Calculator
Example data table
| Group order |G| | Subgroup order |H| | Index [G:H] | Interpretation |
|---|---|---|---|
| 12 | 3 | 4 | Four left cosets of H in G. |
| 24 | 8 | 3 | Three cosets partition the group. |
| 60 | 5 | 12 | Index equals the coset count. |
Formula used
For a finite group G and a subgroup H ≤ G, the subgroup index is:
- [G:H] = |G| / |H|
- |H| = |G| / [G:H]
- |G| = |H| × [G:H]
The index equals the number of left cosets of H in G. In finite groups, |H| must divide |G|.
How to use this calculator
- Select a calculation mode based on the value you want.
- Enter the known integers: |G|, |H|, or [G:H].
- Press Submit to display results above the form.
- If the ratio is not an integer, treat it as invalid for finite subgroups.
- Use CSV or PDF buttons to export the computed summary.
Concept and Notation
The subgroup index [G:H] measures how many equal sized pieces a group splits into when partitioned by a subgroup. For finite G, the index equals |G|/|H| and also equals the number of left cosets gH. Each coset has exactly |H| elements, so the product |H|x[G:H] reconstructs |G|.
Lagrange Link
A key data check is divisibility. Lagrange's theorem implies |H| divides |G| whenever H is a subgroup of a finite group. If your inputs give a non-integer ratio, the calculator flags it because such orders cannot occur together in a finite subgroup relationship. This is why the tool reports both the integer index and the reduced fraction.
Interpreting Results
When the index is 1, the subgroup is the whole group. When the index equals |G|, the subgroup is trivial of order 1. Small indices often signal strong structure: index 2 subgroups are normal, and index p (prime) leads to an action on cosets of size p. The coset count displayed is the same as the index, providing an immediate combinatorial meaning.
Practical Consistency Checks
Use prime factorizations to validate feasibility quickly. For example, if |G|=60=2^2x3x5 and |H|=12=2^2x3, then [G:H]=5, consistent with cancelling factors. If you instead try |H|=8, the ratio 60/8 reduces to 15/2, revealing the mismatch. These factor comparisons are especially helpful for large inputs.
Where It Appears
Index calculations show up in counting arguments, symmetry classifications, and subgroup chains. In finite permutation groups, the index of a stabilizer equals the orbit size. In algebraic computations, indices help compare intermediate substructures and estimate search spaces. Treat the index as both a ratio of orders and a count of distinct cosets, and your reasoning stays transparent. In computational settings, knowing [G:H] bounds the size of coset tables and the work needed for membership tests. If you supply |G| and [G:H], the calculator returns |H|=|G|/[G:H], which is useful when subgroup order is unknown but the number of cosets is known from an action or counting argument. It also supports reverse computation for planning examples. Across many contexts.
FAQs
1. What does [G:H] represent?
It is the number of distinct left cosets of H in G. For finite groups, it equals the ratio |G| divided by |H| and tells how many equal-size blocks the subgroup partitions the group into.
2. Why must |H| divide |G| for finite groups?
Lagrange's theorem states that the order of any subgroup of a finite group divides the order of the group. Because every coset has |H| elements, counting elements by cosets forces |G|=|H|x[G:H].
3. What if |G|/|H| is not an integer?
Then the pair of orders cannot describe a subgroup inside a finite group. The calculator still shows the reduced fraction, but it flags the index as invalid for a finite subgroup relationship.
4. Does a small index tell me something useful?
Often yes. An index of 1 means H=G, and an index of |G| means |H|=1. Index 2 subgroups are always normal. Prime index can indicate a natural action on cosets of size p.
5. How do prime factors help here?
Factorization lets you cancel common primes in |G| and |H| to see whether the quotient is an integer. If leftover primes remain in the denominator, divisibility fails. It's a fast way to sanity-check large inputs.
6. Can I use this for infinite groups?
This calculator uses orders as finite integers, so it targets finite groups. For infinite groups, index is defined differently and may be infinite. Use theoretical methods instead of numeric order division.