Loading [MathJax]/jax/output/CommonHTML/jax.js

Cayley table


A Cayley table for a group is essentially the “multiplication table” of the group.11A caveat to novices in group theory: multiplication is usually used notationally to represent the group operationMathworldPlanetmath, but the operation needn’t resemble multiplication in the reals. Hence, you should take “multiplication table” with a grain or two of salt. 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 ∘ 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)=ei∘ej

where ei is the ith element of the group, and ej is the jth element.

Note that for an Abelian groupMathworldPlanetmath, we have ei∘ej=ej∘ei, hence the Cayley table is a symmetric matrixMathworldPlanetmath.

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

0.1 Examples.

  • •

    The Cayley table for ℤ4, the group of integers modulo 4 (under addition), would be

    ([0][1][2][3][0][0][1][2][3][1][1][2][3][0][2][2][3][0][1][3][3][0][1][2])
  • •

    The Cayley table for S3, the permutation groupMathworldPlanetmath of order 3, is

    ((1)(123)(132)(12)(13)(23)(1)(1)(123)(132)(12)(13)(23)(123)(123)(132)(1)(13)(23)(12)(132)(132)(1)(123)(23)(12)(13)(12)(12)(23)(13)(1)(132)(123)(13)(13)(12)(23)(123)(1)(132)(23)(23)(13)(12)(132)(123)(1))
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