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Named after the 19th century
British British may refer to: Peoples, culture, and language * British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies. ** Britishness, the British identity and common culture * British English, ...

British
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

mathematician
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

Arthur Cayley
, a Cayley table describes the structure of a
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
by arranging all the possible products of all the group's elements in a square table reminiscent of an
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
or
multiplication table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication binary operation, operation for an algebraic system. The decimal multiplication table was traditionally taug ...

multiplication table
. Many properties of a groupsuch as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's
center Center or centre may refer to: Mathematics *Center (geometry) In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the speci ...
can be discovered from its Cayley table. A simple example of a Cayley table is the one for the group under ordinary
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
:


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 illustrativethey 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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:


Properties and uses


Commutativity

The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, a group is abelian
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
its Cayley table's values are
symmetric Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

symmetric
along its diagonal axis. The cyclic group of order 3, above, and 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, does not have a symmetric Cayley table.


Associativity

Because
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, which does not assume associativity as an axiom (indeed, Cayley tables can be used to characterize the operation of any finite
magma Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
). 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, (ab)c=a(bc), 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

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 equalthat is to say, row ''a'' contained the same element twice, our hypothesisthen ''ax'' would equal ''ay''. But because the cancellation law holds, we can conclude that if ''ax'' = ''ay'', then ''x'' = ''y'', a
contradiction In traditional logicIn philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Phil ...
. 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 La ...
. Another, maybe simpler proof: the
cancellation property In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
implies that for each x in the group, the one variable function of y f(x,y)= xy must be a one to one map. And one to one maps on finite sets are permutations.


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 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 diagonaleither 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphic
, though the converse is not true (for instance, the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
''C8'' and the
quaternion group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
''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. Any group in which every element is its own inverse is abelian: let ''a'' and ''b'' be elements of the group, then ''ab'' = (''ab'')−1 = ''b''−1''a''−1 = ''ba''.


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 is done there are several possible options on how to proceed. We will focus on the value of ''ab''. Since each element of the group must appear once and only once in each row, and once and only once in each column, the only possibly valid values of ''ab'' are ''c'', ''d'', or ''f''. However we can see that swapping around the two elements ''d'' and ''f'' would result in exactly the same table as we already have, save for arbitrarily selected labels. We would therefore expect both of these two options to result in the same outcome, up to isomorphism, and so we need only consider one of them. It is also important to note that one or several of the values may (and do, in our case) later lead to contradictionmeaning simply that they were in fact not vaild values at all.


''ab'' = ''c''

By alternatingly multiplying on the left and on the right it is possible to extend one equation into a loop of equations where any one implies all the others: *Multiplying ''ab'' = ''c'' on the left by ''a'' gives ''b'' = ''ac'' *Multiplying ''b'' = ''ac'' on the right by ''c'' gives ''bc'' = ''a'' *Multiplying ''bc'' = ''a'' on the left by ''b'' gives ''c'' = ''ba'' *Multiplying ''c'' = ''ba'' on the right by ''a'' gives ''ca'' = ''b'' *Multiplying ''ca'' = ''b'' on the left by ''c'' gives ''a'' = ''cb'' *Multiplying ''a'' = ''cb'' on the right by ''b'' gives ''ab'' = ''c'' Filling in all of these products, the Cayley table now looks like this (new elements in red): We will now focus on the value of ''ad''. Since each element of the group must appear once and only once in each row, and once and only once in each column, the only possibly valid value of ''ad'' is ''f''. By alternatingly multiplying on the left and on the right it is possible to extend one equation into a loop of equations where any one implies all the others: *Multiplying ''ad'' = ''f'' on the left by ''a'' gives ''d'' = ''af'' *Multiplying ''d'' = ''af'' on the right by ''d'' gives ''d''2 = ''a'' *Multiplying ''d''2 = ''a'' on the left by ''f'' gives ''d'' = ''fa'' *Multiplying ''d'' = ''fa'' on the right by ''a'' gives ''da'' = ''f'' *Multiplying ''da'' = ''f'' on the left by ''f'' gives ''a'' = ''f''2 *Multiplying ''a'' = ''f''2 on the right by ''d'' gives ''ad'' = ''f'' Filling in all of these products, the Cayley table now looks like this (new elements in blue): We will now focus on the value of ''bd''. Unfortunately, since each element of the group must appear once and only once in each row, and once and only once in each column, there are no valid values of ''bd''. This means that the option we selected (''ab'' = ''c'') has lead us to a point where no value can be assigned to ''bd'' without causing contradictions. We have therfore showed that ''ab'' ≠ ''c''. If we in a similar way show that all options lead to contradictions, then we must conclude that no group of order 6 exists with the identity skeleton that we started with.


''ab'' = ''d''

By alternatingly multiplying on the left and on the right it is possible to extend one equation into a loop of equations where any one implies all the others: *Multiplying ''ab'' = ''d'' on the left by ''a'' gives ''b'' = ''ad'' *Multiplying ''b'' = ''ad'' on the right by ''f'' gives ''bf'' = ''a'' *Multiplying ''bf'' = ''a'' on the left by ''b'' gives ''f'' = ''ba'' *Multiplying ''f'' = ''ba'' on the right by ''a'' gives ''fa'' = ''b'' *Multiplying ''fa'' = ''b'' on the left by ''d'' gives ''a'' = ''db'' *Multiplying ''a'' = ''db'' on the right by ''b'' gives ''ab'' = ''d'' Filling in all of these products, the Cayley table now looks like this (new elements in red): We will now focus on the value of ''ac''. Since each element of the group must appear once and only once in each row, and once and only once in each column, the only possibly valid value of ''ac'' is ''f''. By alternatingly multiplying on the left and on the right it is possible to extend one equation into a loop of equations where any one implies all the others: *Multiplying ''ac'' = ''f'' on the left by ''a'' gives ''c'' = ''af'' *Multiplying ''c'' = ''af'' on the right by ''d'' gives ''cd'' = ''a'' *Multiplying ''cd'' = ''a'' on the left by ''c'' gives ''d'' = ''ca'' *Multiplying ''d'' = ''ca'' on the right by ''a'' gives ''da'' = ''c'' *Multiplying ''da'' = ''c'' on the left by ''f'' gives ''a'' = ''fc'' *Multiplying ''a'' = ''fc'' on the right by ''c'' gives ''ac'' = ''f'' Filling in all of these products, the Cayley table now looks like this (new elements in blue): We will now focus on the value of ''bc''. Since each element of the group must appear once and only once in each row, and once and only once in each column, the only possibly valid value of ''bc'' is ''d''. By alternatingly multiplying on the left and on the right it is possible to extend one equation into a loop of equations where any one implies all the others: *Multiplying ''bc'' = ''d'' on the left by ''b'' gives ''c'' = ''bd'' *Multiplying ''c'' = ''bd'' on the right by ''f'' gives ''cf'' = ''b'' *Multiplying ''cf'' = ''b'' on the left by ''c'' gives ''f'' = ''cb'' *Multiplying ''f'' = ''cb'' on the right by ''b'' gives ''fb'' = ''c'' *Multiplying ''fb'' = ''c'' on the left by ''d'' gives ''b'' = ''dc'' *Multiplying ''b'' = ''dc'' on the right by ''c'' gives ''bc'' = ''d'' Filling in all of these products, the Cayley table now looks like this (new elements in green): Finally, since each element of the group must appear once and only once in each row, and once and only once in each column, the only possibly valid value of ''d''2 is ''f'', and the only possibly valid value of ''f''2 is ''d''. Filling in these products, the Cayley table now looks like this (new elements in orange): As we have managed to fill in the whole table without obtaining a contradiction, we have found a group of order 6, and inspection reveals it to be non-abelian. This group is in fact the smallest non-abelian group, the
dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
''D''3:


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 ''n''th column corresponds to the inverse of the element in the ''n''th 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. This particular example lets us create six
permutation matrices In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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. {, border="2" cellpadding="5" align="center" !style="background:#efefef;", !style="background:#efefef;", e !style="background:#efefef;", a !style="background:#efefef;", b !style="background:#efefef;", c !style="background:#efefef;", f !style="background:#efefef;", d , - !style="background:#efefef;", e , 0 , , 1 , , 0 , , 0 , , 0 , , 0 , - !style="background:#efefef;", a , 1 , , 0 , , 0 , , 0 , , 0 , , 0 , - !style="background:#efefef;", b , 0 , , 0 , , 0 , , 0 , , 1 , , 0 , - !style="background:#efefef;", c , 0 , , 0 , , 0 , , 0 , , 0 , , 1 , - !style="background:#efefef;", d , 0 , , 0 , , 1 , , 0 , , 0 , , 0 , - !style="background:#efefef;", f , 0 , , 0 , , 0 , , 1 , , 0 , , 0 This shows us directly that any group of order ''n'' is a subgroup of the
permutation group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''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
semigroup In mathematics, a semigroup is an algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
s,
quasigroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, and
magmas Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the other ...
, but some of the properties above do not hold.


See also

*
Latin square In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 La ...
*
Sudoku Sudoku (; ja, 数独, sūdoku, digit-single; originally called Number Place) is a logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic ...

Sudoku


References

*
Cayley, Arthur
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
Cayley, Arthur
. "On the Theory of Groups", ''
American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press The Johns Hopkins University Press (also referred to as JHU Press or JHUP) is the publishing division of Johns Hopkins Unive ...
'', Vol. 11, No. 2 (Jan 1889), pp. 139–157
Available at JSTOR.
Finite groups