Calculator
Formula used
A subgroup generated by a set S is written ⟨S⟩. It is the smallest subgroup containing S, obtained by closure: include the identity, all inverses, and all finite products of elements.
- Closure step: if x,y ∈ H then xy ∈ H.
- Inverse step: if x ∈ H then x⁻¹ ∈ H.
- Index: [G:H] = |G| / |H| for finite groups.
How to use
- Select a group family and enter its parameters.
- Choose “full” lattice or the lattice inside ⟨S⟩.
- Optionally enter generators S to focus the scope.
- Press Submit to display the diagram and tables.
- Use CSV/PDF buttons to export your results.
Example data table
Example for the cyclic group C12. Cyclic groups have exactly one subgroup for each divisor of n.
| Divisor d of 12 | Subgroup order | One generator | Index |
|---|---|---|---|
| 1 | 1 | 0 | 12 |
| 2 | 2 | 6 | 6 |
| 3 | 3 | 4 | 4 |
| 4 | 4 | 3 | 3 |
| 6 | 6 | 2 | 2 |
| 12 | 12 | 1 | 1 |
In additive notation, “generator” means an element whose repeated sums produce the subgroup.