Cayley table
A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group — such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center — can be discovered from its Cayley table.
A simple example of a Cayley table is the one for the group {1, −1} under ordinary multiplication:
× | 1 | −1 |
---|---|---|
1 | 1 | −1 |
−1 | −1 | 1 |
Contents
History
Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation θ^{ n} = 1". In that paper they were referred to simply as tables, and were merely illustrative — they came to be known as Cayley tables later on, in honour of their creator.
Structure and layout
Because many Cayley tables describe groups that are not abelian, the product ab with respect to the group's binary operation is not guaranteed to be equal to the product ba for all a and b in the group. In order to avoid confusion, the convention is that the factor that labels the row (termed nearer factor by Cayley) comes first, and that the factor that labels the column (or further factor) is second. For example, the intersection of row a and column b is ab and not ba, as in the following example:
* | a | b | c |
---|---|---|---|
a | a^{2} | ab | ac |
b | ba | b^{2} | bc |
c | ca | cb | c^{2} |
Cayley originally set up his tables so that the identity element was first, obviating the need for the separate row and column headers featured in the example above. For example, they do not appear in the following table:
a | b | c |
b | c | a |
c | a | b |
In this example, the cyclic group Z_{3}, a is the identity element, and thus appears in the top left corner of the table. It is easy to see, for example, that b^{2} = c and that cb = a. Despite this, most modern texts — and this article — include the row and column headers for added clarity.
Properties and uses
Commutativity
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table is symmetric along its diagonal axis. The cyclic group of order 3, above, and {1, −1} under ordinary multiplication, also above, are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.
Associativity
Because associativity is taken as an axiom when dealing with groups, it is often taken for granted when dealing with Cayley tables. However, Cayley tables can also be used to characterize the operation of a quasigroup, which does not assume associativity as an axiom (indeed, Cayley tables can be used to characterize the operation of any finite magma). Unfortunately, it is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table, as it is with commutativity. This is because associativity depends on a 3 term equation, , while the Cayley table shows 2-term products. However, Light's associativity test can determine associativity with less effort than brute force.
Permutations
Because the cancellation property holds for groups (and indeed even quasigroups), no row or column of a Cayley table may contain the same element twice. Thus each row and column of the table is a permutation of all the elements in the group. This greatly restricts which Cayley tables could conceivably define a valid group operation.
To see why a row or column cannot contain the same element more than once, let a, x, and y all be elements of a group, with x and y distinct. Then in the row representing the element a, the column corresponding to x contains the product ax, and similarly the column corresponding to y contains the product ay. If these two products were equal — that is to say, row a contained the same element twice, our hypothesis — then ax would equal ay. But because the cancellation law holds, we can conclude that if ax = ay, then x = y, a contradiction. Therefore, our hypothesis is incorrect, and a row cannot contain the same element twice. Exactly the same argument suffices to prove the column case, and so we conclude that each row and column contains no element more than once. Because the group is finite, the pigeonhole principle guarantees that each element of the group will be represented in each row and in each column exactly once.
Thus, the Cayley table of a group is an example of a latin square.
Constructing Cayley tables
Because of the structure of groups, one can very often "fill in" Cayley tables that have missing elements, even without having a full characterization of the group operation in question. For example, because each row and column must contain every element in the group, if all elements are accounted for save one, and there is one blank spot, without knowing anything else about the group it is possible to conclude that the element unaccounted for must occupy the remaining blank space. It turns out that this and other observations about groups in general allow us to construct the Cayley tables of groups knowing very little about the group in question.
The "identity skeleton" of a finite group
Because in any group, even a non-abelian group, every element commutes with its own inverse, it follows that the distribution of identity elements on the Cayley table will be symmetric across the table's diagonal. Those that lie on the diagonal are their own inverse; those that do not have another, unique inverse.
Because the order of the rows and columns of a Cayley table is in fact arbitrary, it is convenient to order them in the following manner: beginning with the group's identity element, which is always its own inverse, list first all the elements that are their own inverse, followed by pairs of inverses listed adjacent to each other.
Then, for a finite group of a particular order, it is easy to characterize its "identity skeleton", so named because the identity elements on the Cayley table are clustered about the main diagonal — either they lie directly on it, or they are one removed from it.
It is relatively trivial to prove that groups with different identity skeletons cannot be isomorphic, though the converse is not true (for instance, the cyclic group C_{8} and the quaternion group Q are non-isomorphic but have the same identity skeleton).
Consider a six-element group with elements e, a, b, c, d, and f. By convention, e is the group's identity element. Because the identity element is always its own inverse, and inverses are unique, the fact that there are 6 elements in this group means that at least one element other than e must be its own inverse. So we have the following possible skeletons:
- all elements are their own inverses,
- all elements save d and f are their own inverses, each of these latter two being the other's inverse,
- a is its own inverse, b and c are inverses, and d and f are inverses.
In our particular example, there does not exist a group of the first type of order 6; indeed, simply because a particular identity skeleton is conceivable does not in general mean that there exists a group that fits it.
It is noteworthy (and trivial to prove) that any group in which every element is its own inverse is abelian.
Filling in the identity skeleton
Once a particular identity skeleton has been decided on, it is possible to begin filling out the Cayley table. For example, take the identity skeleton of a group of order 6 of the second type outlined above:
e | a | b | c | d | f | |
---|---|---|---|---|---|---|
e | e | |||||
a | e | |||||
b | e | |||||
c | e | |||||
d | e | |||||
f | e |
Obviously, the e row and the e column can be filled out immediately. Once this has been done, it may be necessary (and it is necessary, in our case) to make an assumption, which may later lead to a contradiction — meaning simply that our initial assumption was false. We will assume that ab = c. Then:
e | a | b | c | d | f | |
---|---|---|---|---|---|---|
e | e | a | b | c | d | f |
a | a | e | c | |||
b | b | e | ||||
c | c | e | ||||
d | d | e | ||||
f | f | e |
Multiplying ab = c on the left by a gives b = ac. Multiplying on the right by c gives bc = a. Multiplying ab = c on the right by b gives a = cb. Multiplying bc = a on the left by b gives c = ba, and multiplying that on the right by a gives ca = b. After filling these products into the table, we find that the ad and af are still unaccounted for in the a row; as we know that each element of the group must appear in each row exactly once, and that only d and f are unaccounted for, we know that ad must equal d or f; but it cannot equal d, because if it did, that would imply that a equaled e, when we know them to be distinct. Thus we have ad = f and af = d.
Furthermore, since the inverse of d is f, multiplying ad = f on the right by f gives a = f^{2}. Multiplying this on the left by d gives us da = f. Multiplying this on the right by a, we have d = fa.
Filling in all of these products, the Cayley table now looks like this:
e | a | b | c | d | f | |
---|---|---|---|---|---|---|
e | e | a | b | c | d | f |
a | a | e | c | b | f | d |
b | b | c | e | a | ||
c | c | b | a | e | ||
d | d | f | e | |||
f | f | d | e | a |
Because each row must have every element of the group represented exactly once, it is easy to see that the two blank spots in the b row must be occupied by d or f. However, if one examines the columns containing these two blank spots — the d and f columns — one finds that d and f have already been filled in on both, which means that regardless of how d and f are placed in row b, they will always violate the permutation rule. Because our algebraic deductions up until this point were sound, we can only conclude that our earlier, baseless assumption that ab = c was, in fact, false. Essentially, we guessed and we guessed incorrectly. We, have, however, learned something: ab ≠ c.
The only two remaining possibilities then are that ab = d or that ab = f; we would expect these two guesses to each have the same outcome, up to isomorphism, because d and f are inverses of each other and the letters that represent them are inherently arbitrary anyway. So without loss of generality, take ab = d. If we arrive at another contradiction, we must assume that no group of order 6 has the identity skeleton we started with, as we will have exhausted all possibilities.
Here is the new Cayley table:
e | a | b | c | d | f | |
---|---|---|---|---|---|---|
e | e | a | b | c | d | f |
a | a | e | d | |||
b | b | e | ||||
c | c | e | ||||
d | d | e | ||||
f | f | e |
Multiplying ab = d on the left by a, we have b = ad. Right multiplication by f gives bf = a, and left multiplication by b gives f = ba. Multiplying on the right by a we then have fa = b, and left multiplication by d then yields a = db. Filling in the Cayley table, we now have (new additions in red):
e | a | b | c | d | f | |
---|---|---|---|---|---|---|
e | e | a | b | c | d | f |
a | a | e | d | b | ||
b | b | f | e | a | ||
c | c | e | ||||
d | d | a | e | |||
f | f | b | e |
Since the a row is missing c and f and since af cannot equal f (or a would be equal to e, when we know them to be distinct), we can conclude that af = c. Left multiplication by a then yields f = ac, which we may multiply on the right by c to give us fc = a. Multiplying this on the left by d gives us c = da, which we can multiply on the right by a to obtain ca = d. Similarly, multiplying af = c on the right by d gives us a = cd. Updating the table, we have the following, with the most recent changes in blue:
e | a | b | c | d | f | |
---|---|---|---|---|---|---|
e | e | a | b | c | d | f |
a | a | e | d | f | b | c |
b | b | f | e | a | ||
c | c | d | e | a | ||
d | d | c | a | e | ||
f | f | b | a | e |
Since the b row is missing c and d, and since b c cannot equal c, it follows that b c = d, and therefore b d must equal c. Multiplying on the right by f this gives us b = cf, which we can further manipulate into cb = f by multiplying by c on the left. By similar logic we can deduce that c = fb and that dc = b. Filling these in, we have (with the latest additions in green):
e | a | b | c | d | f | |
---|---|---|---|---|---|---|
e | e | a | b | c | d | f |
a | a | e | d | f | b | c |
b | b | f | e | d | c | a |
c | c | d | f | e | a | b |
d | d | c | a | b | e | |
f | f | b | c | a | e |
Since the d row is missing only f, we know d^{2} = f, and thus f^{2} = d. As we have managed to fill in the whole table without obtaining a contradiction, we have found a group of order 6: inspection reveals it to be non-abelian. This group is in fact the smallest non-abelian group, the dihedral group D_{3}:
* | e | a | b | c | d | f |
---|---|---|---|---|---|---|
e | e | a | b | c | d | f |
a | a | e | d | f | b | c |
b | b | f | e | d | c | a |
c | c | d | f | e | a | b |
d | d | c | a | b | f | e |
f | f | b | c | a | e | d |
Permutation matrix generation
The standard form of a Cayley table has the order of the elements in the rows the same as the order in the columns. Another form is to arrange the elements of the columns so that the nth column corresponds to the inverse of the element in the nth row. In our example of D_{3}, we need only switch the last two columns, since f and d are the only elements that are not their own inverses, but instead inverses of each other.
e | a | b | c | f=d^{−1} | d=f^{−1} | |
---|---|---|---|---|---|---|
e | e | a | b | c | f | d |
a | a | e | d | f | c | b |
b | b | f | e | d | a | c |
c | c | d | f | e | b | a |
d | d | c | a | b | e | f |
f | f | b | c | a | d | e |
This particular example lets us create six permutation matrices (all elements 1 or 0, exactly one 1 in each row and column). The 6x6 matrix representing an element will have a 1 in every position that has the letter of the element in the Cayley table and a zero in every other position, the Kronecker delta function for that symbol. (Note that e is in every position down the main diagonal, which gives us the identity matrix for 6x6 matrices in this case, as we would expect.) Here is the matrix that represents our element a, for example.
e | a | b | c | f | d | |
---|---|---|---|---|---|---|
e | 0 | 1 | 0 | 0 | 0 | 0 |
a | 1 | 0 | 0 | 0 | 0 | 0 |
b | 0 | 0 | 0 | 0 | 1 | 0 |
c | 0 | 0 | 0 | 0 | 0 | 1 |
d | 0 | 0 | 1 | 0 | 0 | 0 |
f | 0 | 0 | 0 | 1 | 0 | 0 |
This shows us directly that any group of order n is a subgroup of the permutation group S_{n}, order n!.
Generalizations
The above properties depend on some axioms valid for groups. It is natural to consider Cayley tables for other algebraic structures, such as for semigroups, quasigroups, and magmas, but some of the properties above do not hold.
See also
References
- Cayley, Arthur. "On the theory of groups, as depending on the symbolic equation θ^{ n} = 1", Philosophical Magazine, Vol. 7 (1854), pp. 40–47. Available on-line at Google Books as part of his collected works.
- Cayley, Arthur. "On the Theory of Groups", American Journal of Mathematics, Vol. 11, No. 2 (Jan 1889), pp. 139–157. Available at JSTOR.