Cayley table

A Cayley table (or group table) is a square table used to describe the structure of a finite group. It shows the result of combining any two elements of the group using the group operation.

• Each row and column corresponds to a group element.
• The entry in the row $a$ and column $b$ shows the product $ab$.

This table helps visualize properties like identity and inverses.

Interestingly, the rows and columns of this table cannot have repeated entries, since if $gx=gy$ then we have $x=y$. That is, the rows (and columns) are permutations of the group elements. A Cayley table is a Latin square.

Cayley’s Theorem gives a deeper perspective on this:
Every group is isomorphic to a group of permutations, specifically, the group of permutations of its own elements induced by left multiplication. The Cayley table makes this isomorphism explicit: each row represents a permutation of the group elements, showing how the group element permutes the set via left multiplication.

Example: Rotations of a Square

The group of rotations of a square (denoted as C₄, the cyclic group of order 4) consists of four elements:

• R₀: Rotation by 0° (identity)
• R₉₀: Rotation by 90° clockwise
• R₁₈₀: Rotation by 180°
• R₂₇₀: Rotation by 270°

This table illustrates that the group is closed, has an identity (R₀), and that each element has an inverse.

Related: Cayley graph.