Calculator
Cyclic: integers modulo n, such as
4, 6.Dihedral:
e, r, r^2, s, rs, r^3s, and also forms like sr^2.Klein four:
e, a, b, c.Quaternion:
1, -1, i, -i, j, -j, k, -k.
Formula Used
The normal closure of a subset A in a group G is
the smallest normal subgroup containing A.
Algebraically,
NclG(A) = ⟨ g a g⁻¹ | g ∈ G, a ∈ A ⟩.
The calculator first computes every conjugate gag⁻¹.
It then generates the subgroup formed by those conjugates.
For abelian groups, conjugation changes nothing, so the normal closure equals the ordinary subgroup generated by the chosen subset.
How to Use This Calculator
- Select a supported finite group family.
- Enter the group parameter
nwhen the family needs it. - Type one or more subset elements, separated by commas.
- Press Calculate Normal Closure.
- Review the conjugates, subgroup size, normal closure, and index.
- Use the CSV or PDF buttons to export the result.
- Inspect the graph to compare ambient size and closure size.
Example Data Table
| Group | Subset | Normal Closure | Closure Order | Notes |
|---|---|---|---|---|
| C₁₂ | 4, 6 | {0, 2, 4, 6, 8, 10} | 6 | Abelian case, so closure equals generated subgroup. |
| D₅ | s | Whole group D₅ | 10 | A reflection normally generates the full group here. |
| D₆ | r^3 | {e, r^3} | 2 | The half-turn is central, so its closure stays small. |
| V₄ | a | {e, a} | 2 | Every subgroup is normal in this abelian group. |
| Q₈ | i | {1, -1, i, -i} | 4 | Conjugation can add inverses, but not always the whole group. |
FAQs
1. What does normal closure mean?
It is the smallest normal subgroup containing the chosen subset. You get it by taking all conjugates of the subset and generating a subgroup from them.
2. Why can the normal closure be larger than the subgroup from my inputs?
In non-abelian groups, conjugation can move elements into new positions. Those extra conjugates may force the smallest containing normal subgroup to grow.
3. Why do abelian groups behave differently?
In abelian groups, every element commutes. Therefore, each conjugate equals the original element, and the normal closure is simply the subgroup generated by the input subset.
4. Which groups does this calculator support?
This page supports cyclic groups, dihedral groups, the Klein four group, and the quaternion group. These cover several important finite-group examples for study and practice.
5. What does the index value show?
The index is the quotient of ambient group order by normal closure order. It shows how many cosets the normal closure has inside the full group.
6. Why are conjugacy classes shown separately?
They explain where the normal closure comes from. Since normal closures are generated by conjugates, seeing each conjugacy class makes the computation more transparent.
7. Can I export my result?
Yes. Use the CSV button for structured tabular data and the PDF button for a printable report of the result section and graph summary.
8. What notation should I use for dihedral groups?
Use rotations like e, r, r^2 and reflections like s, rs, r^2s. The calculator also accepts entries such as sr^2 and normalizes them internally.