# Groups [ch.6]

## 6.1. Normal Subgroups and Quotient Groups
- **Definition 6.1: Set of Cosets**
- **Definition 6.5: g-conjugate, normal subgroup**
Note:
- Two subgroups $H_1, H_2\in G$ being conjugate means that $H_1=g^{-1}H_2g$ for some $g \in G$
- **Remark 6.7: Simple Group**
Note:
- In mathematics, the word “simple” does not have the usual meaning of uncomplicated or easily understood. Instead, it denotes an object made of only one part; i.e., an object that cannot be decomposed into smaller (simpler) pieces.
- **Proposition 6.9**
- **Proposition 6.10**
- **Lemma 6.11**
- **Theorem 6.12**

## 6.2. Groups Acting on Sets
- **Definition 6.13: Group Action**
- **Definition 6.15: Orbit & Stabilizer**
- **Proposition 6.19**
Note:
- Equivalence Relation have 3 properties, which are **reflexive**, **Symmetric**, and **transitive**. Check proof in the textbook.
- **Definition 6.20: Transitive Action**

## 6.3. The Orbit-Stabilizer Counting Theorem
- **Theorem 6.21: Orbit-Stabilizer Counting Theorem**
- **Definition 6.23: Center of G**
- **Theorem 6.25**
- **Corollary 6.26**
- **Definition 6.27: Centralizer**
Note:
- **Stabilizer**, **center of G**, **cetralizer**, and **normalizer** are subgroup of G. It seems that things end with "er" are subgroup.
- **Definition 6.28: Normalizer**

## 6.4. Sylow’s Theorem
- **Theorem 6.29: The First Part of Sylow’s Theorem**
- **Lemma 6.30**
- **Definition 6.31: p-Sylow subgroup of G**
- **Definition 6.35: Sylow’s Theorem**
Note: check example 6.36 for Sylow’s Theore.

## 6.5. Two Counting Lemmas
- **Lemma 6.38**
Note: This lemma counts the number of subgroups of $G$ that conjugates to $H$
- **Lemma 6.39**

## 6.6. Double Cosets and Sylow’s Theorem
- **Definition 6.40: Double Coset**
Note:
- Only 2 cases
- $H_1g_1H_2=H_1g_2H_2$
- $H_1g_1H_2\cap H_1g_2H_2=\emptyset$
- check [wiki](https://en.wikipedia.org/wiki/Double_coset) for more details
- **Lemma 6.41**
- **Theorem 6.42: Sylow Theorem**
Note: [better proof of Sylow Theorem by MIT](https://ocw.mit.edu/courses/res-18-011-algebra-i-student-notes-fall-2021/mit18_701f21_lect23.pdf)