Quotient group

🔗

Ultimate explanation: https://math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426

🔗

Does not have to be isomorphic to a subgroup:

This is one of the reasons why the analogy between simple groups of finite groups and prime numbers is limited.

🔗

Table of contents | 51, 2k, 33
- 1. Subquotient | 64 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
- 2. Relationship between the quotient group and direct products | 308 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
- 3. Normal subgroup | 149, 1k, 30 | 🔗 link | 🗖 nosplit | ↑ parent "Quotient group"
  - 3.1. Simple group | 48, 81, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Normal subgroup"
    - 3.1.1. How to show that a group is simple | 33 | 🔗 link | 🗖 nosplit | ↑ parent "Simple group"
  - 3.2. Classification of finite groups | 69, 1k, 27 | 🔗 link | 🗖 nosplit | ↑ parent "Normal subgroup"
    - 3.2.1. List of finite groups | 4, 13, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"
      - 3.2.1.1. GroupNames | 9 | 🔗 link | 🗖 nosplit | ↑ parent "List of finite groups"
    - 3.2.2. Classification of finite simple groups | 186, 949, 23 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"
      - 3.2.2.1. Group of Lie type | 199, 267, 2 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
        
        3.2.2.1.1. Chevalley group | 0, 68, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Group of Lie type"
        
        3.2.2.1.1.1. Chevalley groups $A_{n} (q)$ | 68 | 🔗 link | 🗖 nosplit | ↑ parent "Chevalley group"
      - 3.2.2.2. Sporadic group | 8, 482, 17 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
        
        3.2.2.2.1. Mathieu group | 239, 427, 13 | 🔗 link | 🗖 nosplit | ↑ parent "Sporadic group"
        
        3.2.2.2.1.1. k-transitive group | 19, 174, 7 | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.2.2.2.1.1.1. Classification of k-transitive groups | 22, 155, 6 | 🔗 link | 🗖 nosplit | ↑ parent "k-transitive group"
        
        3.2.2.2.1.1.1.1. 2-transitive group | 0, 0, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.2.2.2.1.1.1.1.1. Classification of 2-transitive groups | 🔗 link | 🗖 nosplit | ↑ parent "2-transitive group"
        
        3.2.2.2.1.1.1.2. Classification of 3-transitive groups | 9 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.2.2.2.1.1.1.3. Classification of 4-transitive groups | 19 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.2.2.2.1.1.1.4. Classification of 5-transitive groups | 95 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.2.2.2.1.1.1.5. Classification of 6-transitive groups | 10 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of k-transitive groups"
        
        3.2.2.2.1.2. Mathieu group $M_{1} 1$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.2.2.2.1.3. Mathieu group $M_{1} 2$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.2.2.2.1.4. Mathieu group $M_{2} 2$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.2.2.2.1.5. Mathieu group $M_{2} 3$ | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.2.2.2.1.6. Mathieu group $M_{2} 4$ | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Mathieu group"
        
        3.2.2.2.2. Janko group | 🔗 link | 🗖 nosplit | ↑ parent "Sporadic group"
        
        3.2.2.2.3. Monster group | 33, 47, 1 | 🔗 link | 🗖 nosplit | ↑ parent "Sporadic group"
        
        3.2.2.2.3.1. Monstrous moonshine | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Monster group"
      - 3.2.2.3. Jordan-Holder Theorem | 14 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
      - 3.2.2.4. Composition series | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite simple groups"
    - 3.2.3. Group extension problem | 110 | 🔗 link | 🗖 nosplit | ↑ parent "Classification of finite groups"

🔗

Ancestors

Subgroup | 0, 2k, 35
Group | 0, 5k, 89
Algebra | 0, 8k, 171
Mathematics | 17, 28k, 633
Ciro Santilli's Homepage | 262, 218k, 4k

🔗

Incoming links

Fundamental theorem on homomorphisms | 31
Normal subgroup | 149, 1k, 30
Projective linear group | 19, 48, 5
Relationship between the quotient group and direct products | 308