---
# System prepended metadata

title: 'Groups [ch.6]'
tags: [abstract algebra]

---

# Groups [ch.6]

## 6.1. Normal Subgroups and Quotient Groups
- **Definition 6.1: Set of Cosets**
![Screenshot 2024-11-24 at 7.36.44 PM](https://hackmd.io/_uploads/B166ZtemJe.png)
- **Definition 6.5: g-conjugate, normal subgroup**
![Screenshot 2024-11-24 at 7.39.46 PM](https://hackmd.io/_uploads/B1_tfFlQkx.png)
![Screenshot 2024-11-24 at 7.41.40 PM](https://hackmd.io/_uploads/SkrgXKgmyl.png)
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**
![Screenshot 2024-11-24 at 7.40.55 PM](https://hackmd.io/_uploads/HJBAGtlQke.png)
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**
![Screenshot 2024-11-24 at 7.42.04 PM](https://hackmd.io/_uploads/ryhb7Kemyx.png)
- **Proposition 6.10**
![Screenshot 2024-11-24 at 7.43.12 PM](https://hackmd.io/_uploads/Sk7UXYlm1x.png)
- **Lemma 6.11**
![Screenshot 2024-11-24 at 7.44.55 PM](https://hackmd.io/_uploads/HJ92QFe7yx.png)
- **Theorem 6.12**
![Screenshot 2024-11-24 at 7.45.32 PM](https://hackmd.io/_uploads/HJZkEKg7yl.png)

## 6.2. Groups Acting on Sets
- **Definition 6.13: Group Action**
![Screenshot 2024-11-24 at 7.46.30 PM](https://hackmd.io/_uploads/HyImVYeQ1g.png)
![Screenshot 2024-11-24 at 7.47.16 PM](https://hackmd.io/_uploads/rJnBNKgm1x.png)
- **Definition 6.15: Orbit & Stabilizer**
![Screenshot 2024-11-24 at 7.50.20 PM](https://hackmd.io/_uploads/HJ1QBYx71e.png)
![Screenshot 2024-11-24 at 7.50.45 PM](https://hackmd.io/_uploads/r1y7HYgQke.png)
- **Proposition 6.19**
![Screenshot 2024-11-24 at 7.58.48 PM](https://hackmd.io/_uploads/HkbWwteXkg.png =90%x)
Note: 
    - Equivalence Relation have 3 properties, which are **reflexive**, **Symmetric**, and **transitive**. Check proof in the textbook.
- **Definition 6.20: Transitive Action**
![Screenshot 2024-11-24 at 7.55.15 PM](https://hackmd.io/_uploads/B1AXIKlQkl.png)

## 6.3. The Orbit-Stabilizer Counting Theorem
- **Theorem 6.21: Orbit-Stabilizer Counting Theorem**
![Screenshot 2024-11-24 at 7.54.30 PM](https://hackmd.io/_uploads/HyY-UKem1x.png)
- **Definition 6.23: Center of G**
![Screenshot 2024-11-24 at 8.14.49 PM](https://hackmd.io/_uploads/BkRaqFxQ1e.png)
- **Theorem 6.25**
![Screenshot 2024-11-24 at 8.15.28 PM](https://hackmd.io/_uploads/BkUzotlQyg.png)
- **Corollary 6.26**
![Screenshot 2024-11-24 at 8.16.11 PM](https://hackmd.io/_uploads/BkgfiKlQkl.png)
- **Definition 6.27: Centralizer**
![Screenshot 2024-11-24 at 8.16.57 PM](https://hackmd.io/_uploads/BynriKgm1l.png)
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**
![Screenshot 2024-11-24 at 8.31.08 PM](https://hackmd.io/_uploads/H1SjRFx71g.png)

## 6.4. Sylow’s Theorem
- **Theorem 6.29: The First Part of Sylow’s Theorem**
![Screenshot 2024-11-24 at 8.20.32 PM](https://hackmd.io/_uploads/HJbQhFxX1x.png)
- **Lemma 6.30**
![Screenshot 2024-11-24 at 8.21.29 PM](https://hackmd.io/_uploads/HkAH3tl71g.png)
- **Definition 6.31: p-Sylow subgroup of G**
![Screenshot 2024-11-24 at 8.22.01 PM](https://hackmd.io/_uploads/SkaAhKl7Jg.png)
![Screenshot 2024-11-24 at 8.23.36 PM](https://hackmd.io/_uploads/r1gJ6Flm1g.png)
![Screenshot 2024-11-24 at 8.23.43 PM](https://hackmd.io/_uploads/Hkl1TFg7yg.png)
![Screenshot 2024-11-24 at 8.23.51 PM](https://hackmd.io/_uploads/Bygy6KlQJg.png)
- **Definition 6.35: Sylow’s Theorem**
![Screenshot 2024-11-24 at 8.22.47 PM](https://hackmd.io/_uploads/HkQjhFlQke.png)
Note: check example 6.36 for Sylow’s Theore.

## 6.5. Two Counting Lemmas
- **Lemma 6.38**
![Screenshot 2024-11-24 at 8.33.47 PM](https://hackmd.io/_uploads/ryDEk9xXJx.png)
Note: This lemma counts the number of subgroups of $G$ that conjugates to $H$
- **Lemma 6.39**
![Screenshot 2024-11-24 at 9.19.26 PM](https://hackmd.io/_uploads/S1Nkq5eXkg.png)

## 6.6. Double Cosets and Sylow’s Theorem
- **Definition 6.40: Double Coset**
![Screenshot 2024-11-24 at 9.25.22 PM](https://hackmd.io/_uploads/HJ-ws5lQ1e.png)
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**
![Screenshot 2024-11-24 at 9.29.20 PM](https://hackmd.io/_uploads/B12N2ceXyl.png)
- **Theorem 6.42: Sylow Theorem**
![Screenshot 2024-11-24 at 9.29.59 PM](https://hackmd.io/_uploads/SJWd25xmyx.png)
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)
