Calculator Form
Use any two finite quantities to calculate the third. Strict mode enforces whole-number divisibility expected in finite group theory.
Example Data Table
These examples show valid finite-group order relationships where subgroup order divides group order exactly.
| Example Group | Group Order |G| | Subgroup Order |H| | Index [G:H] | Quick Note |
|---|---|---|---|---|
| C12 | 12 | 3 | 4 | Four cosets of a subgroup of order 3. |
| S3 | 6 | 2 | 3 | Each subgroup of order 2 has index 3. |
| D8 | 8 | 4 | 2 | A subgroup of order 4 has two cosets. |
| A4 | 12 | 3 | 4 | Order-3 subgroups split the group into four cosets. |
| Q8 | 8 | 2 | 4 | An order-2 subgroup gives index 4. |
Formula Used
Primary index formula: [G:H] = |G| / |H|
Rearranged group order formula: |G| = [G:H] × |H|
Rearranged subgroup order formula: |H| = |G| / [G:H]
For finite groups, Lagrange’s theorem says the order of a subgroup divides the order of the group. That means the subgroup index is the exact number of cosets and should be a positive whole number in valid finite cases.
This calculator checks those relationships, computes the missing quantity, and reports whether the entered values behave like a valid finite subgroup configuration.
How to Use This Calculator
- Select whether you want to find the index, group order, or subgroup order.
- Enter any two known finite values in the form fields.
- Set the number of decimal places for displayed output.
- Keep strict mode on for proper finite-group divisibility checking.
- Press the calculate button and review the result, graph, and export tools.
Frequently Asked Questions
1. What does subgroup index mean?
The subgroup index counts how many distinct cosets of H fit inside G. For finite groups, it equals the group order divided by the subgroup order.
2. Why must the subgroup order divide the group order?
Lagrange’s theorem states that every finite subgroup order divides the finite group order. If divisibility fails, the input does not describe a valid finite subgroup setup.
3. Can the subgroup index be 1?
Yes. Index 1 means the subgroup is the whole group itself, so there is only one coset and no smaller partition occurs.
4. Can a subgroup order be larger than the group order?
No. A subgroup is contained inside the group, so its order cannot exceed the group order in finite group theory.
5. What happens if the ratio is not an integer?
In strict mode, the calculator flags the input as invalid for finite groups. In relaxed mode, it still shows the ratio for exploratory checking.
6. Do left and right cosets always give the same index?
Yes. Even when the subgroup is not normal, the number of left cosets equals the number of right cosets, so the index is the same.
7. Can this calculator handle infinite groups?
No. This version is designed for finite-group order calculations, where divisibility and exact subgroup sizes are meaningful numerical inputs.
8. What is special about prime index?
Prime index often signals stronger structural consequences in group theory. The calculator labels that case so you can spot it immediately.