Cayley table
A Cayley table is a representation of a product
defined on a set G. It is a group-theoretic generalization of an addition or a multiplication table. The product is a mapping, so that if a * b = c and a * b = d then c = d. Cayley tables are named after the mathematician Arthur Cayley.
If the order of G is n then the Cayley table for G will have n+1 rows and n+1 columns: the zeroth row will house the column headers, and the zeroth column will house the row headers.
The row headers are conventionally in the same order as the column headers but in transposed position. The column headers are elements of set G: each element shows up exactly once as one of the column headers.
The zeroth row and zeroth column have no labels (headers), so they are unlabeled. Thus there are n labeled columns and n labeled rows. (From now refer to labeled columns as just "columns" and to labeled rows as just "rows". "Elements" of a row/column will henceforward exclude the zeroth (header) element.)
Each row represents the element in its header as the first factor of a product, and each column represents the element in its header as the second factor of a product.
The intersection of a row and a column corresponds to a product of the row header's element by the column header's element, and the intersection contains an element of G which is the product's result.
If G is a group and its identity element is known then it is customary to let the first column and first row represent the identity element.
Cayley table of a group
If a, x, y, b ∈ G then if a * x = b and a * y = b then x = y, or if x * a = b and y * a = b then x = y.
This is true because, since a∈G, then a−1∈G, so a − 1 * a * x = a − 1 * b, x = a − 1 * b and a − 1 * a * y = a − 1 * b, y = a − 1 * b therefore x = y.
This means that the elements of a row must all be different from each other. (If a row represents element a, and a pair of different columns represent elements x and y, then if a * x = a * y then x = y and the different columns must be the same column, contradiction.) Likewise, the elements of a column must all be different from each other. Each element is unique within its own row and its own column.
Since the number of elements in a row is equal to o(G), and since the elements are all different from each other but all belong to G, it follows that the each row (or column) contains a permutation of G.
[Henceforwards, group products will be represented by juxtaposition, i.e. 
.]
Another property of groups is this: if a b = e and if b c = e, then a = c. This is true because (a b) c = e c = c, a (b c) = a e = a, but since a group is associative, (a b) c = a (b c) therefore a = c.
What this means is that if a pair of elements have the identity elements as their product, then the pair of elements commute with each other. I.e. a b = e if and only if b a = e.
It is possible for an element in a group to be its own inverse, so that when multiplied by itself it yields the identity element. If a is such an element then a2 = e. If b ≠ a then a b ≠ e.
Therefore each non-identity element is either its own inverse or it pairs up with another element which is its unique inverse. It is possible to rearrange the order of the labeled columns such that the first columns to appear, reading from left to right, are those representing self-inverse elements, starting first with the identity element which is always a self-inverse. After all the columns of self-inverses should follow the columns representing pairs of inverses: each pair of inverses should be represented by a pair of adjacent columns.
If a table is initially left empty except for the intersections containing the identity element, the resulting arrangement of identity elements can be viewed as an "identity skeleton" of the Cayley table.
Groups with different identity skeletons cannot be isomorphic, but an identity skeleton boils down simply to numbers of self-inverses versus number of inverse pairs.
As an example, groups with six elements could only possibly have three different identity skeletons (up to isomorphism) (however, there is no six-element group with the first identity skeleton):
| 1 | e | a | b | c | d | f |
|---|---|---|---|---|---|---|
| <i>e | <i>e | |||||
| a | <i>e | |||||
| b | <i>e | |||||
| c | <i>e | |||||
| d | <i>e | |||||
| f | <i>e |
| 2 | <i>e | a | b | c | d | f |
|---|---|---|---|---|---|---|
| <i>e | <i>e | |||||
| a | <i>e | |||||
| b | <i>e | |||||
| c | <i>e | |||||
| d | <i>e | |||||
| f | <i>e |
| 3 | <i>e | a | b | c | d | f |
|---|---|---|---|---|---|---|
| <i>e | <i>e | |||||
| a | <i>e | |||||
| b | <i>e | |||||
| c | <i>e | |||||
| d | <i>e | |||||
| f | <i>e |
If an element x of a group is a self-inverse, so that x2 = e, then for any elements a and b, x has the following properties:
- x a = b ↔ x b = a,
- a x = b ↔ b x = a.
This is true because if x a = b then x x a = x b, e a = x b, a = x b. If a x = b then a x x = b x, a e = b x, a = b x. (The swapping of factor and product may be referred to here as "trans-equality commutation" or "trans-commutation" for short.)
If all the elements of a group are self-inverses, then the group must be abelian (its product is commutative and its Cayley table has reflective symmetry with respect to its "diagonal of squares": its backward slash diagonal). This is true because if, for any a, b, c ∈ G,
- a b = c
then since a is self-inverse, then b and c can "trans-commute":
- a c = b,
but since c is also a self-inverse, then a and b can trans-commute:
- b c = a,
and since b is a self-inverse, a and c can trans-commute:
- b a = c,
therefore
- a b = b a
and since this is true for all pairs a-b in the group, then the group is abelian.
If a pair of elements x and y form an inverse pair, so that x y = y x = e, then for any elements a and b the x-y pair has the following properties (which may be referred to here as "trans-mutations"):
- x a = b ↔ y b = a,
- a x = b ↔ b y = a.
This is true because if x a = b then y x a = y b, e a = y b, a = y b. If a x = b then a x y = b y,
a e = b y, a = b y.
Example
These properties can be useful for constructing Cayley tables of groups. E.g. if it is desired to find a six-element group with the second identity skeleton, then start out by filling the identity row and column; and suppose a b = c,
| <i>e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| <i>e | <i>e | a | b | c | d | f |
| a | a | <i>e | c | |||
| b | b | <i>e | ||||
| c | c | <i>e | ||||
| d | d | <i>e | ||||
| f | f | <i>e |
Since a is self-inverse and a b = c then a c = b by trans-commute. Since c is self-inverse and a c = b then b c = a by trans-commute. Since b is self-inverse and since a b = c then c b = a, and since b c = a then b a = c. Since a is self-inverse and since b a = c then c a = b. Since a d ≠ d because a ≠ e, then a d = f (f is the only element in G left). Then since the a-row contains all elements except d, then a f = d.
Since d is inverse pair with f, and since a d = f, then f f = a, and then d a = f. Then since the a-column contains all elements except d, then f a = d.
| <i>e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| <i>e | <i>e | a | b | c | d | f |
| a | a | <i>e | c | b | f | d |
| b | b | c | <i>e | a | ||
| c | c | b | a | <i>e | ||
| d | d | f | <i>e | |||
| f | f | d | <i>e | a |
The b-row is already occupied with {e, a, b, c} and is only missing {d, f}. Its only open intersections are b d, b f; but the d- and f-columns are already occupied with {d, f}, so the missing intersections cannot be filled with anything. The construction has crashed.
Start over, this time letting a b = d.
| <i>e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| <i>e | <i>e | a | b | c | d | f |
| a | a | <i>e | d | |||
| b | b | <i>e | ||||
| c | c | <i>e | ||||
| d | d | <i>e | ||||
| f | f | <i>e |
Since a is self-inverse and a b = d then a d = b by trans-commute. Since d pairs with f and since a d = b then b f = a by trans-mute. Then since b is self-inverse and b f = a then b a = f by trans-commute. Since a is self-inverse and since b a = f then f a = b. Since f pairs with d and since f a = b then d b = a by trans-mute.
| <i>e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| <i>e | <i>e | a | b | c | d | f |
| a | a | <i>e | d | b | ||
| b | b | f | <i>e | a | ||
| c | c | <i>e | ||||
| d | d | a | <i>e | |||
| f | f | b | <i>e |
Since a-row is missing {c, f} and since a f ≠ f (because e f = f) then a f = c. Then since a is self-inverse and since a f = c then a c = f by trans-commute. Also, since f pairs with d and since a f = c then c d = a by trans-mute. Since c is self-inverse and since a c = f then f c = a. Since f pairs with d and since f c = a, then d a = c by trans-mute. Since a is self-inverse and since d a = c then c a = d by trans-commute.
| <i>e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| <i>e | <i>e | a | b | c | d | f |
| a | a | <i>e | d | f | b | c |
| b | b | f | <i>e | a | ||
| c | c | d | <i>e | a | ||
| d | d | c | a | <i>e | ||
| f | f | b | a | <i>e |
Since b-row is missing {c, d}, and since b c ≠ c, then b c = d, therefore b d = c (by trans-commutation or by row completion). Since d pairs with f and since b d = c then c f = b by trans-mutation. Since c is self-inverse and since c f = b, then c b = f by trans-commute. Since b is self-inverse and since c b = f, then f b = c by trans-commute. Since f pairs with d and since f b = c, then d c = b by trans-mute.
| <i>e | a | b | c | d | f | |
|---|---|---|---|---|---|---|
| <i>e | <i>e | a | b | c | d | f |
| a | a | <i>e | d | f | b | c |
| b | b | f | <i>e | d | c | a |
| c | c | d | f | <i>e | a | b |
| d | d | c | a | b | <i>e | |
| f | f | b | c | a | <i>e |
Since d-row is missing only f, then let d d = f. Since d pairs with f and since d d = f, then f f = d by trans-mute. The completed Cayley table of a six-element non-abelian group, isomorphic to the dihedral group D3, is shown below:
| * | <i>e | a | b | c | d | f |
|---|---|---|---|---|---|---|
| <i>e | <i>e | a | b | c | d | f |
| a | a | <i>e | d | f | b | c |
| b | b | f | <i>e | d | c | a |
| c | c | d | f | <i>e | a | b |
| d | d | c | a | b | f | <i>e |
| f | f | b | c | a | <i>e | d |
Quod erat faciendum.
See also: Klein four-group.