Group Theory

3.

Fundamental Examples of Groups

3.1.

Groups of Numbers

3.1.1.

Integers under Addition

3.1.1.1.

Structure of (ℤ, +)

3.1.1.2.

Properties and Subgroups

3.1.2.

Rational Numbers under Addition

3.1.2.1.

Structure of (ℚ, +)

3.1.3.

Non-zero Rational Numbers under Multiplication

3.1.3.1.

Structure of (ℚ*, ×)

3.1.4.

Real Numbers under Addition

3.1.4.1.

Structure of (ℝ, +)

3.1.5.

Non-zero Real Numbers under Multiplication

3.1.5.1.

Structure of (ℝ*, ×)

3.1.6.

Complex Numbers under Addition

3.1.6.1.

Structure of (ℂ, +)

3.1.7.

Non-zero Complex Numbers under Multiplication

3.1.7.1.

Structure of (ℂ*, ×)

3.1.8.

Integers Modulo n under Addition

3.1.8.1.

Definition of Modular Addition

3.1.8.2.

Structure of ℤ/nℤ

3.1.8.3.

Cayley Tables for Small n

3.1.9.

Group of Units Modulo n under Multiplication

3.1.9.1.

Definition of Units

3.1.9.2.

Structure of (ℤ/nℤ)ˣ

3.1.9.3.

Euler's Totient Function

3.1.9.4.

Computing Units

3.2.

Groups of Symmetries

3.2.1.

Definition of Symmetry Group

3.2.2.

Geometric Transformations

3.2.2.1.

3.2.2.2.

3.2.2.3.

Compositions of Transformations

3.2.3.

The Dihedral Groups

3.2.3.1.

Definition and Presentation

3.2.3.2.

General Structure of Dₙ

3.2.3.3.

Symmetries of Regular Polygons

3.2.3.4.

Symmetries of a Triangle (D₃)

3.2.3.4.1.

Elements and Structure

3.2.3.4.2.

3.2.3.5.

Symmetries of a Square (D₄)

3.2.3.5.1.

Elements and Structure

3.2.3.5.2.

3.2.3.6.

Symmetries of Regular Pentagon (D₅)

3.3.

3.3.1.

General Linear Group

3.3.1.1.

Definition of GL(n, ℝ)

3.3.1.2.

Properties and Structure

3.3.1.3.

Determinant and Invertibility

3.3.2.

Special Linear Group

3.3.2.1.

Definition of SL(n, ℝ)

3.3.2.2.

Properties and Structure

3.3.2.3.

Relationship to GL(n, ℝ)

3.3.3.

Orthogonal Group

3.3.3.1.

Definition of O(n)

3.3.3.2.

Properties and Structure

3.3.3.3.

Preservation of Inner Products

3.3.4.

Special Orthogonal Group

3.3.4.1.

Definition of SO(n)

3.3.4.2.

Properties and Structure

3.3.4.3.

Rotations in n-dimensions

3.3.5.

Matrix Groups over Other Fields

3.3.5.1.

3.3.5.2.

3.3.5.3.

GL(n, ℤ/pℤ)

3.4.

The Klein Four-Group

3.4.1.

Definition and Structure

3.4.2.

3.4.3.

Geometric Interpretation

3.4.4.

Subgroup Structure

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2. Definition of a Group

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4. Order of Elements and Groups