From e7dc80a673041d06e3989f21103d94abb2f6947b Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Tue, 17 Feb 2026 22:17:44 -0600 Subject: [PATCH] udpates? --- content/Math4302/Exam_reviews/Math4302_E1.md | 124 +++++++++++++++++++ content/Math4302/Math4302_L13.md | 2 +- content/Math4302/Math4302_L5.md | 2 +- 3 files changed, 126 insertions(+), 2 deletions(-) create mode 100644 content/Math4302/Exam_reviews/Math4302_E1.md diff --git a/content/Math4302/Exam_reviews/Math4302_E1.md b/content/Math4302/Exam_reviews/Math4302_E1.md new file mode 100644 index 0000000..fc7e058 --- /dev/null +++ b/content/Math4302/Exam_reviews/Math4302_E1.md @@ -0,0 +1,124 @@ +# Math 4302 Exam 1 Review + +> [!NOTE] +> +> This is a review for definitions we covered in the classes. It may serve as a cheat sheet for the exam if you are allowed to use it. + +## Groups + +### Basic definitions + +#### Definition for group + +A group is a set $G$ with a binary operation $*$ that satisfies the following axioms: + +1. Closure: $\forall a,b\in G, a* b\in G$ (automatically guaranteed by definition of binary operation). +2. Associativity: $\forall a,b,c\in G, (a* b)* c=a* (b* c)$. +3. Identity: $\exists e\in G, \forall a\in G, e* a=a* e=a$. +4. Inverses: $\forall a\in G, \exists a^{-1}\in G, a* a^{-1}=a^{-1}* a=e$. + +- Identity element: If $X$ has an identity element, then it is unique. +- Composition of function is associative. + +#### Order of a element + +The order of an element $a$ in a group $G$ is the size of the smallest subgroup generated by $a$, we denote such subgroup as $\langle a\rangle$. + +Equivalently, the order of $a$ is the smallest positive integer $n$ such that $a^n=e$. + +#### Order of a group + +The order of a group $G$ is the size of $G$. + +#### Definition of subgroup + +A subgroup $H$ of a group $G$ is a subset of $G$ that is closed under the group operation. Denoted as $H\leq G$. + +#### Left and right cosets + +If $H$ is a subgroup of $G$, then $aH$ is a coset of $H$ for all $a\in G$. We call $aH$ a left coset of $H$ for $a$. + +$$ +aH=\{x|a\sim x\}=\{x\in G|a^{-1}x\in H\}=\{x|x=ah\text{ for some }h\in H\} +$$ + +Similarly, $Ha$ is a right coset of $H$ for $a$. + +$$ +Ha=\{x|x\sim'a\}=\{x\in G|xa^{-1}\in H\}=\{x|x=ha\text{ for some }h\in H\} +$$ + +- Usually, the left coset and right cosets will give different partitions of $G$. +- Use to prove lagrange theorem (partition of $G$ into cosets) + +#### Definition of normal subgroup + +A subgroup $H$ of a group $G$ is normal if $aH=Ha$ for all $a\in G$. + +### Isomorphism and homomorphism + +#### Definition of isomorphism + +Two groups $G$ and $G'$ are isomorphic if there exists a function $f:G\to G'$ such that + +- Homomorphism property is satisfied: $f(a*b)=f(a)f(b),\forall a,b\in G$ +- $f$ is injective: $f(a)=f(b)\implies a=b$ +- $f$ is surjective: $\forall a\in G',\exists b\in G$ such that $f(b)=a$ + +#### Definition of homomorphism + +A homomorphism is a function that satisfies the homomorphism property. + +If $\phi:G\to G'$ is a homomorphism, then + +- $\phi(e)=e'$, where $e$ is the identity of $G$ and $e'$ is the identity of $G'$. +- $\phi(a^{-1})=(\phi(a))^{-1}$ for all $a\in G$. +- If $H\leq G$ is a subgroup, then $\phi(H)\leq G'$ is a subgroup. +- If $K\leq G'$ is a subgroup, then $\phi^{-1}(K)\leq G$ is a subgroup. +- $\phi$ is surjective if and only if $\operatorname{ker}(\phi)=\{e\}$ (the trivial subgroup of $G$). + +### Basic groups + +#### Trivial group + +The group $(\{e\},*)$ is called the trivial group. + +#### Abelian group + +A group $G$ is abelian if $a*b=b*a$ for all $a,b\in G$. + +- The smallest non-abelian group is $S_3$ (order 6). +- Every abelian group is isomorphic to some direct product of cyclic groups of the form: + $$ + \mathbb{Z}_{p_1^{n_1}}\times \mathbb{Z}_{p_2^{n_2}}\times \cdots \times \mathbb{Z}_{p_k^{n_k}}\times\underbrace{\mathbb{Z}\times \ldots \times \mathbb{Z}}_{m\text{ times}} + $$ + +#### Cyclic group + +A group $G$ is cyclic if $G$ is a subgroup generated by $a\in G$. (may be infinite) + +- The smallest non-cyclic group is Klein 4-group (order 4). +- Every group with prime order is cyclic. +- Every cyclic group is abelian. +- If $G$ has order $n$, then $G$ is isomorphic to $(\mathbb{Z}_n,+)$. +- If $G$ is infinite, then $G$ is isomorphic to $(\mathbb{Z},+)$. +- If $G=\langle a\rangle$ and $H=\langle a^k\rangle$, then $|H|=\frac{|G|}{d}$ where $d=\operatorname{gcd}(|G|,|H|)$. +- Every subgroup of cyclic group is also cyclic. + +#### Dihedral group + +The dihedral group $D_n$ is the group of all symmetries of a regular polygon with $n$ sides. + +- $|D_n|=2n$. +- It is finitely generated by $\{\rho,\phi\}$, where $\rho$ is a rotation of a regular polygon by $\frac{2\pi}{n}$, and $\phi$ is a reflection of a regular polygon with respect to $x$-axis. + +#### Symmetric group + +The symmetric group $S_n$ is the group of all permutations of $n$ objects. + +- $S_n$ has order $n!$. +- Every group $G$ is isomorphic to $S_A$ for some $A$. +- Odd and even permutations + - Every permutation can be written as a product of transpositions. + - $A_n$ is the alternating group with order $\frac{n!}{2}$ consisting of all even permutations. + - A non trivial homomorphism from $S_n$ to $(\Z_2,+)$ is given by $\sigma\mapsto \begin{cases} 0 & \text{if } \sigma\text{ is even} \\ 1 & \text{if } \sigma\text{ is odd} \end{cases}$ \ No newline at end of file diff --git a/content/Math4302/Math4302_L13.md b/content/Math4302/Math4302_L13.md index e49524b..8bcee14 100644 --- a/content/Math4302/Math4302_L13.md +++ b/content/Math4302/Math4302_L13.md @@ -12,7 +12,7 @@ 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 diff --git a/content/Math4302/Math4302_L5.md b/content/Math4302/Math4302_L5.md index abf6dc1..c8699c6 100644 --- a/content/Math4302/Math4302_L5.md +++ b/content/Math4302/Math4302_L5.md @@ -80,7 +80,7 @@ $D_n\leq S_n$ ($S_n$ is the symmetric group of $n$ elements). If $H\subseteq G$ is a non-empty subset of a group $G$. -then ($H$ is a subgroup of $G$) if and only if ($a,b\in H\implies ab^-1\in H$). +then ($H$ is a subgroup of $G$) if and only if ($a,b\in H\implies ab^{-1}\in H$).
Proof