Quotient Groups

Quotient Groups

Definition of Quotient Group

Well-Defined Operation

The operation is well-defined because is normal: if and , then .

Properties

Examples

Example 1: (cyclic group of order ).

Example 2: (circle group) via .

Example 3: via the determinant.

First Isomorphism Theorem

Proof Outline

Define by . This is:

1. Well-defined: if , then , so , hence
2. A homomorphism:
3. Injective:
4. Surjective: follows from being surjective onto its image

Second Isomorphism Theorem

for and .

Third Isomorphism Theorem

for .

Correspondence Theorem

Simple Groups

Classification

• Cyclic groups ( prime) are simple
• is simple for
• Lie groups of certain types are simple
• Classification of finite simple groups (completed 2004)