Enter elements, then paste or generate the table. See checks, warnings, and clear property summaries. Export results as CSV or PDF for sharing fast.
This example uses elements 0,1,2,3 with addition modulo 4.
| ⋆ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 | 3 | 0 | 1 | 2 |
This calculator evaluates a finite algebraic structure described by a Cayley table. You provide a set of symbols and an operation table, and the tool checks whether every table entry belongs to the set. It flags closure violations early, because a single out-of-set product prevents any group classification. Row and column expectations reduce formatting mistakes and keep symbols consistent across tests.
When closure holds, the tool searches for an identity element e satisfying e⋆a=a and a⋆e=a for all a. It then scans each element for a two-sided inverse a⁻¹ with a⋆a⁻¹=e and a⁻¹⋆a=e. The inverse map supports quick sanity checks and helps confirm cancelation behavior. If an element lacks an inverse, the output highlights that the structure is not a group, even if other properties seem plausible.
Associativity is the hardest axiom to verify from raw data. The calculator tests all triples (a,b,c) and reports concrete counterexamples when (a⋆b)⋆c differs from a⋆(b⋆c). Showing explicit witnesses makes debugging your table practical, especially for hand-built operations or classroom exercises. A small list of failures is usually enough to locate the inconsistent row, because associativity breaks propagate through repeated products.
If the structure is a group, the tool tests commutativity and computes element orders by repeated multiplication until the identity appears. Order statistics often reveal patterns: an element of order n suggests a cyclic group, while many elements of order two can suggest a Klein-type structure. The output provides a cautious likely label based on these indicators. Use the quick generator to compare your table with examples like addition modulo n or unit multiplication modulo n.
For groups up to size twelve, the calculator enumerates candidate subgroups, verifies closure and inverses inside each subset, and then tests normality using conjugation g⋆h⋆g⁻¹. If only {e} and the whole group are normal, it flags the group as simple. This supports rapid exploration of examples and strengthens intuition about structure and symmetry. The normal subgroup list is also useful for quotient-group planning, because normality is the key condition for well-defined cosets and factor operations.
Enter a comma-separated element list and a square operation table using the same order. You can also generate a standard table with the quick generator for common modular examples.
Yes. Elements are treated as tokens, so you may use letters or symbols like e,a,b,c. Just keep spelling consistent in both the element list and every table cell.
Closure only checks membership of products, not how products associate. A table can stay inside the set yet still violate (a⋆b)⋆c=a⋆(b⋆c). The reported counterexample shows the exact triple causing failure.
A group is simple when it has no nontrivial normal subgroups. The tool tests normality by conjugation and flags simplicity when only {e} and the whole group are normal.
Subgroup enumeration grows as 2^n, so it becomes expensive quickly. The calculator limits full subgroup and normality listing to n≤12 to keep results fast in a browser.
CSV export saves a summary plus the inverse and order table for spreadsheets. PDF export creates a printable report using an in-page generator, and includes the inverse and order table on a separate page.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.