Group Theory

16.

Solvable and Nilpotent Groups

16.1.

Commutator Subgroups

16.1.1.

Definition of Commutator

16.1.2.

Derived Subgroup

16.1.3.

Properties of Derived Subgroups

16.2.

Solvable Groups

16.2.1.

Definition of Solvable Group

16.2.2.

16.2.3.

Solvable Length

16.2.4.

Examples of Solvable Groups

16.2.5.

Properties of Solvable Groups

16.3.

Nilpotent Groups

16.3.1.

Definition of Nilpotent Group

16.3.2.

Lower Central Series

16.3.3.

Upper Central Series

16.3.4.

Nilpotency Class

16.3.5.

Examples of Nilpotent Groups

16.3.6.

Properties of Nilpotent Groups

16.4.

16.4.1.

Nilpotent Implies Solvable

16.4.2.

p-Groups are Nilpotent

16.4.3.

Finite Nilpotent Groups

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17. Composition Series and Jordan-Hölder Theorem