Calculator Inputs
Large screens show three columns, smaller screens show two, and mobile shows one.
Example Data Table
| Normal Group | Quotient Group | |N| | |Q| | Action Type | H² Classes | Resulting Diagnostic |
|---|---|---|---|---|---|---|
| C3 | C4 | 3 | 4 | Trivial | 1 | Split direct product of order 12. |
| C5 | C4 | 5 | 4 | Nontrivial | 1 | Split semidirect product of order 20. |
| C2 | C2 | 2 | 2 | Trivial | 2 | Central-extension candidate of order 4. |
Formula Used
1 → N → G → Q → 1
|G| = |N| × |Q|
If gcd(|N|, |Q|) = 1, then the extension splits.
Split + trivial action gives a direct product. Split + nontrivial action gives a semidirect product.
Candidate action–cohomology pairs = known action families × known H² classes.
The cohomology count and action-family count must come from external algebra work or software. They cannot be deduced from group orders alone.
How to Use This Calculator
- Enter labels for the normal subgroup, extension group, and quotient group.
- Provide the orders of the normal subgroup and quotient.
- Enter a center order if you want central-extension diagnostics.
- Add known action-family and H² class counts from your separate algebra analysis.
- Choose whether the action is trivial, nontrivial, or still unknown.
- Mark abelian and solvable properties if you already know them.
- Submit the form to display the result summary above the calculator.
- Use the CSV or PDF buttons to export the result table.
Frequently Asked Questions
1. What does this calculator actually compute?
It computes finite extension diagnostics from supplied orders, action data, solvability flags, and known cohomology counts. It summarizes likely structure, splitting behavior, order, and basic inheritance statements.
2. Can it fully classify every extension?
No. Full classification usually requires explicit group actions, cocycles, and equivalence testing. This page organizes those inputs and reports mathematically safe conclusions from them.
3. Why is the H² class count entered manually?
Second cohomology depends on the chosen action and module structure. Orders alone do not determine H², so the tool asks for a count you already computed elsewhere.
4. What does a split extension mean here?
A split extension means the quotient has a complement inside the full group. In finite coprime-order cases, that split is guaranteed by standard extension theory.
5. When does the calculator show a direct product?
It shows a direct-product diagnosis when the extension splits and the action is marked trivial. Under those conditions, the extension behaves like N × Q.
6. What is the central-extension candidate flag?
That flag appears when the action is trivial, the center order exceeds one, and multiple H² classes are supplied. Those conditions can signal non-split central extensions.
7. Why does solvability matter?
If both the normal subgroup and the quotient are solvable, then the whole extension is solvable. This is a useful inheritance rule for finite groups.
8. What does the p-group diagnostic tell me?
If both supplied orders are powers of the same prime, the extension order is also a power of that prime. Then the full group is a finite p-group.