quat} [1] {\tilde{\mathbf{#1}}} % quaternion symbol \newcommand {\uquat} [1] {\check{\mathbf{#1}}} % versor symbol \newcommand {\gquat} [1] {\tilde{\boldsymbol{#1}}} % greek quaternion symbol \newcommand {\guquat}[1] {\check{\boldsymbol{#1}}} % greek versor symbol % vectors \renewcommand {\vec} [1] {\mathbf{#1}} % vector symbol \newcommand {\uvec} [1] {\hat{\mathbf{#1}}} % unit vector symbol \newcommand {\gvec} [1] {\boldsymbol{#1}} % greek vector symbol \newcommand {\guvec} [1] {\hat{\boldsymbol{#1}}} % greek unit vector symbol % special math vectors \renewcommand {\r} {\vec{r}} % r vector [m] \newcommand {\R} {\vec{R}} % R = r - r' difference vector [m] \newcommand {\ur} {\uvec{r}} % r unit vector [#] \newcommand {\uR} {\uvec{R}} % R unit vector [#] \newcommand {\ux} {\uvec{x}} % x unit vector [#] \newcommand {\uy} {\uvec{y}} % y unit vector [#] \newcommand {\uz} {\uvec{z}} % z unit vector [#] \newcommand {\urho} {\guvec{\rho}} % rho unit vector [#] \newcommand {\utheta} {\guvec{\theta}} % theta unit vector [#] \newcommand {\uphi} {\guvec{\phi}} % phi unit vector [#] \newcommand {\un} {\uvec{n}} % unit normal vector [#] % vector operations \newcommand {\inner} [2] {\left\langle {#1} , {#2} \right\rangle} % <,> \newcommand {\outer} [2] {{#1} \otimes {#2}} \newcommand {\norm} [1] {\mm{#1}} \renewcommand {\dot} {\cdot} % dot product \newcommand {\cross} {\times} % cross product % matrices \newcommand {\mat} [1] {\mathbf{#1}} % matrix symbol \newcommand {\gmat} [1] {\boldsymbol{#1}} % greek matrix symbol % ordinary derivatives \newcommand {\od} [2] {\q{d #1}{d #2}} % ordinary derivative \newcommand {\odn} [3] {\q{d^{#3}{#1}}{d{#2}^{#3}}} % nth order od \newcommand {\odt} [1] {\q{d{#1}}{dt}} % time od % partial derivatives \newcommand {\de} {\partial} % partial symbol \newcommand {\pd} [2] {\q{\de{#1}}{\de{#2}}} % partial derivative \newcommand {\pdn} [3] {\q{\de^{#3}{#1}}{\de{#2}^{#3}}} % nth order pd \newcommand {\pdd} [3] {\q{\de^2{#1}}{\de{#2}\de{#3}}} % 2nd order mixed pd \newcommand {\pdt} [1] {\q{\de{#1}}{\de{t}}} % time pd % vector derivatives \newcommand {\del} {\nabla} % del \newcommand {\grad} {\del} % gradient \renewcommand {\div} {\del\dot} % divergence \newcommand {\curl} {\del\cross} % curl % differential vectors \newcommand {\dL} {d\vec{L}} % differential vector length [m] \newcommand {\dS} {d\vec{S}} % differential vector surface area [m^2] % special functions \newcommand {\Hn} [2] {H^{(#1)}_{#2}} % nth order Hankel function \newcommand {\hn} [2] {H^{(#1)}_{#2}} % nth order spherical Hankel function % transforms \newcommand {\FT} {\mathcal{F}} % fourier transform \newcommand {\IFT} {\FT^{-1}} % inverse fourier transform % signal processing \newcommand {\conv} [2] {{#1}\ast{#2}} % convolution \newcommand {\corr} [2] {{#1}\star{#2}} % correlation % abstract algebra \newcommand {\lie} [1] {\mathfrak{#1}} % lie algebra % other \renewcommand {\d} {\delta} % optimization %\DeclareMathOperator* {\argmin} {arg\,min} %\DeclareMathOperator* {\argmax} {arg\,max} \newcommand {\argmin} {\fn{arg\,min}} \newcommand {\argmax} {\fn{arg\,max}} % waves \renewcommand {\l} {\lambda} % wavelength [m] \renewcommand {\k} {\vec{k}} % wavevector [rad/m] \newcommand {\uk} {\uvec{k}} % unit wavevector [#] \newcommand {\w} {\omega} % angular frequency [rad/s] \renewcommand {\TH} {e^{j \w t}} % engineering time-harmonic function [#] % classical mechanics \newcommand {\F} {\vec{F}} % force [N] \newcommand {\p} {\vec{p}} % momentum [kg m/s] % \r % position [m], aliased \renewcommand {\v} {\vec{v}} % velocity vector [m/s] \renewcommand {\a} {\vec{a}} % acceleration [m/s^2] \newcommand {\vGamma} {\gvec{\Gamma}} % torque [N m] \renewcommand {\L} {\vec{L}} % angular momentum [kg m^2 / s] \newcommand {\mI} {\mat{I}} % moment of inertia tensor [kg m^2/rad] \newcommand {\vw} {\gvec{\omega}} % angular velocity vector [rad/s] \newcommand {\valpha} {\gvec{\alpha}} % angular acceleration vector [rad/s^2] % electromagnetics % fields \newcommand {\E} {\vec{E}} % electric field intensity [V/m] \renewcommand {\H} {\vec{H}} % magnetic field intensity [A/m] \newcommand {\D} {\vec{D}} % electric flux density [C/m^2] \newcommand {\B} {\vec{B}} % magnetic flux density [Wb/m^2] % potentials \newcommand {\A} {\vec{A}} % vector potential [Wb/m], [C/m] % \F % electric vector potential [C/m], aliased % sources \newcommand {\I} {\vec{I}} % line current density [A] , [V] \newcommand {\J} {\vec{J}} % surface current density [A/m] , [V/m] \newcommand {\K} {\vec{K}} % volume current density [A/m^2], [V/m^2] % \M % magnetic current [V/m^2], aliased, obsolete % materials \newcommand {\ep} {\epsilon} % permittivity [F/m] % \mu % permeability [H/m], aliased \renewcommand {\P} {\vec{P}} % polarization [C/m^2], [Wb/m^2] % \p % electric dipole moment [C m], aliased \newcommand {\M} {\vec{M}} % magnetization [A/m], [V/m] \newcommand {\m} {\vec{m}} % magnetic dipole moment [A m^2] % power \renewcommand {\S} {\vec{S}} % poynting vector [W/m^2] \newcommand {\Sa} {\aa{\vec{S}}_t} % time averaged poynting vector [W/m^2] % quantum mechanics \newcommand {\bra} [1] {\left\langle {#1} \right|} % <| \newcommand {\ket} [1] {\left| {#1} \right\rangle} % |> \newcommand {\braket} [2] {\left\langle {#1} \middle| {#2} \right\rangle} $

Table of Contents

Set Theory

Groups

Mathematical groups are ubiquitous abstract algebra structures. They algebraically capture the notion of symmetry. Groups are rich structures despite their simple rules.

Axioms

A group is a 2-tuple $(G, \oplus)$ consisting of set $G$ and a binary operation on $G$ called the group operation $\oplus$ such that for $a, b, c \in G$ $$ \begin{align} &\oplus : G \times G \rightarrow G & \text{closure} \\ &[a \oplus b] \oplus c = a \oplus [b \oplus c] & \text{associative} \\ &\exists 0 \in G \; [a \oplus 0 = a] & \text{existence of identity} \\ &\exists [-a] \in G \; [a \oplus [-a] = 0] & \text{existence of inverse} . \end{align} $$ Groups are usually referred to by the same symbol as their underlying set, in this case $G$.

Symmetric Group

Consider a set $A$ with $n$ elements. A permutation is a bijection from set $A$ to itself. The n-th degree symmetric group $(S_n, \circ)$ is the set $S_n$ of all permutations on $A$ with functional composition $\circ$ as the group operation.

A transposition is a permutation where only two elements swap places. Every permutation can be constructed by functionally composing a sequence of transpositions. If a permutation is written from an even (odd) number of transpositions, then the permutation is said to be even (odd). The sign of permutation $\sigma \in S_n$ is defined as $$ \fn{sgn}(\sigma) \equiv \begin{cases} +1 & \text{if $\sigma$ is even} \\ -1 & \text{if $\sigma$ is odd}\end{cases} . $$

Fields

Fields are abstract algebraic structures motivated by the arithmetic operations of addition, subtraction, multiplication, and division on real numbers.

Axioms

A field is a 3-tuple $(F, \oplus, \odot)$ consisting of set $F$ and two binary operations on $F$ called field addition $\oplus$ and field multiplication $\odot$ such that for $a, b, c \in F$ $$ \begin{align} &\oplus : F \times F \rightarrow F &\text{closure} \\ &[a \oplus b] \oplus c = a \oplus [b \oplus c] &\text{associative} \\ &a \oplus b= b \oplus a &\text{commutative} \\ &\exists 0 \in F \; [a \oplus 0 = a] &\text{existence of additive identity} \\ &\exists [-a] \in F \; [a \oplus [-a] = 0] &\text{existence of additive inverse} \\ &\nonumber \\ &\odot : F \times F \rightarrow F &\text{closure} \\ &[a \odot b] \odot c = a \odot [b \odot c] &\text{associative} \\ &a \odot b = b \odot a &\text{commutative} \\ &\exists 1 \in F \; [a \odot 1 = a] &\text{existence of multiplicative identity} \\ &\exists [a^{-1}] \in F \; [a \odot [a^{-1}] = 1] \;\text{for}\; a \neq 0 &\text{existence of multiplicative inverse} \\ &\nonumber \\ & a \odot [b \oplus c] = [a \odot b] \oplus [a \odot c] &\text{distributive} . \end{align} $$ The symbol $[-a]$ represents the additive inverse of $a \in F$, while $[a^{-1}]$ represents the multiplicative inverse of $a \in F - \cc{0}$. Subtraction is shorthand for the addition of an additive inverse $$ a - b \equiv a \oplus [-b] $$ and division is shorthand for multiplication by a multiplicative inverse $$ \q{a}{b} \equiv a \odot [b^{-1}] . $$ Examples of fields are real numbers $\setR$, complex numbers $\setC$, and rational numbers $\setQ$. Integers $\setZ$ are not a field because most multiplicative inverses fail to exist (not in $\setZ$).

Identities

The field axioms imply familiar arithmetic identities (proofs omitted) for $a,b,c,d \in F$ $$ \begin{split} &0 \neq 1; \quad -0 = 0; \quad 1^{-1} = 1; \quad -[-a] = a; \quad [a^{-1}]^{-1} = a; \quad [-a] = -1 a; \\ & [-a] b = a [-b] = -[a b]; \quad \q{a}{b} \q{c}{d} = \q{a c}{b d}; \quad \q{a / b}{c / d} = \q{a d}{b c}; \quad \q{a}{b} + \q{c}{d} = \q{ad + bc}{bd}. \end{split} $$

Index Notation

Summation notation sums symbols indexed by elements of index set $I$ $$ \sum_{i \in I} a_i . $$ When the index set is an interval of integers $I = [m,n] \subset \setZ$ the sum is written $$ \sum_{i = m}^n a_i = a_m + a_{m+1} + \dots + a_{n - 1} + a_n . $$ The index set can be omitted when it is clear from context $$ \sum_i a_i . $$ For expressions with several summations, each summation only applies to the multiplicative factors that immediately follow it. This convention and the field axioms imply the following summation identities $$ \sum_i [a b_i + c d_i] = a \sum_i b_i + c \sum_i d_i $$ $$ \sum_i a_i \sum_j b_j = \sum_i \sum_j a_i b_j = \sum_j \sum_i a_i b_j . $$

Product notation multiplies symbols indexed with elements from index set $I$

$$ \prod_{i \in I} a_i . $$ When the index set is an interval of integers $I = [m,n] \subset \setZ$ the product is written $$ \prod_{i = m}^n a_i = a_m a_{m+1} \dots a_{n-1} a_n . $$ For expressions with several products, each product only applies to the multiplicative factors that immediately follow it. The field axioms imply the following product notation identity $$ \prod_i a_i b_i = \bb{\prod_i a_i} \bb{\prod_i b_i} . $$

Exponentiation and Logarithms

$$ a^n = \underbrace{a a a \ldots a}_{n \, \text{factors}} = \prod_{i=1}^n a $$ $$ a^m a^n = a^{m + n} $$ $$ [a^m]^n = a^{m n} $$ $$ [a b]^n = a^n b^n $$ $$ a^{-n} = [a^{-1}]^n % = \q{1}{a^n} $$ $$ a^n a^{-n} = 1 $$ $$ a^0 \equiv 1 \quad\text{for}\quad a \neq 0 $$ $$ \bb{a^{\q{1}{n}}}^n = 1 $$ $$ \sqrt[n]{a} \equiv a^{\q{1}{n}} $$ $$ a^b \equiv \lim_{\q{m}{n} \rightarrow b} a^{\q{m}{n}} $$

logarithms

$$ b^x = y $$ $$ \log_b y = x $$ $$ b^{\log_b y} = y $$ $$ \log_b(b^x) = x $$ $$ \log_b(x y) = \log_b x + \log_b y $$ $$ \log_b \pp{\q{y}{x}} = \log_b{y} - \log_b{x} $$ $$ \log_b (x^n) = n \log_b x $$ $$ \log_a x = \q{\log_b x}{\log_b a} $$ $$ \ln x \equiv \log_e x $$

Complex Numbers

The field of complex numbers $\setC$ is of fundamental importance to algebra. The real numbers $\setR$ are a subset of $\setC$, and complex arithmetic is similar to real arithmetic.

The fundamental complex identity defines the imaginary unit $i \in \setC$ as $$ i^2 \equiv -1 . $$ A complex number $\z{z} \in \setC$ can be represented in rectangular form as the sum of a real part $x \in \setR$ and imaginary part $y \in \setR$ multiplied by the imaginary unit $i$ $$ \z{z} = x + i y . $$ The tilde is used to distinguish complex numbers from real numbers. Addition and multiplication of complex numbers $\z{z}, \z{z}' \in \setC$ in rectangular form are $$ \z{z} + \z{z}' = [x + i y] + [x' + i y'] = [x + x'] + i[y + y'] $$ $$ \z{z} \z{z}' = [x + i y] [x' + i y'] = x x' + i x y' + i x' y - y y' = [x x' - y y'] + i [x y' + x' y] . $$ An important unary operation is complex conjugation $\conj{[\cdot]} : \setC \rightarrow \setC$ which flips the sign of the imaginary part of a complex number $$ \label{eqn:complexconjugation} \conj{\z{z}} = \conj{[x + i y]} = x - i y . $$ Real and imaginary part functions $\Re : \setC \rightarrow \setR$ and $\Im : \setC \rightarrow \setR$ can be defined abstractly in terms of conjugation $$ \Re(\z{z}) \equiv \q{1}{2}[\z{z} + \conj{\z{z}}] = x $$ $$ \Im(\z{z}) \equiv \q{1}{2 i}[\z{z} - \conj{\z{z}}] = y . $$ The modulus (magnitude) $|\cdot| : \setC \rightarrow \setR$ of a complex number is defined as $$ \label{eqn:complexmodulus} \nn{\z{z}} \equiv [\conj{\z{z}} \z{z}]^{\q{1}{2}} = [x^2 + y^2]^{\q{1}{2}} . $$ Complex division is understood as complex multiplication divided by a real number $$ \q{\z{z}'}{\z{z}} = \q{\conj{\z{z}}}{\conj{\z{z}}} \q{\z{z}'}{\z{z}} = \q{\conj{\z{z}} \z{z}'}{\nn{\z{z}}^2} . $$ The above definitions are easily shown to satisfy the field axioms.

Another complex number representation is polar form which uses magnitude and angle. The argument (angle) $\arg : \setC \rightarrow [-\pi, \pi] \subset \setR$ of a complex number is defined as $$ \arg(\z{z}) \equiv \atantwo(\Im(\z{z}), \Re(\z{z})) = \phi . $$ The rectangular form of a complex number in terms of magnitude and angle is $$ \z{z} = \nn{\z{z}}[\cos \phi + i \sin \phi] $$ and this is rewritten using Euler's identity $$ e^{i \phi} = \cos \phi + i \sin \phi $$ as a complex exponential in polar form $$ \z{z} = |\z{z}| e^{i \phi} . $$ A benefit of polar form is that complex multiplication is algebraically easy $$ \z{z} \z{z}' = \nn{\z{z}} e^{i \phi} \nn{\z{z}'} e^{i \phi'} = \nn{\z{z}} \nn{\z{z}'} e^{i[\phi + \phi']} . $$

The tilde is suppressed when values are understood to be complex.