Calculator Inputs
Use the single-column page layout below. The fields inside the calculator use a responsive three-column, two-column, and one-column grid.
Formula Used
These formulas power the calculator:
- Cyclic group: |Cₙ| = n
- Dihedral group: |Dₙ| = 2n
- Symmetric group: |Sₙ| = n!
- Alternating group: |Aₙ| = n!/2 for n ≥ 2
- Direct product: |Cₘ × Cₙ| = mn
- General linear group: |GL(n, q)| = ∏k=0n-1(qn − qk)
- Klein four group: |V₄| = 4
Lagrange’s theorem says subgroup orders must divide the full group order. Element orders also divide |G|, although not every divisor must occur.
How to Use This Calculator
- Choose a finite group family from the dropdown.
- Enter the needed parameters such as n, m, or q.
- Press Calculate Group Order.
- Read the exact order, formula, divisor note, and graph.
- Download a CSV summary or a PDF report if needed.
Example Data Table
| Family | Input | Formula | Order | Comment |
|---|---|---|---|---|
| Cyclic | C₁₂ | |Cₙ| = n | 12 | One generator exists when gcd(k, 12) = 1. |
| Dihedral | D₈ | |Dₙ| = 2n | 16 | Symmetries of a regular octagon. |
| Symmetric | S₅ | |Sₙ| = n! | 120 | All permutations on five symbols. |
| Alternating | A₅ | |Aₙ| = n!/2 | 60 | Simple non-abelian group example. |
| Direct product | C₄ × C₆ | |Cₘ × Cₙ| = mn | 24 | Multiply the component orders. |
| General linear | GL(2, 3) | (3²−1)(3²−3) | 48 | Invertible 2×2 matrices over F₃. |
Frequently Asked Questions
1) What is the order of a finite group?
The order of a finite group is its number of elements. This calculator returns exact orders for several common finite group families.
2) What does Dₙ mean here?
Here Dₙ means the symmetry group of a regular n-gon, so its order is 2n. Some books use different notation conventions.
3) Why are subgroup-order candidates based on divisors?
Lagrange’s theorem says every subgroup order must divide the full group order. That rule is necessary, but not every divisor is guaranteed to occur.
4) Are all divisors actual element orders?
No. Every element order divides |G|, but many divisors may never appear as actual element orders inside a specific group.
5) Why must q be a prime power in GL(n, q)?
GL(n, q) is defined over a finite field with q elements. Finite fields exist only when q is a prime power.
6) Why does the graph use log10(|G|)?
Some group orders grow extremely quickly. A log scale keeps the chart readable and still shows relative growth clearly.
7) What do the export buttons include?
The CSV download includes a result summary table. The PDF export captures the visible result section so you can save or print it.
8) Which inputs are most practical?
Cyclic, dihedral, and direct products handle larger values easily. Symmetric, alternating, and GL options grow faster, so moderate values render best.