symmetry-group-identifier

# Symmetry Group Identifier

## What Is It?

This skill helps you **map identified symmetries to mathematical groups**. Once you know what transformations should leave your predictions unchanged, this skill formalizes them into the language of group theory.

**Why groups matter**: Neural network architectures are built around specific symmetry groups. Knowing your group tells you exactly which architecture patterns to use.

## 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](#group-taxonomy) to guide classification. For mathematical foundations, see [Group Theory Primer](./resources/group-theory-primer.md).

**Step 3: Match to specific groups using taxonomy**

Use the [Discrete Groups](#discrete-groups) and [Continuous Groups](#continuous-groups-lie-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). Fo