Generate and explore Cayley tables for common groups (Z_n, S_n, D_n). Verify group axioms and find subgroups visually.
Group Cayley Table Generator lays out every product for a selected cyclic, symmetric, or dihedral group. Choose the family and its parameter, then read each result where an element's row meets another element's column. The table supports checks of the group axioms and visual identification of subgroups. A common surprise is that n is not always the table width: Z_n has n elements, while S_n has n! elements. Dihedral notation also varies between sources, so confirm the displayed element count.
A concrete input and expected output from the current implementation.
Input
Choose Z_n, enter n = 3, and use addition.
Expected output
Table:
+ | 0 1 2
0 | 0 1 2
1 | 1 2 0
2 | 2 0 1
All group axioms pass. The subgroups are {0} and {0, 1, 2}. Each cell is the row label plus the column label modulo 3, so 1 + 2 becomes 0 and 2 + 2 becomes 1. Zero is the identity, while 1 and 2 are mutual inverses. Since the group has prime order 3, it has no nontrivial proper subgroup.
The selected group family determines both the elements and the operation. For Z_n, the elements are residue classes and each entry is addition modulo n. For S_n, the elements are permutations and entries are permutation compositions. For D_n, entries combine rotations and reflections. Row and column labels identify the two operands, while their intersection contains the resulting group element.
Closure requires every cell to contain an element already in the group. An identity row and column reproduce the headers, and each element must pair with an inverse to produce that identity. Associativity compares (a*b)*c with a*(b*c) for all triples. A highlighted subset is a subgroup only when it contains the identity and remains closed under products and inverses.
With identical row and column ordering, symmetry across the main diagonal means a*b = b*a for every pair, so the group is abelian. Z_n under addition is symmetric. S_n is noncommutative from n = 3 onward, and polygon dihedral groups are noncommutative for n at least 3, so their tables need not be symmetric.
Not from a similar appearance, equal order, or matching colors alone. Two finite groups are isomorphic when one Cayley table can be transformed into the other by a consistent relabeling of elements while preserving every product. The generator can supply tables for comparison, but a visual resemblance without a product-preserving correspondence is not a proof.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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