Explore group symmetry with orbit and stabilizer tools. Compute sizes, list orbits, and export results. See relationships clearly and learn faster with examples included.
This tool supports both the orbit-stabilizer size relationship and orbit enumeration from permutation generators written as one-line mappings.
These examples illustrate typical integer outputs of the orbit–stabilizer relationship.
| Scenario | Group order |G| | Orbit size |Orb(x)| | Stabilizer size |Stab(x)| | Interpretation |
|---|---|---|---|---|
| Permutations of 3 symbols, fix one symbol | 6 | 3 | 2 | Orbit is all symbols; stabilizer swaps remaining two |
| Square symmetries acting on vertices | 8 | 4 | 2 | Two symmetries keep a chosen vertex fixed |
| Rotations of 12-gon acting on vertices | 12 | 12 | 1 | Only the identity rotation fixes a vertex |
For a group action of G on a set and an element x, the orbit is the set of values reachable from x. The stabilizer is the subset of group elements that keep x unchanged.
The orbit enumerator uses one-line permutations. Each generator maps i → p[i], and the orbit is built by repeatedly applying generators.
In a finite group action, an orbit collects every image of an element x under all symmetries. The stabilizer is the set of symmetries that keep x fixed, measuring symmetry that remains when x is preserved. These views are complementary: orbits describe motion across the set, while stabilizers describe constraints at a point. For actions on permutations, the orbit is a subset of {1,…,n}.
The calculator applies the identity |G| = |Orb(x)| × |Stab(x)| and rearranges it to solve any missing quantity. If |G| = 24 and |Stab(x)| = 4, then |Orb(x)| = 6, and 24 = 6 × 4 verifies consistency. If you instead measure an orbit size of 8, the implied stabilizer becomes 3, suggesting a different structure. Because sizes in finite actions are integers, noninteger outputs signal rechecking inputs.
When you provide generators as one line mappings, the tool repeatedly applies each generator to every discovered element and records any new images. This breadth first exploration continues until no new elements appear, producing the orbit for the subgroup generated by the supplied permutations. The output includes a sorted list, a count, and an optional stabilizer estimate using |G|/|Orb(x)|. The orbit limit setting, default 500, prevents runaway exploration for large n.
A noninteger stabilizer estimate usually indicates that the group order entered is not the true order of the acting group, or that the orbit was computed for a smaller generated subgroup. It can also happen if a permutation line is not a bijection, which input validation catches by rejecting duplicates or out of range values. Practical fixes include confirming the intended action, adding generators, or recomputing |G| from two reliable sizes.
Use the size solver for textbook exercises, symmetry counting, and quick checks in combinatorics. Use the enumerator to study reachability in permutation groups, compare generating sets, and identify invariant blocks when disjoint orbits appear. Exported CSV supports spreadsheets, while PDF snapshots support reports and assignments. Together these outputs help document assumptions, reproduce calculations, and explain results clearly. It also supports auditing answers during revision sessions.
Orbit size counts how many distinct elements you can reach from x using the group action. In permutation actions, it is the number of positions x can move to under the generated symmetries.
Enter |G| when you know the size of the full acting group and want a stabilizer estimate or to solve sizes directly. If you only know generators, you can still list the orbit without |G|.
Provide a one line mapping of length n, where the i-th number is the image of i. Values must be a rearrangement of 1 through n, separated by spaces or commas.
Some inputs generate very large orbits, especially when n is big. The limit stops excessive computation and protects the page. Increase it only when you expect the true orbit to be larger.
Noninteger estimates usually mean |G| does not match the group that produced the orbit, or the orbit comes from a smaller generated subgroup. Verify the correct group order and generators, then recompute.
Exports capture the results table shown after calculation, including sizes, lists, and notes. CSV is convenient for spreadsheets, while PDF provides a formatted snapshot for sharing or printing.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.