|
|
|
|
Cayley table
|
(Definition)
|
|
|
A Cayley table for a group is essentially the “multiplication table” of the group.1 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  be our group, with operation  the group operation. Let  be the Cayley table for the group, with  denoting the element at row  and column  . Then
where  is the  th element of the group, and  is the  th element.
Note that for an Abelian group, we have
 , 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).
Footnotes
- 1
- A caveat to novices in group theory: multiplication is usually used notationally to represent the group operation, but the operation needn't resemble multiplication in the reals. Hence, you should take “multiplication table” with a grain or two of salt.
|
"Cayley table" is owned by akrowne.
|
|
(view preamble)
| Other names: |
Cayley-table |
|
|
Cross-references: order, permutation group, addition, integers, ordering, labeling, invariant, isomorphic, isomorphic groups, symmetric matrix, abelian group, cells, matrix, rows, columns, reals, operation, group operation, represent, multiplication, theory, group
There are 2 references to this entry.
This is version 8 of Cayley table, born on 2002-10-22, modified 2005-03-03.
Object id is 3540, canonical name is CayleyTable.
Accessed 9203 times total.
Classification:
| AMS MSC: | 20A99 (Group theory and generalizations :: Foundations :: Miscellaneous) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle \left(\begin{array}{c\vert cccc} & [0] & [1] & [2] & [3] \ \hli... ... & [2] & [3] & [0] & [1] \ \; [3] & [3] & [0] & [1] & [2] \end{array}\right) $](http://images.planetmath.org:8080/cache/objects/3540/l2h/img14.png "$\displaystyle \left(\begin{array}{c\vert cccc} & [0] & [1] & [2] & [3] \ \hli... ... & [2] & [3] & [0] & [1] \ \; [3] & [3] & [0] & [1] & [2] \end{array}\right) $")
