Choose a structure
Cayley table
Row first, column second. The entry in row a and column b is a * b. For permutations that is a ∘ b: first b, then a.
LaTeX code
An array for display mathematics: put it between \[ and \]. Labels made of words use \text, from the amsmath package.
Verification report
Your own table
Rename the elements in the top row, then choose a result for every cell. The row element comes first: the cell in row a and column b is a * b. Nothing is filled in or repaired for you.
Paste or import a table
Paste a table as CSV (commas, semicolons or tabs) or as JSON saved from this page, or choose a file. In CSV the first row holds a corner cell and then the column labels; each further row starts with its own label. An empty cell is a missing entry.
Structure
Powers of an element
Cayley graph
A Cayley graph needs a choice of generators. Its vertices are the elements, and for each chosen s there is an arrow g → g * s (multiplication on the right).
Generators (choose up to four):
Help: tables, axioms and graphs
What a Cayley table is. A finite set with one operation is described completely by a square table: one row and one column for each element, and in row a, column b the result a * b. The row element is always the left operand. That order matters as soon as the operation is not commutative: in S₃, (1 2 3) ∘ (1 2) = (1 3) but (1 2) ∘ (1 2 3) = (2 3).
The group axioms. A group is a set with an operation that is closed (every result is in the set), associative ((a * b) * c = a * (b * c) for all three), has a two-sided identity e (e * a = a = a * e), and gives every element a two-sided inverse (a * b = e = b * a). Commutativity is not required; a commutative group is called abelian.
Why associativity needs care. It is the one axiom that cannot be read off the table at a glance: it concerns triples, so a table with n elements needs n³ checks, and this page makes every one of them. A table can look perfectly regular, with every element once in each row and column (a Latin square), and still fail it: subtraction modulo 3 is an example.
Weaker structures. A closed operation alone is a magma; an associative one is a semigroup; a semigroup with an identity is a monoid; a monoid in which every element has an inverse is a group. Multiplication modulo 4 is a commutative monoid and not a group, because 0 and 2 have no inverses. A one-sided identity or a one-sided inverse is not enough, and the report says which side holds.
Element orders. In a group, the order of a is the least k ≥ 1 with aᵏ = e: walk a, a², a³, … through the table until the identity comes back. It always divides the order of the group. A group is cyclic when some element has order equal to the size of the group.
Cayley graphs. Choose a set S of elements and join each g to g * s for every s in S. If S generates the group the graph is connected, and a path from e to g spells g as a product of generators. If not, the graph falls into pieces: the cosets of the subgroup that S generates.
Limits. Built-in structures have at most 240 elements and are checked exhaustively. The table is shown whole up to 36 elements and in a 36 × 36 window beyond that; the editor takes up to 20 elements, imports up to 100; colours are offered up to 24 elements and the Cayley graph up to 48. Everything is computed in your browser.