Associativity A property of operations. An operation * is called associative if: a * (b * c) = (a * b) * c for every possible a, b, and c. Axiom For Greek geometry, an axiom was a 'self-evident truth'. Axioms acted as the foundation from which all true statements should be deduced. In modern mathematics, an axiom is a property or feature used as part of a definition. Groups are defined by axioms. Deductions from these axioms lead to true statements about all groups. Cancellativity A property of operations. An operation * is called cancellative if: only when a = b for every possible a, band c. Cayley table The 'multiplication table' of a group operation is called its Cayley table. Cayley tables are Latin squares. However, not every Latin square represents the Cayley table of a group. Classification A classification of groups is a logical scheme capable, in principle, of identifying all possible groups. The approach to classification taken in this book is to identify groups, as far as possible, according to the number of elements that they contain. Colouring and colouring type When marking a geometric object according to certain rules (e.g. painting the six faces of a cube black or white), a colouring is a particular way of performing the marking with the object fixed. For example, a cube with the top face black and all others white is a colouring. A colouring type is a collection of colourings obtained from any particular colouring by rotating the object. For example, the six colourings of the cube which has one black face and five white faces form a single colouring type. Commutativity A property of operations. An operation* is called commutative if: for every possible a and b. Composition Composition is an operation on transformations (permutations, symmetries, etc), usually denoted by the symbol 0 The composition X o Y of two 124
transformations X and Y is the action of doing X first, then doing Y. Composition is automatically associative: X o (Yo Z) means doing X, then doing Y followed by Z, while (X o Y) o Z means doing X followed by Y, then doing Z. Cyclic group A group which consists entirely of the positive and negative powers of a single element is called a cyclic group. Generators A set of some elements of a group, from which all elements can be obtained by taking products and inverses, is called a generating set for the group. If a group has a single element which generates the group on its own, the group is cyclic. Group A group is any mathematical structure which satisfies the group axioms: (1) There is a set of objects (called the group elements). (2) There is an operation on these objects. (3) The operation is associative. (4) There is an identity element for the operation. (5) Each element has an inverse for the operation. Identity Where a mathematical structure involves an operation * on its elements, one of the elements (often called e) may have the property that: for every x. Such an element is called an identity element. For instance, 0 is an identity for addition in the integers. The possession of an identity is one of the defining features of a group. In groups, the identity is unique. Where the operation is composition (on a set of transformations), the identity is the transformation which leaves everything where it started. Integers The integers are the whole numbers:... '-3, -2, -1, 0, 1, 2, 3,... The system of integers is denoted l. This system, with the operation of addition, forms a group. It is a semigroup, but not a group, with the operation of multiplication. Inverse Where a mathematical structure involves an operation * on its elements, it may have an identity e. If it does, and two elements a and bare such that: then b is called an inverse for a (and a is also an inverse for b). For instance, -2 is an inverse for 2 in the integers with addition. The possession of inverses for all elements is one of the defining features of a group. In groups, the inverse of each element is unique. Isometry An isometry is a non-distorting transformation of a geometrical object. In two dimensions, in addition to the identity transformations, there are three kinds of isometry: rotations, reflections and translations. Isomorphism An isomorphism is a one-to-one correspondence between the elements of two groups, which is capable of 'translating' the Cayley table of one into the Cayley table of the other. For instance, one isomorphism between the groups l 2 and z: (see Modulo) is the correspondence: 125
which interconverts the two tables: fo ++ 1 \1 ++ 2 and If there is an isomorphism between two groups, they are called isomorphic; as far as group theory is concerned, they are then practically equal. Latin square A Latin square is an n x n table in which each of n elements appears exactly once in each column and exactly once in each row. The Cayley table of a group is always a Latin square. Matrices A matrix is a rectangular array, often of real numbers. The system of all n-by-n real matrices is called Mn(IR). It is a group with the operation of matrix addition. It is a semi group, but not a group, with the operation of matrix multiplication. The system of all invertible n-by-n matrices, denoted GLn(IR), is a group under matrix multiplication. Modulo The integers modulo n are the whole numbers: 0, 1, 2, 3,..., n - 1 The system of integers modulo n is denoted ln. This system forms a group with the operation of addition modulo n: a +n b = { a+ a+ b- b n (if a + b < n) (if a + b ;;::. n) It is a semigroup, but not a group, under the operation of multiplication modulo n: a xn b = lab ab- n ab ~ 2n (if ab < n) (if n ::::;;; ab < 2n) (if 2n ::::;;; ab < 3n) The system of non-zero integers modulo n is denoted l.~. This system is a group under multiplication modulo n when n is prime (and only then). Natural numbers The natural numbers are the non-negative whole numbers: 0, 1, 2, 3,... ' The system of natural numbers is denoted 1'\1. This system, with the operation of addition, forms a semigroup but not a group. It is also a semigroup, but not a group, with the operation of multiplication. Operation An operation is a way of combining any two members of a set of objects to result in a third member. For example, subtraction is an operation on the set of integers but not on the set of natural numbers - because combinations like 2-5 do not represent a natural number. Permutations A permutation is a rearrangement of a set of positioned objects. A 126
permutation can be represented by noting the destination point of each object. For I th. ( 1 2 3 ) h b.... 1.. 3 examp e, e permutation 3 2 1 moves t eo Ject m pos1t10n to pos1t1on, that at position 2 remains at position 2, and that at position 3 moves to position 1. The set of permutations of {1, 2,..., n} is denoted sn. This is a group with the operation of composition. Platonic solids There are five solids whose faces are identical regular polygons and in which each vertex is the junction of the same number of faces. These are the Platonic solids: the regular tetrahedron (equilateral triangles, three to a vertex); the cube (squares, three to a vertex); the regular octahedron (equilateral triangles, four to a vertex); the regular dodecahedron (regular pentagons, three to a vertex); the regular icosahedron (equilateral triangles, five to a vertex). Rational numbers The rational number system is the set of fractions min (with m and n integers and n =I= 0). This system is denoted 0. With the operation of addition, it forms a group. It is a semigroup, but not a group, with the operation of multiplication. The system consisting of all rationals apart from zero is denoted by 0*. This is a group under multiplication but not even a semigroup under addition. Real numbers The real number system is the set of infinite decimals, m a 1a 2a 3a 4 This system is denoted lit With the operation of addition, it forms a group. It is a semigroup, but not a group, with the operation of multiplication. The system consisting of all reals apart from zero is denoted by R *. It is a group under multiplication but not even a semigroup under addition. Semigroup A semigroup is any mathematical structure which satisfies the semigroup axioms: (1) There is a set of objects (called semigroup elements). (2) There is an operation on these objects. (3) The operation is associative. A group is automatically a semigroup. However, some semigroups are not groups (such as the natural numbers with the operation of addition). Subgroup Within a group, it sometimes happens that a subset of elements also forms a group under the same group operation. If this is the case, this subset is called a subgroup. For example, the set of even integers:..., -4, -2, 0, 2, 4,..., forms a subgroup of the group of integers under addition. Symmetry An isometry is said to be a symmetry of a particular geometrical object if the object appears unchanged by the transformation. The set of symmetries of an object X is denoted Sym(X). It is a group with the operation of composition. Rot(X) is the set of all symmetries of X which are rotations. It is a subgroup of Sym(X). Tiling A tiling is any figure in the plane consisting of joined straight-line segments. This includes simple polygons, but also includes infinite figures. The symmetry group of a regular infinite tiling can include translations, and can itself be infinite. Crystallography is the study of three-dimensional 'tilings'.
Trivial group A group with only one element is said to be trivial. The one element is necessarily the identity. Vectors A vector is an ordered collection, often of real numbers. The system of all real vectors of length n is called IRn. It is a group with the operation of vector addition. It is not a semigroup with the operation of vector product. 128
INDEX abelian group 70 addition in number systems 1-2, 6, 8, 10, 15, 24, 67-8 matrix 14-15, 17-18 modulo n 19-22, 24, 68-9, 75--(), 86, 126 table 21-2, 24 vector 11-14 associative (operation) 2-4, 6, 8, 10, 12-16, 18-22, 24, 30, 36, 61, 63, 65, 67, 70, 71, 78, 80--1, 93, 95, 99-101, 108-9, 124, 125, axioms abelian group 70 group 62, 70, 71, 78, 92, 93, 96, 99, 102, 108-9, 110, 125 nature of 64-6, 70, 124 semigroup 70 cancellative (operation) 6~7, 79, 98, 124 Cayley table as Latin square 73-4, 98-101 definition 70--1, 78, 124 used as a basis for classification 9~ 109 used to find subgroups 90--3 central inversion 54, 56 classification of groups 95-110, 124 closure (for an operation) 68-9 colouring 112-23, 124 colouring types group formula for number of 114-15, 120--1, 122, 123 of coloured cubes 121-2 of tiles 112-17,120,122-3 commutative (operation) 1-2, 3, 4, 6, 8, 10, 12-15, 17-21, 24, 32, 36, 65, 70, 78-80, 85, 124 composition and matrix multiplication 47-9, 54, 63 of permutations 28-36, 50, 124-5, of symmetries 47-52, 54, 60--61, 63, 124-5, properties of 30--2, 69-70, 124-5 cube colourings of 121-2 symmetries of 56, 58-9, 121-2 cyclic group 85-7, 92, 93, 96, 125 always commutative 85 division as inverse for multiplication 14, 18, 19 by zero 7, 9, 18, 68, in integers modulo p 22-4, 68-9 in rationals 7-9, 68 in reals 10, 68 element (of a group) 65 exponentiation (not commutative or associative) 34 fixing of a colouring 113-22 of a shape 41, 61 generators 82-5, 86, 92, 93-4, 125 group 129
INDEX abelian 70 axioms 62, 70, 71, 78, 92, 93, 96, 99, 102, 108-9, 110, 125 Cayley table of a 70--1, 73-4, 78, 90--3, 96--109, 124 classification 95-110, 124 cyclic 85-7, 92, 93-4, 125 examples 64, 66--70, 110, 125-8 generators 82-5, 86, 92, 93-4, 125 isomorphism 75-7, 86--9, 96, 99, 101-7, 109-10, 125-6 permutation 32-6, 64, 69, 77, 90-- 3, 103, 106 semi- 70, 125-8 sub- 87-94, symmetry 50--2, 64, 60--70, 77-8, 103, trivial 33, 99, 128 'hunt the pea' (permutation game) 25-9, 31, 32-3 identity for addition in number systems 4, 10, 20--1, 24, 67-8, 82 for matrix addition 15, 69, 88 for matrix multiplication 16--17, 69 for multiplication in number systems 4, 10, 20--1, 24, 68-9 for vector addition 12-14, 69 for vector product 14 of a group 65, 66--7, 71-4, 78, 81, 90, 95, 97, 99, 125, 128 permutation 31, 36, 69 transformation (isometry) 45, 50-4, 56, 58-63, 66, 69-70, 113-14, 116--17 uniqueness, in a group 66, 67, 79 integers 5-7, 68, 82-5, 125 even 88-9, modulo n 19-24, 68-9, 75-6, 84, 85, 89, 102-3, 106--8, 126 modulo p 22-4, 68-9, 109, 126 inverse matrix (multiplicative) 18-19 of a group element 65-7, 71-3, 78, 81-3, 87, 94, 95, 99, 108, 125 permutation 31-2, 36 transformation (isometry) 57, 61, 63 uniqueness, in a group 66, 67, 79 invertible matrices definition 18-19 set of, as a group 69, 126 isometry 40--2, 43, 45-7, 49, 84--5, 125, isomorphism definition of 88-9, 125-6 of any cyclic group and zn or z 85-6, 96, 109 of S 3 and Sym(~) 77 of Z 4 and z; 75-6, 86--7, 103 use in classification 96, 99, 101-6, 107, 109-10 Latin square property definition 73, 126 of Cayley tables 73-4, 98-101, 103-5, 106, 110, 124 of tables which are not Cayley tables 74, 108-9, 110, 124 matrices and symmetries 14, 42-7, 48-9, 53-4, 63 invertible 18-19, 69, 126 systems of 15-19, 69, 87-8, 126 modulo arithmetic 19-24, 68-9 with prime base 22-4, 68-9 multiplication in number systems 2-3, 6, 8, 10, 24, 68, 79-80 matrix 14--15, 16--18, 19, 24, 44--7, 48-9, 69 modulo n 19-24, 68-9, 75-6, 86--7, 126 of vectors 12-14 'multiplication table' 21-4, 33-6, 50--2, 61; see also Cayley table natural numbers 1-4, 5, 7, 24, 56--7, 67-8, 89, 126 non-zero integers modulo p 68-9, 75--6, 126 rationals 68, 79-80, 87, 89, 93, reals 57, 68, 87, numbers integers 5-7, 68, 82-5, 125 natural 1-4, 5, 7, 24, 56--7, 67-8, 89, 126 rational 7-9, 24, 68, 83, 87, 89, 93, real 9-11, 24, 56--7, 68, 80, 81, 88, 130
INDEX systems of 1-11, 24, 64, 67-9 operation 1-2, 6, 11, 16, 19, 29-32, 39-41, 46, 61, 64, 7G--8, 89, 105, 124--7; see also addition, composition, multiplication; associative, cancellative, commutative permutation composition 28-36, 50, 124--5, full description 27, identity 31, 36, 69 inverse 31-2, 36 system 32-6, 64, 69, 77, 9G--3, 103, 106 Platonic solids 55--6; see also cube powers (of a group element) 8G--2, 83, 85-6, 93, 109, 125 rationals (rational numbers) 7-9, 24, 68, 83, 87, 89, 93, reals (real numbers) 9-11, 24, 56-7, 68, 80, 81, 88, reflection 12, 4G--2, 45, 48-9, 5G--2, 53-4, 60, 62-3, 84--5 rotation 14, 4G--5, 47-9, 5G--2, 53-4, 57--60, 62, 63, 84--5, 113-23 fixing a colouring 113-22 semigroup 70, 125-8 'snark' 95--6 solids Platonic 55-6; see also cube symmetries of 52-6, 58--60, 63, 115, 121-2 subgroups definition 87-8, extracted using Cayley table 9{}-3 of S 3, complete list 9G--3 subtraction (inverse for addition) 5, 8, 10, 12, 14, 15-16, 21, 22, 24 symmetry and matrices 14, 42-7, 48-9, 53-4, 63 definition 37-42, identity 45, 50-4, 56, 58--63, 66, 69-70, 113-14, 116-17 inverse 57, 61, 63 of coloured tiles 113-17 of quadrilaterals 38-9, 42, 46, 51-2, 78, 103, 106 of solids 52-6, 58--60, 63, 115, 121-2 of tilings 6{}-3 of triangles 37-8, 39-40, 42, 47-9, 5{}-1,57, 77, 84,103 systems 5G--2, 64, 6G--70, 77-8, 103, system infinite 19, 24, 6G--1, 63, of matrices 15-19, 69, 87-8, 126 of numbers 1-11, 24, 64, 67-9 of permutations 32--6, 64, 69, 77, 9{}-3, 103, 106 of rotational symmetries 56--60, 113-21, 123, of symmetries 5G--2, 64, 6G--70, 77-8, 103, tiles, colouring types of 112-17, 120, 122-3 tiling 6{}-3, translation 40, 45, 6{}-3 trivial group 33, 99, 128 vectors 11-14, 15, 44, 69, 88-9, 128 product of 12-14, 128 zero as an additive identity 4, 10, 12, 15, 2G--1, 24, 67-8, 82, 89 division by 7, 9, 18, 68, matrix 15, 69, 88 vector 12, 69 131