Example data table
| Label | Meaning | Quick note |
|---|---|---|
| e | identity | Leaves every element unchanged. |
| r | rotation (120°) | r ⊗ r ⊗ r = e. |
| r² | rotation (240°) | Inverse of r. |
| s | reflection | s ⊗ s = e. |
| sr | reflection then rotation | Noncommuting with r. |
| sr² | reflection then rotation² | Creates asymmetric products. |
Formula used
- Closure: every table entry must be one of the n elements.
- Identity: find e such that e⊗a=a and a⊗e=a for all a.
- Inverse: for each a, find b with a⊗b=e and b⊗a=e.
- Associativity: verify (a⊗b)⊗c = a⊗(b⊗c) for all triples.
- Nonabelian test: check whether a⊗b ≠ b⊗a for some pair.
How to use this calculator
- Choose n, and optionally provide n comma-separated labels.
- Click Build Table to generate the operation grid.
- Fill each cell with the product of row ⊗ column elements.
- Press Check Nonabelian Property to evaluate axioms.
- Download CSV or PDF to keep the table and results.
Data inputs and table design
A finite group can be represented by an n×n Cayley table where each cell stores the product of a row element with a column element. This calculator accepts n from 2 to 12, optional labels, and a complete operation grid. Labels must match n, otherwise default indices are used for consistency. The header row and first column remind you that row ⊗ column defines multiplication direction.
Axiom checks and computational cost
The checker performs closure, identity, inverse, and associativity tests over the provided table. Closure scans n² entries and rejects any value outside the element set. Identity tries each candidate e and verifies 2n conditions, giving O(n²) work. Inverses search n candidates per element, again O(n²). Associativity is the heaviest step, comparing (a⊗b)⊗c with a⊗(b⊗c) for all triples, which is O(n³). The routine stops at the first failure and stores a witness triple for troubleshooting.
Identity and inverse interpretation
When a valid identity is found, it is reported using your labels, helping you interpret the structure without renaming elements. Two‑sided inverses are required: a⊗b=e and b⊗a=e. This matters because some algebraic systems have one‑sided identities or inverses; the calculator distinguishes those from true group behavior and highlights the first element lacking an inverse. For valid groups, the inverse list is computed implicitly and can be verified by scanning the identity row and column.
Noncommutativity evidence reporting
After confirming the group axioms, commutativity is tested by scanning all pairs and checking a⊗b versus b⊗a. If a mismatch is detected, the tool prints a concrete witness pair and both products. This “counterexample” style output is useful for proofs, homework write‑ups, and debugging a table constructed from permutations or matrices. If no witness exists, the calculator reports the structure as abelian, meaning every pair commutes.
Exportable outputs for audit trails
CSV export captures the summary flags and the labeled table, making it easy to paste into spreadsheets or share with collaborators. PDF export produces a compact report with the same summary and a formatted grid suitable for printing. Together, these exports support reproducible checking of small algebraic models and preserve the exact data that generated the conclusion. Securely.
FAQs
What does “nonabelian” mean here?
It means the structure is a valid group and at least one pair of elements does not commute. The report shows a concrete pair a,b where a⊗b and b⊗a produce different results.
Why must every table cell be filled?
Group verification needs a defined product for every ordered pair. Missing cells prevent closure and associativity checks, so the calculator flags the first missing or out‑of‑range entry.
What happens if my label list count is wrong?
If the comma‑separated labels are not exactly n items, the calculator uses default numeric labels. This keeps the table consistent while you correct the labeling.
How is associativity verified?
The checker evaluates both (a⊗b)⊗c and a⊗(b⊗c) for all triples of elements. If any mismatch appears, it stops and reports a witness triple that fails.
Can it validate infinite or symbolic groups?
No. This tool works only with finite tables where every product is explicitly listed. For infinite groups, you typically prove axioms from a formula rather than enumerating all products.
What should I do with the witness pair?
Use it as a direct counterexample when arguing the group is not commutative. It also helps debug a mistaken table entry by showing which products disagree.