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Dihedral groups

Below is a clear, simple, and exam-oriented explanation of Dihedral Groups — perfect for BCA/MCA/Engineering/Discrete Mathematics & Abstract Algebra exams.

⭐ DIEDRAL GROUPS (Dn / D2n)

A dihedral group is the group of symmetries of a regular n-sided polygon.

It includes:

✔ Rotations
✔ Reflections

These are rigid motions that keep the shape unchanged.

⭐ 1. Definition

The dihedral group of order 2n (denoted by (D_n) or (D_{2n})) is:

[
D_n = \text{Group of all symmetries of a regular n-gon}
]

The symmetries include:

  • n rotations
  • n reflections

Total:

[
|D_n| = 2n
]

⭐ 2. Elements of Dihedral Group

Let the regular n-gon be centered at the origin.

✔ 1️⃣ Rotations (n of them)

Rotation by:

[
0^\circ,\ \frac{360}{n}^\circ,\ \frac{720}{n}^\circ,\ \ldots,\ (n-1)\frac{360}{n}^\circ
]

Let r be rotation by (360^\circ/n).
Then:

[
r^n = e \quad (\text{identity})
]

So rotation subgroup is cyclic of order n.

✔ 2️⃣ Reflections (n of them)

Let s be a reflection in one axis.

Each reflection can be expressed as:

[
sr^k \quad (k = 0,1,2,\dots,n-1)
]

⭐ 3. Presentation of Dihedral Group (VERY IMPORTANT)

[
D_n = \langle r, s \mid r^n = e,\ s^2 = e,\ srs = r^{-1} \rangle
]

This presentation summarizes the structure:

1️⃣ Rotation repeated n times gives identity
[
r^n = e
]

2️⃣ Reflection squared is identity
[
s^2 = e
]

3️⃣ Conjugating rotation by reflection reverses direction
[
srs = r^{-1}
]

⭐ 4. Multiplication Rules in Dihedral Groups

These rules help compute products:

✔ Rotation × Rotation

[
r^i r^j = r^{i+j}
]

✔ Rotation × Reflection

[
r^i s = sr^{-i}
]

✔ Reflection × Rotation

[
sr^i = r^{-i}s
]

✔ Reflection × Reflection

[
(sr^i)(sr^j) = r^{j-i}
]

⭐ 5. Example: Dihedral Group D₃ (Symmetry of Triangle)

Regular triangle → n=3
[
|D_3| = 6
]

Elements:

  1. Rotations:
    • (e)
    • (r) = 120°
    • (r^2) = 240°
  2. Reflections:
    • (s)
    • (sr)
    • (sr^2)

This group is non-abelian.

⭐ 6. Example: Dihedral Group D₄ (Square)

n=4 → |D4| = 8.

Rotations:

  • (e)
  • (r) = 90°
  • (r^2) = 180°
  • (r^3) = 270°

Reflections:

  • Vertical (s)
  • Horizontal (sr^2)
  • Diagonal (sr)
  • Other diagonal (sr^3)

⭐ 7. Properties of Dihedral Groups

✔ 1️⃣ Order = 2n

Always even.

✔ 2️⃣ Non-abelian

For all (n \ge 3).

✔ 3️⃣ Contains cyclic subgroup

Rotation subgroup:

[
\langle r \rangle \cong \mathbb{Z}_n
]

✔ 4️⃣ Contains n involutions (elements of order 2)

All reflections satisfy:

[
(sr^k)^2 = e
]

✔ 5️⃣ Has 2 conjugacy classes for reflections

Helpful in applications.

✔ 6️⃣ D2 ≅ Klein Four Group

For n=2, D2 is abelian.

⭐ 8. Subgroups of Dihedral Groups

✔ Rotation subgroup

[
\langle r \rangle
]

✔ Reflection subgroups

Each reflection alone generates order-2 subgroup.

✔ Other subgroups (divisors of n)

If (d \mid n):

[
\langle r^{n/d} \rangle
]

⭐ 9. Application of Dihedral Groups

✔ Computer Graphics (rotational symmetries)
✔ Robotics (motion and configurations)
✔ Chemistry (molecular symmetry)
✔ Physics (crystal symmetries)
✔ Coding Theory
✔ Designing patterns and digital geometry
✔ Cryptography (using permutation groups)
✔ Automata theory and semigroup theory

⭐ Exam-Oriented Summary

Definition:

(D_n) is the group of symmetries of a regular n-gon.

Order:

[
|D_n| = 2n
]

Elements:

  • (n) rotations: (e, r, r^2, \dots, r^{n-1})
  • (n) reflections: (s, sr, sr^2, \dots, sr^{n-1})

Presentation:

[
r^n = e,\ s^2 = e,\ srs = r^{-1}
]

Structure:

  • Contains cyclic subgroup ≤
  • Non-abelian for n ≥ 3
  • Many involutions

Examples:

  • D3 (triangle) → 6 elements
  • D4 (square) → 8 elements