學海無涯，思境無維；數乃六藝，理之始也。
或有一得足矣 愚千慮

$$$$

Abstract Algebra Theory and Applications

https://open.umn.edu/opentextbooks/textbooks/abstract-algebra-theory-and-applications

Preliminaries

Equivalence Relation

• Rational Number Equivalence
• Function Equivalence
• Equivalence Classes (form a partition) of a Set
• Matrix Equivalence

Matrix Equivalence (similar relation)

$$\begin{gathered} A,B\in Matrix \\ A\sim B⟺\mathrm{\exists }P,PA{P}^{-1}=B \end{gathered}$$

$A$ and

$B$ are smiliar.

• Points on the same circle
  
  $\{\begin{array}{l}{p}_1=(x_1,y_1)\\ p_2=(x_2,y_2)\end{array},{x_1}^2+{y_1}^2={x_2}^2+{y_2}^2$
• Integer Modulo
  $n$
  
  $r\equiv s(\mathrm{mod}n)⟺r{\equiv }_{n}s⟺r-s=kn\mid k\in \mathbb{Z}$
  
  $r$ is congruent to
  $s$ modulo
  $n$ (r 與 s 同餘)

The Division Algorithm

$$\begin{gathered} \{\begin{array}{l}a,b\in \mathbb{Z}\\ S=\{am+bn\mid m,n\in \mathbb{Z},am+bn>0\}⟹am+bn\in \mathbb{N}⟹S\text{ is well-ordered}\\ d\text{ is the least element of S},\mathrm{\exists }x,y\in \mathbb{Z},d=ax+by,d>0\end{array} \\ a=q_{1}d+r_1,0\le r_1<d⟹\begin{array}{ll}{r}_1& =a-q_{1}d\\ & =a-q_1(ax+by)\\ & =a(1-q_{1}x)+b(-q_{1}y)\end{array} \\ {r}_1>0⟹r_1\in S\wedge r_1<d,\text{contradiction to }d\text{ is the least element of }S \\ \therefore r_1=0,a=q_{1}d⟹d\mid a \\ b=q_{2}d+r_2,0\le r_2<d⟹\begin{array}{ll}{r}_2& =b-q_{2}d\\ & =b-q_2(ax+by)\\ & =a(-q_{2}x)+b(1-q_{2}y)\end{array} \\ {r}_2>0⟹r_2\in S\wedge r_2<d,\text{contradiction to }d\text{ is the least element of }S \\ \therefore r_2=0,b=q_{2}d⟹d\mid b \\ d\mid a\wedge d\mid b⟹d\text{ is the common divisor of }a\text{ and }b \\ assume\mathrm{\exists }{d}^{\prime }\text{ is the other common divisor of }a\text{ and }b \\ \{\begin{array}{l}a=k_a{d}^{\prime },k_a\in \mathbb{Z}\\ b=k_b{d}^{\prime },k_b\in \mathbb{Z}\end{array} \\ \begin{array}{ll}d& =ax+by\\ & =k_a{d}^{\prime }x+k_b{d}^{\prime }y\\ & =d^{\prime }(k_{a}x+k_{b}y)\end{array} \\ \therefore d^{\prime }\mid d⟹d=gcd(a,b) \\ gcd(a,b)=\text{the least element of }\{am+bn\mid m,n\in \mathbb{Z},am+bn\in \mathbb{N}\} \end{gathered}$$

Integers

Prime Number

$$p\in prime⟹\{\begin{array}{l}p\in \mathbb{N}\\ p\nmid 1\\ \mathrm{\forall }a\in \mathbb{N},a<p⟹gcd(a,p)=1\\ \mathrm{\forall }{p}_1,p_2\in prime,gcd(p_1,p_2)=1\end{array}$$

Euclid

Let a and b be integers and p be a prime number. If p | ab, then either p|a or p|b.

$$a,b\in \mathbb{Z},p\in prime,p\mid ab⟹p\mid a\vee p\mid b$$

Proof:

$$\begin{gathered} ifp\nmid a⟹gcd(a,p)=1⟹\mathrm{\exists }x,y\in \mathbb{Z},ax+py=1 \\ b=b\cdot 1=b(ax+py)=(ab)x+p(by) \\ p\mid ab\wedge p\mid p⟹p\mid b \\ \text{for the same reason},ifp\nmid b⟹p\mid a \end{gathered}$$

Infinite Prime Numbers

Assume there're only finite prime numbers

$p_1,p_2,\dots ,p_n$
Let

$P=p_1{p}_2\cdot p_n+1$, then

$\mathrm{\exists }{p}_i,1\le i\le n,p_i\mid P$

$$\{\begin{array}{l}{p}_i\mid (p_1{p}_2\cdot p_n)\\ p_i\mid P\end{array}⟹p_i\mid (P-p_1{p}_2\cdot p_n)⟹p_i\mid 1$$
Since all prime numbers can't divide 1, contradiction to

$p_i\in prime$.
Therefore exists infinite prime numbers.

$ab\equiv 1(\mathrm{mod}n)$

Let a be a nonzero integer. Then

$gcd(a,n)=1$ if and only if there exists a multiplicative inverse

$b$ for

$a(\mathrm{mod}n)$; that is, a nonzero integer

$b$ such that

$ab\equiv 1(\mathrm{mod}n)$

$$\begin{gathered} {\mathbb{Z}}^{\mathbb{\ast }}=\mathbb{Z}\setminus \{0\} \\ \mathrm{\forall }a,n\in {\mathbb{Z}}^{\mathbb{\ast }},gcd(a,n)=1⟺\mathrm{\exists }b\in {\mathbb{Z}}^{\mathbb{\ast }},ab\equiv 1(\mathrm{mod}n) \end{gathered}$$
Proof:

$⟹)$

$$\begin{gathered} gcd(a,n)=1⟹\mathrm{\exists }r,s\in {\mathbb{Z}}^{\mathbb{\ast }},ar+ns=1 \\ ar=1-ns⟹ar\equiv 1(\mathrm{mod}n) \\ \therefore b=r(\mathrm{mod}n)\in {\mathbb{Z}}_{\mathbb{n}} \end{gathered}$$

$⟸)$

$$\begin{gathered} ab\equiv 1(\mathrm{mod}n)⟹n\mid ab-1 \\ \mathrm{\exists }k\in {\mathbb{Z}}^{\mathbb{\ast }},ab-1=kn⟹ab-kn=1 \\ letd=gcd(a,n)⟹d\mid a\wedge d\mid n \\ \therefore d\mid (ab-kn)⟹d\mid 1⟹d=1 \\ gcd(a,n)=1 \end{gathered}$$

Groups

Group of Units of

${\mathbb{Z}}_{\mathbb{n}}$

Unit of

${\mathbb{Z}}_{\mathbb{n}}$

$$\begin{gathered} \mathrm{\forall }a\in {\mathbb{Z}}_{\mathbb{n}},gcd(a,n)=1\text{ aka. }a,n\text{ are relatively prime or coprime}, \\ a\text{ is a unit of }{\mathbb{Z}}_{\mathbb{n}} \end{gathered}$$

U(n) : the set of all units of

${\mathbb{Z}}_{\mathbb{n}}$ form a group called the group of units of

${\mathbb{Z}}_{\mathbb{n}}$ under

${\times }_n$

$$U(n)=\{u\mid u\in unitof{\mathbb{Z}}_{\mathbb{n}}\}$$

Cayley Table of

$({\mathbb{Z}}_{\mathbb{8}},{\times }_8)$

$$\begin{gathered} U(8)=\{1,3,5,7\} \\ \begin{array}{ccccc}{\times }_8& 1& 3& 5& 7\\ 1& 1& 3& 5& 7\\ 3& 3& 1& 7& 5\\ 5& 5& 7& 1& 3\\ 7& 7& 5& 3& 1\end{array} \end{gathered}$$

General Linear Group

$G{L}_n(\mathbb{R})$

$$\begin{gathered} \{\begin{array}{l}{M}_n(\mathbb{R})⟺n\times nmatrix\\ I_n⟺n\times nidentitymatrix\end{array} \\ G{L}_n(\mathbb{R})⟺\mathrm{\forall }a\in M_n(\mathbb{R}),\mathrm{\exists }b\in M_n(\mathbb{R}),ab=ba=I_n \end{gathered}$$

Quaternion Group

$$\begin{gathered} 1=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),I=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right) \\ J=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),K=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right) \\ \begin{array}{lcccc}\cdot & 1& I& J& K\\ 1& 1& I& J& K\\ I& I& -1& K& -J\\ J& J& -K& -1& I\\ K& K& J& -I& -1\end{array}⟺\{\begin{array}{l}{i}^2=-1\\ I^2=J^2=K^2=-1\\ IJ=K,JI=-K\\ JK=I,KJ=-I\\ KI=J,IK=-J\end{array} \\ {Q}_8=\{\pm 1,\pm I,\pm J,\pm K\} \end{gathered}$$

Complex Plane Example:

$$\begin{gathered} {\mathbb{C}}^{\mathbb{\ast }}=\mathbb{C}\setminus \{0\} \\ \mathrm{\forall }z\in {\mathbb{C}}^{\mathbb{\ast }},z=a+bi⟹z^{-1}={\displaystyle \frac{a-bi}{a^2+b^2}} \end{gathered}$$

Heisenberg Group

$$\begin{gathered} a=\left(\begin{array}{ccc}1& x& y\\ 0& 1& z\\ 0& 0& 1\end{array}\right),b=\left(\begin{array}{ccc}1& x^{\prime }& y^{\prime }\\ 0& 1& z^{\prime }\\ 0& 0& 1\end{array}\right)⟹ab=\left(\begin{array}{ccc}1& x+x^{\prime }& y+y^{\prime }+x{z}^{\prime }\\ 0& 1& z+z^{\prime }\\ 0& 0& 1\end{array}\right) \\ Identity=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right) \\ Inverseofa=\left(\begin{array}{ccc}1& -x& xz-y\\ 0& 1& -z\\ 0& 0& 1\end{array}\right) \end{gathered}$$

Subgroups

Subgroup of Complex Number under Multiplication

$$\begin{gathered} H=\{1,-1,i,-i\}\subset {\mathbb{C}}^{\mathbb{\ast }} \\ \begin{array}{crrrr}\cdot & 1& -1& i& -i\\ 1& 1& -1& i& -i\\ -1& -1& 1& -i& i\\ i& i& -i& -1& 1\\ -i& -i& i& 1& -1\end{array} \\ \therefore H⊲{\mathbb{C}}^{\mathbb{\ast }} \end{gathered}$$

Special Linear Subgroup

$S{L}_n(\mathbb{R})$ of

$G{L}_n(\mathbb{R})$

$$S{L}_n(\mathbb{R})⟺\{x\in G{L}_n(\mathbb{R})\mid det(x)=1\}$$

Cyclic Groups

Permutation Groups

Coset and Lagrange's Theorem

Coset

Lagrange's Theorem

Euler's Theorem

Euler's

$\varphi$-function

$$\varphi (n):|\{m\mid 1\le m<n,gcd(m,n)=1\}|$$
Example

$$\begin{gathered} |U(12)|=|\{1,5,7,11\}|=\varphi (12)=4 \\ \mathrm{\forall }p\in prime,\varphi (p)=p-1 \end{gathered}$$

tags: math