Maths - Cayley Table

This is the multiplication table for the group. This completely defines the group although many of the table entries can derived from the others so it contains quite a lot of redundancy.

Cayley tables can be used to define a group as explained on this page or as part of a definition for a ring or associated algebra. They can also be used in a slightly modified form for hypercomplex numbers see this page.

Latin Squares

The Cayley Table is a Latin Square. The properties of a group result in the following properties when a Cayley table represents a group

So each element of the group appears once in each row and each column.

Proof of this - suppose x appears in a row labeled with 'a' twice:

x = a b
x = a c

then we cancel to get: b=c but we use distinct elements to label the columns.

Example

Here is a Cayley table representing a group with the elements '1','a','b','aba','ba' and 'ab' as follows:

  1 a b aba ba ab
1 1 a b aba ba ab
a a 1 ab ba aba b
b b ba 1 ab a aba
aba aba ab ba 1 b a
ba ba b aba a ab 1
ab ab aba a b 1 ba

If we take any element say 'aba' then we can see that it appears in each row and each column only once.

  1 a b aba ba ab
1 1 a b aba ba ab
a a 1 ab ba aba b
b b ba 1 ab a aba
aba aba ab ba 1 b a
ba ba b aba a ab 1
ab ab aba a b 1 ba

 


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see also:
  • For an introduction to cayley tables see this page.
  • Generating cayley tables for hypercomplex numbers see this page.
Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Symmetry and the Monster - This is a popular science type book which traces the history leading up to the discovery of the largest symmetry groups.

Terminology and Notation

Specific to this page here:

 

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