udpates?

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 $\sigma\mapsto \begin{cases} 0 & \text{if } \sigma\text{ is even} \\ 1 & \text{if } \sigma\text{ is odd} \end{cases}$

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
 

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 ($a,b\in H\implies ab^{-1}\in H$).
 
 <details>
 <summary>Proof</summary>