Cayley table

A Cayley table for a group is essentially the “multiplication table” of the group. The columns and rows of the table (or matrix) are labeled with the elements of the group, and the cells represent the result of applying the group operation to the row-th and column-th elements.

Formally, let $G$ be our group, with operation $\circ$ the group operation. Let $C$ be the Cayley table for the group, with $C(i,j)$ denoting the element at row $i$ and column $j$. Then

$$C(i,j)=e_i\circ e_j$$

where $e_i$ is the $i$th element of the group, and $e_j$ is the $j$th element.

Note that for an Abelian group, we have $e_i\circ e_j=e_j\circ e_i$, hence the Cayley table is a symmetric matrix.

All Cayley tables for isomorphic groups are isomorphic (that is, the same, invariant of the labeling and ordering of group elements).

0.1 Examples.

• The Cayley table for ${\mathbb{Z}}_4$, the group of integers modulo 4 (under addition), would be

• The Cayley table for $S_3$, the permutation group of order 3, is

Title	Cayley table
Canonical name	CayleyTable
Date of creation	2013-03-22 13:06:44
Last modified on	2013-03-22 13:06:44
Owner	akrowne (2)
Last modified by	akrowne (2)
Numerical id	11
Author	akrowne (2)
Entry type	Definition
Classification	msc 20A99
Synonym	Cayley-table