Subgroups and Normal Subgroups
Definition of Subgroup
Subgroup Test
Examples
Example 1: for any .
Example 2: (matrices with determinant ).
Example 3: (cyclic subgroup generated by ).
Cosets
Lagrange's Theorem
Normal Subgroups
Properties of Normal Subgroups
Conjugacy Classes
where is the centralizer of .
Center of a Group
Class Equation
Applications
Key Results:
- (alternating group is normal in symmetric group)
- Commutator subgroup
- Sylow subgroups provide structural information
- Simple groups (no non-trivial normal subgroups) are building blocks