From a5ea6f86a70343390bebcd2e039f371e54a13334 Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Tue, 31 Mar 2026 00:51:59 -0500 Subject: [PATCH] updates --- content/Math4302/Exam_reviews/Math4302_E2.md | 440 ++++++++++++++++++- content/Math4302/Math4302_L26.md | 4 +- 2 files changed, 441 insertions(+), 3 deletions(-) diff --git a/content/Math4302/Exam_reviews/Math4302_E2.md b/content/Math4302/Exam_reviews/Math4302_E2.md index d027bfb..74ced44 100644 --- a/content/Math4302/Exam_reviews/Math4302_E2.md +++ b/content/Math4302/Exam_reviews/Math4302_E2.md @@ -1 +1,439 @@ -# Math 4302 Exam 2 Review \ No newline at end of file +# Math 4302 Exam 2 Review + +## Groups + + +### Direct products + +$\mathbb{Z}_m\times \mathbb{Z}_n$ is cyclic if and only if $m$ and $n$ have greatest common divisor $1$. + +More generally, for $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \cdots \times \mathbb{Z}_{n_k}$, if $n_1,n_2,\cdots,n_k$ are pairwise coprime, then the direct product is cyclic. + + +If $n=p_1^{m_1}\ldots p_k^{m_k}$, where $p_i$ are distinct primes, then the group + +$$ +G=\mathbb{Z}_n=\mathbb{Z}_{p_1^{m_1}}\times \mathbb{Z}_{p_2^{m_2}}\times \cdots \times \mathbb{Z}_{p_k^{m_k}} +$$ + +is cyclic. + +### Structure of finitely generated abelian groups + +#### Theorem for finitely generated abelian groups + +Every finitely generated abelian group $G$ is isomorphic to + +$$ +Z_{p_1}^{n_1}\times Z_{p_2}^{n_2}\times \cdots \times Z_{p_k}^{n_k}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}} +$$ + +#### Corollary for divisor size of abelian subgroup + +If $g$ is abelian and $|G|=n$, then for every divisor $m$ of $n$, $G$ has a subgroup of order $m$. + +> [!WARNING] +> +> This is not true if $G$ is not abelian. +> +> Consider $A_4$ (alternating group for $S_4$) does not have a subgroup of order 6. + + +### Cosets + +#### Definition of Cosets + +Let $G$ be a group and $H$ its subgroup. + +Define a relation on $G$ and $a\sim b$ if $a^{-1}b\in H$. + +This is an equivalence relation. + +- Reflexive: $a\sim a$: $a^{-1}a=e\in H$ +- Symmetric: $a\sim b\Rightarrow b\sim a$: $a^{-1}b\in H$, $(a^{-1}b)^{-1}=b^{-1}a\in H$ +- Transitive: $a\sim b$ and $b\sim c\Rightarrow a\sim c$ : $a^{-1}b\in H, b^{-1}c\in H$, therefore their product is also in $H$, $(a^{-1}b)(b^{-1}c)=a^{-1}c\in H$ + +So we get a partition of $G$ to equivalence classes. + +Let $a\in G$, the equivalence class containing $a$ + +$$ +aH=\{x\in G| a\sim x\}=\{x\in G| a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\} +$$ + +This is called the coset of $a$ in $H$. + +#### Definition of Equivalence Class + +Let $a\in H$, and the equivalence class containing $a$ is defined as: + +$$ +aH=\{x|a\simeq x\}=\{x|a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\} +$$ + +#### Properties of Equivalence Class + +$aH=bH$ if and only if $a\sim b$. + +#### Lemma for size of cosets + +Any coset of $H$ has the same cardinality as $H$. + +Define $\phi:H\to aH$ by $\phi(h)=ah$. + +$\phi$ is an bijection, if $ah=ah'\implies h=h'$, it is onto by definition of $aH$. + +#### Corollary: Lagrange's Theorem + +If $G$ is a finite group, and $H\leq G$, then $|H|\big\vert |G|$. (size of $H$ divides size of $G$) + +### Normal Subgroups + +#### Definition of Normal Subgroup + +A subgroup $H\leq G$ is called a normal subgroup if $aH=Ha$ for all $a\in G$. We denote it by $H\trianglelefteq G$ + + +#### Lemma for equivalent definition of normal subgroup + +The following are equivalent: + +1. $H\trianglelefteq G$ +2. $aHa^{-1}=H$ for all $a\in G$ +3. $aHa^{-1}\subseteq H$ for all $a\in G$, that is $aha^{-1}\in H$ for all $a\in G$ + +### Factor group + +Consider the operation on the set of left coset of $G$, denoted by $S$. Define + +$$ +(aH)(bH)=abH +$$ + +#### Condition for operation + +The operation above is well defined if and only if $H\trianglelefteq G$. + +#### Definition of factor (quotient) group + +If $H\trianglelefteq G$, then the set of cosets with operation: + +$$ +(aH)(bH)=abH +$$ + +is a group denoted by $G/H$. This group is called the quotient group (or factor group) of $G$ by $H$. + +#### Fundamental homomorphism theorem (first isomorphism theorem) + +If $\phi:G\to G'$ is a homomorphism, then the function $f:G/\ker(\phi)\to \phi(G)$, ($\phi(G)\subseteq G'$) given by $f(a\ker(\phi))=\phi(a)$, $\forall a\in G$, is an well-defined isomorphism. + +> - If $G$ is abelian, $N\leq G$, then $G/N$ is abelian. +> - If $G$ is finitely generated and $N\trianglelefteq G$, then $G/N$ is finitely generated. + +#### Definition of simple group + +$G$ is simple if $G$ has no proper ($H\neq G,\{e\}$), normal subgroup. + +### Center of a group + +Recall from previous lecture, the center of a group $G$ is the subgroup of $G$ that contains all elements that commute with all elements in $G$. + +$$ +Z(G)=\{a\in G\mid \forall g\in G, ag=ga\} +$$ + +this subgroup is normal and measure the "abelian" for a group. + +#### Definition of the commutator of a group + +Let $G$ be a group and $a,b\in G$, the commutator $[a,b]$ is defined as $aba^{-1}b^{-1}$. + +$[a,b]=e$ if and only if $a$ and $b$ commute. + +Some additional properties: + +- $[a,b]^{-1}=[b,a]$ + +#### Definition of commutator subgroup + +Let $G'$ be the subgroup of $G$ generated by all commutators of $G$. + +$$ +G'=\{[a_1,b_1][a_2,b_2]\ldots[a_n,b_n]\mid a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n\in G\} +$$ + +Then $G'$ is the subgroup of $G$. + +- Identity: $[e,e]=e$ +- Inverse: $([a_1,b_1],\ldots,[a_n,b_n])^{-1}=[b_n,a_n],\ldots,[b_1,a_1]$ + +Some additional properties: + +- $G$ is abelian if and only if $G'=\{e\}$ +- $G'\trianglelefteq G$ +- $G/G'$ is abelian +- If $N$ is a normal subgroup of $G$, and $G/N$ is abelian, then $G'\leq N$. + +### Group acting on a set + +#### Definition for group acting on a set + +Let $G$ be a group, $X$ be a set, $X$ is a $G$-set or $G$ acts on $X$ if there is a map + +$$ +G\times X\to X +$$ +$$ +(g,x)\mapsto g\cdot x\, (\text{ or simply }g(x)) +$$ + +such that + +1. $e\cdot x=x,\forall x\in X$ +2. $g_2\cdot(g_1\cdot x)=(g_2 g_1)\cdot x$ + +#### Group action is a homomorphism + +Let $X$ be a $G$-set, $g\in G$, then the function + +$$ +\sigma_g:X\to X,x\mapsto g\cdot x +$$ + +is a bijection, and the function $\phi:G\to S_X, g\mapsto \sigma_g$ is a group homomorphism. + + +#### Definition of orbits + +We define the equivalence relation on $X$ + +$$ +x\sim y\iff y=g\cdot x\text{ for some }g +$$ + +So we get a partition of $X$ into equivalence classes: orbits + +$$ +Gx\coloneqq \{g\cdot x|g\in G\}=\{y\in X|x\sim y\} +$$ + +is the orbit of $X$. + +$x,y\in X$ either $Gx=Gy$ or $Gx\cap Gy=\emptyset$. + +$X=\bigcup_{x\in X}Gx$. + +#### Definition of isotropy subgroup + +Let $X$ be a $G$-set, the stabilizer (or isotropy subgroup) corresponding to $x\in X$ is + +$$ +G_x=\{g\in G|g\cdot x=x\} +$$ + +$G_x$ is a subgroup of $G$. $G_x\leq G$. + +- $e\cdot x=x$, so $e\in G_x$ +- If $g_1,g_2\in G_x$, then $(g_1g_2)\cdot x=g_1\cdot(g_2\cdot x)=g_1 \cdot x$, so $g_1g_2\in G_x$ +- If $g\in G_x$, then $g^{-1}\cdot g=x=g^{-1}\cdot x$, so $g^{-1}\in G_x$ + +#### Orbit-stabilizer theorem + +If $X$ is a $G$-set and $x\in X$, then + +$$ +|Gx|=(G:G_x)=\text{ number of left cosets of }G_x=\frac{|G|}{|G_x|} +$$ + +#### Theorem for orbit with prime power groups + +Suppose $X$ is a $G$-set, and $|G|=p^n$ for some prime $p$. Let $X_G$ be the set of all elements in $X$ whose orbit has size $1$. (Recall the orbit divides $X$ into disjoint partitions.) Then $|X|\equiv |X_G|\mod p$. + +#### Corollary: Cauchy's theorem + +If $p$ is prime and $p|(|G|)$, then $G$ has a subgroup of order $p$. + +> This does not hold when $p$ is not prime. +> +> Consider $A_4$ with order $12$, and $A_4$ has no subgroup of order $6$. + +#### Corollary: Center of prime power group is non-trivial + +If $|G|=p^m$, then $Z(G)$ is non-trivial. ($Z(G)\neq \{e\}$) + +#### Proposition: Prime square group is abelian + +If $|G|=p^2$, where $p$ is a prime, then $G$ is abelian. + + +### Classification of small order + +Let $G$ be a group + +- $|G|=1$ + - $G=\{e\}$ +- $|G|=2$ + - $G\simeq\mathbb{Z}_2$ (prime order) +- $|G|=3$ + - $G\simeq\mathbb{Z}_3$ (prime order) +- $|G|=4$ + - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2$ + - $G\simeq\mathbb{Z}_4$ +- $|G|=5$ + - $G\simeq\mathbb{Z}_5$ (prime order) +- $|G|=6$ + - $G\simeq S_3$ + - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_2\simeq \mathbb{Z}_6$ +
+Proof + +$|G|$ has an element of order $2$, namely $b$, and an element of order $3$, namely $a$. + +So $e,a,a^2,b,ba,ba^2$ are distinct. + +Therefore, there are only two possibilities for value of $ab$. ($a,a^2$ are inverse of each other, $b$ is inverse of itself.) + +If $ab=ba$, then $G$ is abelian, then $G\simeq \mathbb{Z}_2\times \mathbb{Z}_3$. + +If $ab=ba^2$, then $G\simeq S_3$. +
+ +- $|G|=7$ + - $G\simeq\mathbb{Z}_7$ (prime order) +- $|G|=8$ + - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$ + - $G\simeq\mathbb{Z}_4\times \mathbb{Z}_2$ + - $G\simeq\mathbb{Z}_8$ + - $G\simeq D_4$ + - $G\simeq$ quaternion group $\{e,i,j,k,-1,-i,-j,-k\}$ where $i^2=j^2=k^2=-1$, $(-1)^2=1$. $ij=l$, $jk=i$, $ki=j$, $ji=-k$, $kj=-i$, $ik=-j$. +- $|G|=9$ + - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_3$ + - $G\simeq\mathbb{Z}_9$ (apply the corollary, $9=3^2$, these are all the possible cases) +- $|G|=10$ + - $G\simeq\mathbb{Z}_5\times \mathbb{Z}_2\simeq \mathbb{Z}_{10}$ + - $G\simeq D_5$ +- $|G|=11$ + - $G\simeq\mathbb{Z}_11$ (prime order) +- $|G|=12$ + - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_4$ + - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3$ + - $A_4$ + - $D_6\simeq S_3\times \mathbb{Z}_2$ + - ??? One more +- $|G|=13$ + - $G\simeq\mathbb{Z}_{13}$ (prime order) +- $|G|=14$ + - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_7$ + - $G\simeq D_7$ + + +#### Lemma for group of order $2p$ where $p$ is prime + +If $p$ is prime, $p\neq 2$, and $|G|=2p$, then $G$ is either abelian $\simeq \mathbb{Z}_2\times \mathbb{Z}_p$ or $G\simeq D_p$ + +## Ring + + +### Definition of ring + +A ring is a set $R$ with binary operation $+$ and $\cdot$ such that: + +- $(R,+)$ is an abelian group. +- Multiplication is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$. +- Distribution property: $a\cdot (b+c)=a\cdot b+a\cdot c$, $(b+c)\cdot a=b\cdot a+c\cdot a$. (Note that $\cdot$ may not be abelian, may not even be a group, therefore we need to distribute on both sides.) + +> [!NOTE] +> +> $a\cdot b=ab$ will be used for the rest of the sections. + +#### Properties of rings + +Let $0$ denote the identity of addition of $R$. $-a$ denote the additive inverse of $a$. + +- $0\cdot a=a\cdot 0=0$ +- $(-a)b=a(-b)=-(ab)$, $\forall a,b\in R$ +- $(-a)(-b)=ab$, $\forall a,b\in R$ + +#### Definition of commutative ring + +A ring $(R,+,\cdot)$ is commutative if $a\cdot b=b\cdot a$, $\forall a,b\in R$. + +#### Definition of unity element + +A ring $R$ has unity element if there is an element $1\in R$ such that $a\cdot 1=1\cdot a=a$, $\forall a\in R$. + +#### Definition of unit + +Suppose $R$ is a ring with unity element. An element $a\in R$ is called a unit if there is $b\in R$ such that $a\cdot b=b\cdot a=1$. + +In this case $b$ is called the inverse of $a$. + +#### Definition of division ring + +If every $a\neq 0$ in $R$ has a multiplicative inverse (is a unit), then $R$ is called a division ring. + +#### Definition of field + +A commutative division ring is called a field. + +#### Units in $\mathbb{Z}_n$ is coprime to $n$ + +More generally, $[m]\in \mathbb{Z}_n$ is a unit if and only if $\operatorname{gcd}(m,n)=1$. + +### Integral Domains + +#### Definition of zero divisors + +If $a,b\in R$ with $a,b\neq 0$ and $ab=0$, then $a,b$ are called zero divisors. + +#### Zero divisors in $\mathbb{Z}_n$ + +$[m]\in \mathbb{Z}_n$ is a zero divisor if and only if $\operatorname{gcd}(m,n)>1$ ($m$ is not a unit). + +#### Corollaries of integral domain + +If $R$ is a integral domain, then we have cancellation property $ab=ac,a\neq 0\implies b=c$. + +#### Units with multiplication forms a group + +If $R$ is a ring with unity, then the units in $R$ forms a group under multiplication. + +### Fermat’s and Euler’s Theorems + +#### Fermat’s little theorem + +If $p$ is not a divisor of $m$, then $m^{p-1}\equiv 1\mod p$. + +#### Corollary of Fermat’s little theorem + +If $m\in \mathbb{Z}$, then $m^p\equiv m\mod p$. + +#### Euler’s totient function + +Consider $\mathbb{Z}_6$, by definition for the group of units, $\mathbb{Z}_6^*=\{1,5\}$. + +$$ +\phi(n)=|\mathbb{Z}_n^*|=|\{1\leq x\leq n:gcd(x,n)=1\}| +$$ + +#### Euler’s Theorem + +If $m\in \mathbb{Z}$, and $gcd(m,n)=1$, then $m^{\phi(n)}\equiv 1\mod n$. + +#### Theorem for existence of solution of modular equations + +$ax\equiv b\mod n$ has a solution if and only if $d=\operatorname{gcd}(a,n)|b$ And if there is a solution, then there are exactly $d$ solutions in $\mathbb{Z}_n$. + +### Ring homomorphisms + +#### Definition of ring homomorphism + +Let $R,S$ be two rings, $f:R\to S$ is a ring homomorphism if $\forall a,b\in R$, + +- $f(a+b)=f(a)+f(b)\implies f(0)=0, f(-a)=-f(a)$ +- $f(ab)=f(a)f(b)$ + +#### Definition of ring isomorphism + +If $f$ is a ring homomorphism and a bijection, then $f$ is called a ring isomorphism. diff --git a/content/Math4302/Math4302_L26.md b/content/Math4302/Math4302_L26.md index ccda02a..4e2dbfc 100644 --- a/content/Math4302/Math4302_L26.md +++ b/content/Math4302/Math4302_L26.md @@ -79,7 +79,7 @@ $\phi(8)=|\{1,3,5,7\}|=4$ If $[a]\in \mathbb{Z}_n^*$, then $[a]^{\phi(n)}=[1]$. So $a^{\phi(n)}\equiv 1\mod n$. -#### Theorem +#### Euler’s Theorem If $m\in \mathbb{Z}$, and $gcd(m,n)=1$, then $m^{\phi(n)}\equiv 1\mod n$. @@ -104,7 +104,7 @@ Solution for $2x\equiv 1\mod 3$ So solution for $2x\equiv 1\mod 3$ is $\{3k+2|k\in \mathbb{Z}\}$. -#### Theorem for exsistence of solution of modular equations +#### Theorem for existence of solution of modular equations $ax\equiv b\mod n$ has a solution if and only if $\operatorname{gcd}(a,n)|b$ and in that case the equation has $d$ solutions in $\mathbb{Z}_n$.