Stabilizer Calculator Form
Example Data Table
| Case | Degree | Generators | Target | Group Order | Orbit | Stabilizer Order |
|---|---|---|---|---|---|---|
| Symmetric action on three points | 3 | (1 2 3), (1 2) | 1 | 6 | {1, 2, 3} | 2 |
| Independent swaps on four points | 4 | (1 2), (3 4) | 1 | 4 | {1, 2} | 2 |
| Cyclic action on four points | 4 | (1 2 3 4) | 2 | 4 | {1, 2, 3, 4} | 1 |
Formula Used
This calculator treats the input as a permutation group action on the set
{1, 2, ..., n}.
The stabilizer of a point x is:
Stab(x) = { g in G : g(x) = x }
The orbit of x is:
Orb(x) = { g(x) : g in G }
For an exact closure, the calculator checks the orbit-stabilizer theorem:
|G| = |Orb(x)| × |Stab(x)|
It also measures point stabilizer sizes, full-action fixed points, orbit partitions, and the average support size, which is the average number of points moved by each subgroup element.
How to Use This Calculator
- Choose the degree of the action. This defines the set
{1, ..., n}. - Enter one or more generators in cycle notation, one per line or separated by semicolons.
- Select the target element whose stabilizer you want.
- Set a closure cap large enough for the generated subgroup.
- Press Compute Stabilizer to view the subgroup order, orbit, stabilizer order, theorem check, orbit partition, and graph.
- Use the export buttons to save your result as CSV or PDF.
FAQs
1) What does this stabilizer calculator compute?
It computes the stabilizer subgroup of a chosen point under the subgroup generated by your permutations. It also returns the orbit, subgroup order, orbit partition, fixed points, and theorem check.
2) Which stabilizer concept does this page use?
This page uses the group-action stabilizer from abstract algebra. It is the set of all generated permutations that keep the chosen target element unchanged.
3) How should I enter generators?
Use standard cycle notation such as (1 2 3), (1 2)(3 4), or id. You can place one generator per line or separate several with semicolons.
4) Why is there a maximum closure size?
Some generated subgroups become large very quickly. The cap prevents long computations and memory spikes. If results are partial, raise the limit and recompute.
5) What does the orbit-stabilizer check mean?
For an exact subgroup closure, the theorem states that subgroup order equals orbit size times stabilizer order. A successful check confirms the computation is internally consistent.
6) What are full-action fixed points?
These are points fixed by every subgroup element, not just by one generator. They reveal positions that remain unchanged under the whole generated action.
7) What does the graph show?
The graph plots the stabilizer size for each point in the action set. Larger bars mean more subgroup elements fix that point.
8) When is the result exact?
The result is exact when the subgroup closes before the selected element cap is reached. If the cap is hit, the page labels the output as partial.