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 easily deduced by examining its Cayley table. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... This picture illustrates how the hours on a clock form a group under modular addition. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ... Abelian, in mathematics, is used in many different definitions: In group theory: Abelian group, a group in which the binary operation is commutative Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms Metabelian group is a group where the commutator subgroup is contained in... In mathematics, the inverse of an element x, with respect to an operation *, is an element x such that their compose gives a neutral element. ... In abstract algebra, the centre of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if x and...

A simple example of a Cayley table is the one for the group {1, -1} under ordinary multiplication: In mathematics, multiplication is an elementary arithmetic operation. ...

History

Cayley tables were first presented in Cayley's 1854 paper, On The Theory of Groups, as depending on the symbolic equation θⁿ = 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 which 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 first factor (termed nearer factor by Cayley) in any row of the table is the same, and that the second factor (or further factor) in any column is the same, as in the following example: In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

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:

In this example, the cyclic group Z₃, a is the identity element, and thus appears in the top left corner of the table. It is easy to see, for example, that b² = c and that cb = a. Despite this, most modern texts — and this article — include the row and column headers for added clarity. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

Properties and uses

Commutativity

The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, the Cayley table of an abelian group 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. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... The smallest non-Abelian group has 6 elements. ...

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 is the case with commutativity. In mathematics, associativity is a property that a binary operation can have. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...

Permutations

Owing to the fact that 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. In mathematics, an element a in a magma (M,*) has the left cancellation property (or is left-cancellative) if for all b and c in M, a * b = a * c always implies b = c. ...

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. Reductio ad absurdum (Latin: reduction to the absurd) also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption... The inspiration for the name of the principle: pigeons in holes. ...

Thus, the Cayley table of a group is an example of a latin square. A Latin square is an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column. ...

Constructing Cayley tables

Because of the structure of groups, one can very often "fill in" Cayley tables which are 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 which 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 which 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 of this statement is not true. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

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 which 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:

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:

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 = f². 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:

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:

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):

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:

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):

Since the d row is missing only f, we know d² = f, and thus f² = 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 D₃: This article may be confusing for some readers, and should be edited to enhance clarity. ...

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. In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In abstract algebra, a quasigroup is a algebraic structure resembling a group in the sense that division is always possible. ... In abstract algebra, a magma (also called a groupoid) is a particularly basic kind of algebraic structure. ...

References

See also

• Dihedral group of order 6
• Latin square