Update Math4302_L13.md

content/Math4302/Math4302_L13.md

@@ -1,2 +1,143 @@
 # Math4302 Modern Algebra (Lecture 13)
 
+## Groups
+
+### Cosets
+
+Last time we see that (left coset) $a\sim b$ (to differentiate from right coset, we may denote it as $a\sim_L b$) by $a^{-1}b\in H$ defines an equivalence relation.
+
+#### 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 $a\sim b$.
+
+<details>
+<summary>Proof</summary>
+
+If $aH=bH$, then since $a\in aH, a\in bH$, then for some $h$, $a=bh$, since $b^{-1}a\in H$, so $a^{-1}b\in H$, therefore $a\simeq b$.
+
+If $a\sim b$, then $aH\subseteq bH$, since anything in $aH$ is related to $a$, therefore it is related to $b$ so $a\in bH$.
+
+$bH\subseteq aH$, apply the reflexive property for equivalence relation, therefore $b\in aH$.
+
+So $aH=bH$.
+
+</details>
+
+If $aH\cap bH\neq \emptyset$, then $aH=bH$.
+
+<details>
+<summary>Proof</summary>
+If $x\in aH\cap bH$, then $x\sim a$ and $x\sim b$, so $a\sim b$, so $aH=bH$.
+
+</details>
+
+$aH=H$ if and only if $a\in H$.
+
+<details>
+<summary>Proof</summary>
+$aH=eH$ if and only if $a\sim e$, if and only if $a\in H$.
+
+</details>
+
+$aH$ is called **left coset** of $a$ in $H$.
+
+<details>
+<summary>Examples</summary>
+
+Consider $G=S_3=\{e,\rho,\rho^2,\tau_1,\tau_2,\tau_3\}$.
+
+where $\rho=(123),\rho^2=(132),\tau_1=(12),\tau_2=(23),\tau_3=(13)$.
+
+$H=\{e,\rho,\rho^2\}$.
+
+All the left coset for $H$ is $H=eH=\rho H=\rho^2H$.
+
+$$
+\tau_1\rho=(23)=\tau_2\\
+\tau_1\rho^2=(13)=\tau_3\\
+\tau_2\rho=(31)=\tau_3\\
+\tau_2\rho^2=(12)=\tau_1
+\tau_3\rho=(12)=\tau_1\\
+\tau_3\rho^2=(23)=\tau_2
+$$
+
+$$
+\tau_1H=\{\tau_1,\tau_2,\tau_3\}=\tau_2H=\tau_3H\\
+$$
+
+---
+
+Consider $G=\mathbb{Z}$ with $H=5\mathbb{Z}$.
+
+We have 5 cosets, $H,1+H,2+H,3+H,4+H$.
+
+</details>
+
+#### 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 $G$)
+
+<details>
+<summary>Proof</summary>
+
+Suppose $H$ has $r$ distinct cosets, then $|G|=r|H|$, so $|H|$ divides $|G|$.
+
+</details>
+
+#### Corollary for Lagrange's Theorem
+
+If $|G|=p$, where $p$ is a prime number, then $G$ is cyclic.
+
+<details>
+<summary>Proof</summary>
+
+Prick $e\neq a\in G$, let $H=\langle a\rangle \leq G$, then $|H|$ divides $|G|$, since $p$ is prime, then $|H|=|G|$, so $G=\langle a \rangle$.
+
+</details>
+
+If $G$ is finite and $a\in G$, then $\operatorname{ord}(a)\big\vert|G|$.
+
+<details>
+<summary>Proof</summary>
+
+Since $\operatorname{ord}(a)=|\langle a\rangle|$, and $\langle a\rangle $ is a subgroup, so $\operatorname{ord}(a)\big\vert|G|$.
+
+</details>
+
+#### Definition of index
+
+Suppose $H\leq G$, the number of distinct left cosets of $H$ is called the index of $H$ in $G$. Notation is $(G:H)$.
+
+#### Definition of right coset
+
+Suppose $H\leq G$, define the equivalence relation by $a\sim 'b$ (or $a\sim_R b$ in some textbook) if $a b^{-1}\in H$. (note the in left coset, we use $a^{-1}b \in H$, or equivalently $b^{-1}a \in H$, these are different equivalence relations)
+
+The equivalent class is defined
+
+$$
+Ha=\{x\in G|x\sim'a\}=\{x\in G|xa^{-1}\in H\}=\{x|x=ha\text{ for some }h\in H\}
+$$
+
+Some properties are the same as the left coset
+
+- $Ha=H\iff a\in H$
+- $Ha=Hb$ if and only if $a\sim'b\iff a b^{-1}\in H$.
+- $Ha\cap Hb\neq \emptyset\iff Ha=Hb$.
+
+Some exercises: Find all the left and right cosets of $G=S_3$, there should be 2 left cosets and 2 right cosets (giving different partition of $G$).