From 1ee87c63e75884632227be595f8da4e365bf6319 Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Tue, 4 Feb 2025 11:01:52 -0600
Subject: [PATCH] update

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 pages/Swap/Math401/index.md      |  5 +++
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+# Node 1
+
+_all the materials are recovered after the end of the course. I cannot split my mind away from those materials._
+
+## Recap on familiar ideas
+
+### Group
+
+A group is a set $G$ with a binary operation $\cdot$ that satisfies the following properties:
+
+1. **Closure**: For all $a, b \in G$, the result of the operation $a \cdot b$ is also in $G$.
+2. **Associativity**: For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
+3. **Identity**: There exists an element $e \in G$ such that for all $a \in G$, $e \cdot a = a \cdot e = a$.
+4. **Inverses**: For each $a \in G$, there exists an element $b \in G$ such that $a \cdot b = b \cdot a = e$.
+
+### Ring
+
+A ring is a set $R$ with two binary operations, addition and multiplication, that satisfies the following properties:
+
+1. **Additive Closure**: For all $a, b \in R$, the result of the addition $a + b$ is also in $R$.
+2. **Additive Associativity**: For all $a, b, c \in R$, $(a + b) + c = a + (b + c)$.
+3. **Additive Identity**: There exists an element $0 \in R$ such that for all $a \in R$, $0 + a = a + 0 = a$.
+4. **Additive Inverses**: For each $a \in R$, there exists an element $-a \in R$ such that $a + (-a) = (-a) + a = 0$.
+5. **Multiplicative Closure**: For all $a, b \in R$, the result of the multiplication $a \cdot b$ is also in $R$.
+6. **Multiplicative Associativity**: For all $a, b, c \in R$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
+
+Others not shown since you don't need too much.
+
+## Symmetric Group
+
+### Definition
+
+The symmetric group $S_n$ is the group of all permutations of $n$ elements. Or in other words, the group of all bijections from a set of $n$ elements to itself.
+
+Example:
+
+$$
+e=1,2,3\\
+(12)=2,1,3\\
+(13)=3,2,1\\
+(23)=1,3,2\\
+(123)=3,1,2\\
+(132)=2,3,1
+$$
+
+$(12)$ means that $1\to 2, 2\to 1, 3\to 3$ we follows the cyclic order for the elements in the set.
+
+$S_3 = \{e, (12), (13), (23), (123), (132)\}$
+
+The multiplication table of $S_3$ is:
+
+|Element|e|(12)|(13)|(23)|(123)|(132)|
+|---|---|---|---|---|---|---|
+|e|e|(12)|(13)|(23)|(123)|(132)|
+|(12)|(12)|e|(123)|(13)|(23)|(132)|
+|(13)|(13)|(132)|e|(12)|(23)|(123)|
+|(23)|(23)|(123)|(132)|e|(12)|(13)|
+|(123)|(123)|(13)|(23)|(132)|e|(12)|
+|(132)|(132)|(23)|(12)|(123)|(13)|e|
+
+## Functions defined on $S_n$
+
+### Symmetric Generating Set
+
+
+
+
diff --git a/pages/Swap/Math401/index.md b/pages/Swap/Math401/index.md
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+# Math 401
+
+This is a course about symmetrical group and bunch of applications in other fields of math.
+
+Prof. Fere is teaching this course.