---
# System prepended metadata

title: 'Groups [ch.2]'
tags: [abstract algebra]

---

# Groups [ch.2]
## 2.1. Introduction to Groups
- **Definition 2.3: Permutation**
![Screenshot 2024-11-24 at 4.41.16 PM](https://hackmd.io/_uploads/H1WgtUg7Jg.png)
Note: A permutation is defined as bijective.

## 2.2. Abstract Groups
- **Definition 2.6: Groups**
![Screenshot 2024-11-24 at 4.46.08 PM](https://hackmd.io/_uploads/BJeCtIemkg.png)
![Screenshot 2024-11-24 at 4.47.12 PM](https://hackmd.io/_uploads/H1gM5Ie7kx.png)
- **Proposition 2.9: Basic Properties of Groups**
![Screenshot 2024-11-24 at 4.47.57 PM](https://hackmd.io/_uploads/Skd858xQke.png)
- **Proposition 2.10: Cancellation Rule for Groups**
![Screenshot 2024-11-24 at 4.48.45 PM](https://hackmd.io/_uploads/ryPK5IlXkx.png)
- **Definition 2.11: Order, Cardinality of a Group**
![Screenshot 2024-11-24 at 4.51.30 PM](https://hackmd.io/_uploads/SJzGjIx7yx.png)
- **Definition 2.12**
![Screenshot 2024-11-24 at 4.52.02 PM](https://hackmd.io/_uploads/rJJ8oLgXJg.png)
- **Proposition 2.13**
![Screenshot 2024-11-24 at 4.53.28 PM](https://hackmd.io/_uploads/ByuKjLlmkx.png)

## 2.3. Interesting Examples of Groups
- **Definition 2.17: cyclic groups**
![Screenshot 2024-11-24 at 4.54.33 PM](https://hackmd.io/_uploads/H1gRsUgm1g.png)
- **Examples 2.19: Permutation Groups**
![Screenshot 2024-11-24 at 4.56.33 PM](https://hackmd.io/_uploads/SyH8hLlXkg.png)
- **Remark 2.20**
![Screenshot 2024-11-24 at 4.57.17 PM](https://hackmd.io/_uploads/S1JF2UgmJe.png)
- **Examples 2.20**
![Screenshot 2024-11-24 at 5.01.51 PM](https://hackmd.io/_uploads/r1JKaUgXJe.png)
- **Examples 2.21**![Screenshot 2024-11-24 at 4.59.34 PM](https://hackmd.io/_uploads/SkqWT8l7yx.png)

## 2.4. Group Homomorphisms
- **Definition 2.24: Group Homomorphisms**
![Screenshot 2024-11-24 at 5.02.43 PM](https://hackmd.io/_uploads/Hkpp6Ux7kg.png)
- **Proposition 2.25**
![Screenshot 2024-11-24 at 5.02.52 PM](https://hackmd.io/_uploads/Sk50aUlQJl.png)
- **Example 2.29: Determinant Map**
![Screenshot 2024-11-24 at 5.05.33 PM](https://hackmd.io/_uploads/BJnD0UeXJl.png)
- **Definition 2.30: Group Isomorphisms**
![Screenshot 2024-11-24 at 5.04.29 PM](https://hackmd.io/_uploads/rJF4A8gX1x.png)
![Screenshot 2024-11-24 at 5.06.26 PM](https://hackmd.io/_uploads/BkwqC8lQJe.png)
![Screenshot 2024-11-24 at 5.06.59 PM](https://hackmd.io/_uploads/rkPh0Llmyx.png)

## 2.5. Subgroups, Cosets, and Lagrange’s Theorem
- **Definition 2.34: Subgroups**
![Screenshot 2024-11-24 at 5.08.52 PM](https://hackmd.io/_uploads/SksXkPeXkg.png)
![Screenshot 2024-11-24 at 5.09.31 PM](https://hackmd.io/_uploads/SkTHyPg7Jl.png)
- **Definition 2.41: Kernel**
![Screenshot 2024-11-24 at 5.11.15 PM](https://hackmd.io/_uploads/ByXh1wlQ1e.png)
- **Proposition 2.44**
![Screenshot 2024-11-24 at 5.12.06 PM](https://hackmd.io/_uploads/HJq1lPlQyl.png)
- **Definition: 2.45: Coset**
![Screenshot 2024-11-24 at 5.13.09 PM](https://hackmd.io/_uploads/BJq7xwg7kg.png)
- **Proposition 2.47**
![Screenshot 2024-11-24 at 5.14.37 PM](https://hackmd.io/_uploads/SybYlvlQJe.png)
- **Theorem 2.48: Lagrange’s Theorem**
![Screenshot 2024-11-24 at 5.15.15 PM](https://hackmd.io/_uploads/HyNhewxQ1x.png)
- **Theorem 2.52**
![Screenshot 2024-11-24 at 5.16.18 PM](https://hackmd.io/_uploads/BJhgbDlm1l.png)
- **Theorem 2.53: Sylow’s Theorem**
![Screenshot 2024-11-24 at 5.17.06 PM](https://hackmd.io/_uploads/BJvGWDgQyx.png)

## 2.6. Products of Groups
- **Definition: 2.54: Products of Groups**
![Screenshot 2024-11-24 at 5.20.17 PM](https://hackmd.io/_uploads/H1S0bvgXkx.png)
- **Example 2.55**
![Screenshot 2024-11-24 at 5.22.02 PM](https://hackmd.io/_uploads/HJnSfDxQyx.png)
- **Theorem 2.56: Structure Theorem for Finite Abelian Groups**
![Screenshot 2024-11-24 at 5.24.24 PM](https://hackmd.io/_uploads/rkjTGPgQkl.png)
Note: check section 11.9.1 for more details
- **Example 2.57: Projections and Inclusions**
![Screenshot 2024-11-24 at 5.26.16 PM](https://hackmd.io/_uploads/rJNBmweQJl.png)
Note: check textbooks for other examples in this chapter