# Groups [ch.2]
## 2.1. Introduction to Groups
- **Definition 2.3: Permutation**
Note: A permutation is defined as bijective.

## 2.2. Abstract Groups
- **Definition 2.6: Groups**
- **Proposition 2.9: Basic Properties of Groups**
- **Proposition 2.10: Cancellation Rule for Groups**
- **Definition 2.11: Order, Cardinality of a Group**
- **Definition 2.12**
- **Proposition 2.13**

## 2.3. Interesting Examples of Groups
- **Definition 2.17: cyclic groups**
- **Examples 2.19: Permutation Groups**
- **Remark 2.20**
- **Examples 2.20**
- **Examples 2.21**

## 2.4. Group Homomorphisms
- **Definition 2.24: Group Homomorphisms**
- **Proposition 2.25**
- **Example 2.29: Determinant Map**
- **Definition 2.30: Group Isomorphisms**

## 2.5. Subgroups, Cosets, and Lagrange’s Theorem
- **Definition 2.34: Subgroups**
- **Definition 2.41: Kernel**
- **Proposition 2.44**
- **Definition: 2.45: Coset**
- **Proposition 2.47**
- **Theorem 2.48: Lagrange’s Theorem**
- **Theorem 2.52**
- **Theorem 2.53: Sylow’s Theorem**

## 2.6. Products of Groups
- **Definition: 2.54: Products of Groups**
- **Example 2.55**
- **Theorem 2.56: Structure Theorem for Finite Abelian Groups**
Note: check section 11.9.1 for more details
- **Example 2.57: Projections and Inclusions**
Note: check textbooks for other examples in this chapter