Calculator Inputs
The page uses a single-column section flow, while the input controls switch to three columns on large screens, two on medium, and one on mobile.
Example Data Table
Sample case: addition modulo 12, subgroup H = {0, 4, 8}, representative g = 5.
| Representative g | H | Left Coset g + H | Coset Size |
|---|---|---|---|
| 0 | {0, 4, 8} | {0, 4, 8} | 3 |
| 1 | {0, 4, 8} | {1, 5, 9} | 3 |
| 2 | {0, 4, 8} | {2, 6, 10} | 3 |
| 3 | {0, 4, 8} | {3, 7, 11} | 3 |
Formula Used
General left coset: gH = { g ⊗ h | h ∈ H }
Addition mode: g + H = { (g + h) mod n | h ∈ H }
Multiplication mode: gH = { (g × h) mod n | h ∈ H }
Coset size: |gH| = |H| whenever H is a valid subgroup.
Index of the subgroup: [G:H] = |G| / |H|
How to Use This Calculator
- Select either addition modulo
nor multiplication modulon. - Enter the modulus
n. - Provide subgroup elements as comma-separated integers.
- Enter a representative
g. - Press Calculate Left Coset.
- Review the coset, subgroup checks, index, distinct cosets, and chart.
- Use the export buttons to save results as CSV or PDF.
FAQs
1) What is a left coset?
A left coset is formed by combining one group element with every element of a subgroup. In this calculator, the combination uses modular addition or modular multiplication.
2) Why must H be a subgroup?
Coset theory assumes H is a subgroup, not just any subset. The calculator checks subset membership, identity, closure, and inverses before reporting formal coset results.
3) What groups does this calculator support?
It supports Z_n under addition and U(n) under multiplication. These are common finite modular groups used in introductory and intermediate abstract algebra problems.
4) Why is the multiplicative option limited to U(n)?
Not every nonzero residue has a multiplicative inverse modulo n. U(n) contains exactly the invertible residues, so it forms a proper group for subgroup and coset calculations.
5) Does every coset have the same size?
Yes. For a valid subgroup H, every left coset has exactly |H| elements. This calculator displays subgroup order and coset size so you can compare them directly.
6) What does the subgroup index mean?
The index [G:H] tells how many distinct left cosets partition the whole group. It equals the group order divided by the subgroup order.
7) Are left and right cosets different here?
For the modular groups used here, no. They are abelian groups, so left and right cosets match. In nonabelian groups, the two can differ.
8) Can I use negative inputs?
Yes. The calculator normalizes all entered integers modulo n, so negative representatives or subgroup elements are converted into their standard residue classes.