Maps identified symmetries to mathematical groups (cyclic, dihedral, symmetric, SO(3), SE(3), E(3)) for equivariant neural network architecture design, using taxonomy and foundations from Visual Group Theory. Use when candidate symmetries have been identified and need formalization into group theory language, or when user mentions cyclic groups, dihedral groups, Lie groups, SO(3), SE(3), or permutation groups.
Knowing your symmetry group tells you which equivariant architecture patterns to use. This skill formalizes identified transformations into the language of group theory.
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 to guide classification. For mathematical foundations, see .
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.
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)
| 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 |
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.
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.
What they represent: All permutations of n elements
Elements: n! permutations
Use when: Element ordering is arbitrary (sets, graphs, point clouds).
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.
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.
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.
Elements: SE(3) + Reflections
Structure: E(3) = O(3) ⋊ ℝ³
Use when: SE(3) symmetry PLUS reflection symmetry (most molecules).
E(3) = O(3) ⋊ ℝ³
│ exclude reflections
▼
SE(3) = SO(3) ⋊ ℝ³
│ exclude translations
▼
SO(3)
│ 2D restriction
▼
SO(2)
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)
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
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 |
| 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 |
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