Color-coded Cayley tables that expose patterns, symmetry, and subgroup structure
A multiplication table lists every product in a group. Color it in and hidden patterns jump out.
Choose a group and click cells to reveal products one at a time, predicting each before you click. Notice that no row or column ever repeats an element.
| * | e | (0 1 2) | (0 2 1) |
|---|---|---|---|
| e | ? | ? | ? |
| (0 1 2) | ? | ? | ? |
| (0 2 1) | ? | ? | ? |
Click cells to reveal products. 0 / 9 revealed.
A group's Cayley table (or multiplication table) lists every possible product. Color-coding reveals deep structure: the Latin square property, subgroup patterns, and whether the group is abelian.
While Cayley diagrams show the "shape" of a group, multiplication tables give a complete, unambiguous specification of its operation.
Click cells to reveal products one at a time. Can you predict the result before clicking? Try to fill in the entire table for Z3 or V4.
Key insight: Every row and every column contains each element exactly once. This is the Latin square property, and it follows directly from the group axioms.
Choose a group and switch between the element, diagonal and none coloring modes. Compare Z4, whose table mirrors across the diagonal, with S3, whose table does not.
| * | e | (0 1 2 3) | (0 2)(1 3) | (0 3 2 1) |
|---|---|---|---|---|
| e | e | (0 1 2 3) | (0 2)(1 3) | (0 3 2 1) |
| (0 1 2 3) | (0 1 2 3) | (0 2)(1 3) | (0 3 2 1) | e |
| (0 2)(1 3) | (0 2)(1 3) | (0 3 2 1) | e | (0 1 2 3) |
| (0 3 2 1) | (0 3 2 1) | e | (0 1 2 3) | (0 2)(1 3) |
Apply different coloring modes to reveal hidden structure. Element coloring shows the Latin square pattern. Diagonal mode highlights symmetry -- abelian groups have symmetric tables.
Key insight: A group is abelian if and only if its multiplication table is symmetric across the main diagonal. Switch between Z4 (symmetric) and S3 (not symmetric) to see this clearly.
Compare Z6 and S3, two groups of order 6, side by side. Red outlines mark the cells where a * b differs from b * a.
An abelian group has a symmetric multiplication table (a * b = b * a for all a, b). Compare Z6 (abelian) with S3 (non-abelian). Red outlines show cells where commutativity fails.
| * | e | (0 1 2 3 4 5) | (0 2 4)(1 3 5) | (0 3)(1 4)(2 5) | (0 4 2)(1 5 3) | (0 5 4 3 2 1) |
|---|---|---|---|---|---|---|
| e | e | (0 1 2 3 4 5) | (0 2 4)(1 3 5) | (0 3)(1 4)(2 5) | (0 4 2)(1 5 3) | (0 5 4 3 2 1) |
| (0 1 2 3 4 5) | (0 1 2 3 4 5) | (0 2 4)(1 3 5) | (0 3)(1 4)(2 5) | (0 4 2)(1 5 3) | (0 5 4 3 2 1) | e |
| (0 2 4)(1 3 5) | (0 2 4)(1 3 5) | (0 3)(1 4)(2 5) | (0 4 2)(1 5 3) | (0 5 4 3 2 1) | e | (0 1 2 3 4 5) |
| (0 3)(1 4)(2 5) | (0 3)(1 4)(2 5) | (0 4 2)(1 5 3) | (0 5 4 3 2 1) | e | (0 1 2 3 4 5) | (0 2 4)(1 3 5) |
| (0 4 2)(1 5 3) | (0 4 2)(1 5 3) | (0 5 4 3 2 1) | e | (0 1 2 3 4 5) | (0 2 4)(1 3 5) | (0 3)(1 4)(2 5) |
| (0 5 4 3 2 1) | (0 5 4 3 2 1) | e | (0 1 2 3 4 5) | (0 2 4)(1 3 5) | (0 3)(1 4)(2 5) | (0 4 2)(1 5 3) |
Symmetric (abelian)
| * | e | (0 1) | (0 1 2) | (1 2) | (0 2) | (0 2 1) |
|---|---|---|---|---|---|---|
| e | e | (0 1) | (0 1 2) | (1 2) | (0 2) | (0 2 1) |
| (0 1) | (0 1) | e | (1 2) | (0 1 2) | (0 2 1) | (0 2) |
| (0 1 2) | (0 1 2) | (0 2) | (0 2 1) | (0 1) | (1 2) | e |
| (1 2) | (1 2) | (0 2 1) | (0 2) | e | (0 1 2) | (0 1) |
| (0 2) | (0 2) | (0 1 2) | (0 1) | (0 2 1) | e | (1 2) |
| (0 2 1) | (0 2 1) | (1 2) | e | (0 2) | (0 1) | (0 1 2) |
Not symmetric (non-abelian)
Compare Z6 (abelian, cyclic) with S3 (non-abelian) side by side. Both have order 6, but their multiplication tables look very different. Red outlines mark cells where a * b is different from b * a.
Key insight: Z6 and S3 both have 6 elements, but they are fundamentally different groups. Z6 is generated by a single element of order 6, while S3 requires two generators and has non-commuting elements.
Spot a mistake or have an idea?