Permutation Group Order Calculator

Calculator inputs

Enter permutations in cycle notation (1 2 3)(4 5) or one-line notation 2 3 1 5 4.

Degree (n)

We label symbols as 1..n.

Mode

Use subgroup mode to combine generators.

Element cap

Safety limit for group generation.

Generators (one per line)

Example above generates the full symmetric group S3 when n=3.

Single permutation

Used only in single-permutation mode.

Tip: For generated groups, start with small n (≤8) to avoid huge groups.

Example data

These examples show typical inputs and expected results.

Case	n	Input	Expected output
Permutation order	5	(1 2 3)(4 5)	Order = lcm(3,2) = 6
Generated group	3	(1 2 3) and (1 2)	Group order = 6 (S₃)
Cyclic subgroup	4	(1 2 3 4)	Group order = 4

Formula used

The order of a permutation equals the least positive integer k such that p^k = id. For disjoint cycles, it is:

order(p) = lcm( cycle_length₁, cycle_length₂, … )

The order of a generated subgroup ⟨g₁, …, g_m⟩ is the number of distinct permutations reachable by multiplying generators and closing under composition.

This calculator uses iterative closure with an element cap to keep runtime predictable.

How to use this calculator

Set n to match your symbol set 1..n.
Select Generated subgroup order to combine multiple generators.
Enter each generator on a new line, using cycles or one-line form.
Press Calculate to show results above the form.
Use Download CSV or Download PDF to export.

Why group order matters

In permutation groups, the order measures how many distinct symmetries your generators produce. A small change in generators can shift the order from a cyclic subgroup to the full symmetric group, affecting proofs, counting arguments, and algorithmic complexity. It also indicates how many unique states are reachable under repeated moves for analysis, teaching, and review.

Two calculation modes

This calculator supports a single permutation order and a generated subgroup order. Single mode decomposes the permutation into disjoint cycles and takes the least common multiple of their lengths. Generated mode repeatedly composes generators until closure, then counts the distinct elements found. You can enter permutations using cycle notation or one-line notation.

Cycle data you can verify

When you enter (1 2 3)(4 5) in degree 5, the cycle lengths are 3 and 2, so the order is lcm(3,2)=6. If you enter a 4-cycle (1 2 3 4) in degree 4, the order is 4 because one cycle controls all moved points. Identity components are ignored, so fixed points never inflate the result.

Generator order diagnostics

For subgroup mode, each generator’s order is computed first. These values give quick insight: if all generators have small orders, the subgroup may still be large because their interactions create new elements. The Plotly chart visualizes generator orders beside the computed subgroup size. This helps you spot when one generator dominates.

Performance and safety caps

Group generation can grow factorially with n. For example, S8 has 40320 elements, while S10 has 3628800. The element cap prevents runaway generation by stopping after a chosen limit and marking the result as truncated, which is useful for exploratory work. If truncation appears, reduce n or simplify generators.

Interpreting exported results

CSV export captures the core metrics you see on screen, while PDF export provides a shareable snapshot for notes. Use the example table to sanity check inputs, then increase n gradually, watching both the order and the truncation flag for reliable comparisons. When comparing generator sets, keep n fixed and log the same metrics.

FAQs

Quick answers for common permutation group order questions.

1) What does “order” mean for a permutation?

The order is the smallest positive integer k such that applying the permutation k times returns every element to its original position.

2) Why does the LCM of cycle lengths work?

Disjoint cycles repeat independently. The whole permutation returns to identity exactly when every cycle completes an integer number of rotations, which occurs at the least common multiple.

3) What does “generated subgroup order” compute?

It counts distinct permutations produced by composing your generators until no new elements appear. This is the size of the subgroup ⟨g1,…,gm⟩ within S_n.

4) Why do I see “truncated at cap”?

Your generated subgroup exceeded the safety limit before closure completed. Lower n, reduce generators, or increase the cap carefully to finish generation.

5) Can I mix cycle and one-line notation?

Yes. Each line is parsed independently. Use cycles like (1 2 3) or one-line like 2 3 1. Keep values within 1..n and avoid duplicates.

6) What’s a good workflow for checking results?

Start with small n, verify generator orders, and compare against known cases such as cyclic groups and S3. Then increase complexity gradually while monitoring steps and truncation.