Symmetry Group Identifier

Knowing your symmetry group tells you which equivariant architecture patterns to use. This skill formalizes identified transformations into the language of group theory.

Workflow

Copy this checklist and track your progress:

Group Identification Progress:
- [ ] Step 1: List symmetries from discovery phase
- [ ] Step 2: Classify each as discrete or continuous
- [ ] Step 3: Match to specific groups using taxonomy
- [ ] Step 4: Determine how groups combine
- [ ] Step 5: Verify group properties
- [ ] Step 6: Document final group specification

Step 1: List symmetries from discovery phase

Gather the identified symmetries from the discovery phase. List each identified transformation and whether it requires invariance or equivariance. Note confidence levels. If symmetries haven't been discovered yet, work with user to identify them through domain analysis first.

Step 2: Classify each as discrete or continuous

For each symmetry, determine: Is the transformation set finite (discrete) or infinite (continuous)? Discrete examples: 90° rotations (4 elements), permutations of n items (n! elements). Continuous examples: rotation by any angle, translation by any distance. Use Group Taxonomy to guide classification. For mathematical foundations, see Group Theory Primer.

Step 3: Match to specific groups using taxonomy

Use the Discrete Groups and Continuous Groups reference sections. Identify the specific group name and notation for each symmetry. Common matches: n-fold rotation → Cₙ, rotation+reflection → Dₙ, permutation → Sₙ, 3D rotation → SO(3), rigid motion → SE(3), full Euclidean → E(3). For detailed Lie group information (SO(3), SE(3), E(3)), consult Lie Groups Reference.

Step 4: Determine how groups combine

If multiple symmetries are present, determine how they combine. Direct product (G × H): symmetries act independently. Semidirect product (G ⋊ H): one symmetry "twists" the other (e.g., SE(3) = SO(3) ⋊ ℝ³). Use Combining Groups reference.

Step 5: Verify group properties

Check that identified structure satisfies group axioms: closure, associativity, identity, inverses. Verify important properties: Is it compact? (affects representation theory). Is it abelian? (commutative or not). Is it connected? (affects implementation). Use Group Properties Checklist. For detailed verification methodology, see Methodology.

Step 6: Document final group specification

Create specification using Output Template. Include: group name/notation, dimension/size, key properties, invariance vs equivariance requirements, and recommended architecture family. This specification provides the foundation for architecture design. Quality criteria for this output are defined in Quality Rubric.

Group Taxonomy

Overview Diagram

                    SYMMETRY GROUPS
                          │
          ┌───────────────┴───────────────┐
          │                               │
     DISCRETE                        CONTINUOUS
          │                          (Lie Groups)
          │                               │
    ┌─────┼─────┐               ┌────────┼────────┐
    │     │     │               │        │        │
  Cyclic Dihedral Symmetric   SO(n)   SE(n)    E(n)
   Cₙ     Dₙ      Sₙ         rotations rigid   Euclidean
                              only    motions  (w/ reflect)

Quick Reference Table

Symmetry Type	Group	Notation	Elements	Common Use
n-fold rotation	Cyclic	Cₙ	n	Image rotation (90°, 60°)
Rotation + reflection	Dihedral	Dₙ	2n	Regular polygons
Permutation	Symmetric	Sₙ	n!	Sets, graphs
2D rotation (continuous)	Special orthogonal	SO(2)	∞	Continuous rotation
3D rotation	Special orthogonal	SO(3)	∞	3D orientation
3D rigid motion	Special Euclidean	SE(3)	∞	Robotics, molecules
3D with reflections	Euclidean	E(3)	∞	Chemistry, physics

Discrete Groups

Cyclic Groups (Cₙ)

What they represent: Rotations by multiples of 360°/n

Elements: {e, r, r², ..., rⁿ⁻¹} where rⁿ = e (identity)

Group	Rotations	Example
C₂	0°, 180°	Playing cards
C₄	0°, 90°, 180°, 270°	Square images
C₆	60° increments	Hexagonal patterns

Use when: Rotation symmetry present but NOT reflection symmetry.

Dihedral Groups (Dₙ)

What they represent: Rotations + reflections of regular n-gon

Elements: n rotations + n reflections = 2n total

Group	Elements	Example
D₄	8	Square with diagonals (p4m group)
D₆	12	Regular hexagon

Use when: Both rotation AND reflection symmetry present.

Symmetric Groups (Sₙ)

What they represent: All permutations of n elements

Elements: n! permutations

Use when: Element ordering is arbitrary (sets, graphs, point clouds).

Continuous Groups (Lie Groups)

SO(2) - 2D Rotations

Elements: Rotation by any angle θ ∈ [0, 2π)

Matrix form: R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]]

Use when: Continuous rotation symmetry in 2D.

SO(3) - 3D Rotations

Elements: All rotations in 3D (3 degrees of freedom)

Representations: Rotation matrices, quaternions, Euler angles, axis-angle

Use when: 3D orientation doesn't matter, but handedness does.

SE(3) - 3D Rigid Motions

Elements: Rotations + Translations in 3D

Structure: SE(3) = SO(3) ⋊ ℝ³ (semidirect product)

Use when: Objects can be anywhere and in any orientation, handedness matters.

E(3) - Full Euclidean Group

Elements: SE(3) + Reflections

Structure: E(3) = O(3) ⋊ ℝ³

Use when: SE(3) symmetry PLUS reflection symmetry (most molecules).

Group Hierarchy

E(3) = O(3) ⋊ ℝ³
    │ exclude reflections
    ▼
SE(3) = SO(3) ⋊ ℝ³
    │ exclude translations
    ▼
SO(3)
    │ 2D restriction
    ▼
SO(2)

Combining Groups

Direct Product (G × H)

When to use: Symmetries act independently (neither affects the other).

Example: Image with separate translation and color permutation → SE(2) × S₃

Property: (g₁, h₁) · (g₂, h₂) = (g₁g₂, h₁h₂)

Semidirect Product (G ⋊ H)

When to use: One symmetry "twists" the other (don't commute).

Example: SE(3) = SO(3) ⋊ ℝ³ (rotating then translating ≠ translating then rotating)

Common cases: SE(n) = SO(n) ⋊ ℝⁿ, E(n) = O(n) ⋊ ℝⁿ, Dₙ = Cₙ ⋊ C₂

Group Properties Checklist

For your identified group, verify:

Property	Question	Why It Matters
Compact	Is the group "bounded"?	Affects representation theory
Abelian	Does order matter? (g₁g₂ = g₂g₁?)	Simplifies architecture
Connected	Is group in one piece?	Affects irreducible representations
Finite	Finite number of elements?	Discrete vs continuous architecture

Group Selection by Domain

Domain	Typical Group	Notes
2D Image Classification	C₄ or D₄	p4 or p4m groups
3D Molecular Energy	E(3) × Sₙ	Full Euclidean + atom permutation
3D Molecular Chirality	SE(3) × Sₙ	No reflections
Point Cloud Classification	SO(3) × Sₙ	Rotation + permutation
Graph Classification	Sₙ	Permutation invariant
Robotics	SE(3)	Sometimes with gravity constraint

Output Template

SYMMETRY GROUP SPECIFICATION
============================

Identified Symmetries:
1. [Symmetry] → Group: [name] ([notation])
2. [Symmetry] → Group: [name] ([notation])

Combined Group Structure:
- Full group: [G₁ × G₂] or [G₁ ⋊ G₂]
- Size: [# elements] or [continuous]

Group Properties:
- Compact: [Yes/No]
- Abelian: [Yes/No]
- Connected: [Yes/No]

Symmetry Requirements:
- [Group]: [Invariant/Equivariant] for [task type]

Recommended Architecture Family:
- [Architecture] supporting [group]

NEXT STEPS:
- Empirically validate symmetry hypotheses if not yet confirmed
- Design equivariant architecture based on group specification