F1.3YE2 Revision Notes on Group Theory 1 Introduction By a group we mean a set together with some algebraic operation (such as addition or multiplication of numbers) that satisfies certain rules. There are many examples of groups in Mathematics, so it makes sense to understand their general theory, rather than try to reprove things every time we come across a new example. Common examples of groups include the set of integers together with addition, the set of nonzero real numbers together with multiplication, the set of invertible n n matrices together with matrix multiplication. × 2 Binary Operations The formal definition of a group uses the notion of a binary operation.A binary operation on a set A is a map A A A, written (a; b) a b. Examples include most of the∗ standard arithmetic operations× ! on the real or7! complex∗ numbers, such as addition (a + b), multiplication (a b), subtraction (a b). Other examples of binary operations (on suitably defined sets)× are exponentiation− ab (on the set of positive reals, for example), composition of functions, matrix addition and multiplication, subtraction, vector addition, vector procuct of 3-dimensional vectors, and so on. Definition A binary operation on a set A is commutative if a b = b a a; b A. ∗ ∗ ∗ 8 2 Addition and multiplication of numbers is commutative, as is addition of matri- ces or vectors, union and intersection of sets, etc. Subtraction of numbers is not commutative, nor is matrix multiplication. Definition A binary operation on a set A is associative if a (b c) = (a b) c a; b; c A. ∗ ∗ ∗ ∗ ∗ 8 Addition2 and multiplication (of numbers and matrices) are associative. Examples of nonassociative binary operations are subtraction (of anything), exponentiation of positive reals, and vector product. Definition An identity for a binary operation on a set A is an element e A such that e a = a = a e a A. ∗ 2 Examples∗ are 0∗ for8 addition2 of numbers, 1 for multiplication of numbers, the iden- tity n n matrix for matrix multiplication. Not all binary operations have identities, however:× an example is subtraction of numbers. Definition Let be a binary operation on a set A and let a A. An inverse for a (with respect to ∗ ) is an element b A such that a b and b 2a are identities for . ∗ 2 ∗ ∗ ∗ 1 2 Thus for example 5 is an inverse for 5 with respect to addition of integers; 3 is 3 − an inverse for 2 with respect to multiplication of positive real numbers. Other ex- amples are matrix inverses (matrix multiplication) and appropriately defined inverse functions (function composition). Lemma If a binary operation on a set has an identity, then this identity is unique. Proof. Suppose that e and f are both identities for a binary operation on a set A. Then e = e f = f. The first equality holds because f is an identity.∗ The second holds because∗ e is an identity. Lemma If a A has an inverse with respect to an associative binary operation on A, then the inverse2 is unique. ∗ Proof. Suppose that b and c are both inverses for a. Then b = b (a c) = (b a) c = c. The first equaliy holds because a c is an identity, the second because∗ ∗ is associative,∗ ∗ and the third because b a is an∗ identity. ∗ ∗ In the last result, the associativity of is definitely used in the proof. In fact the result is not in general true for nonassociative∗ binary operations. 3 Groups We are now ready to give a definition of a group. Definition A group is a nonempty set G together with a binary operation on G which is associative and such that every element of G has an inverse. If the∗ binary operation is also commutative, then G is called a commutative, or abelian group (after a 19th∗ century Norwegian mathematician Niels Abel). In particular, G has an identity, since there is at least one element a G, which has an inverse b, and a b is an identity. By the above results the identity2 in G is ∗ unique. We often denote it e = eG or 1 = 1G. The inverse of an element a is also 1 unique. We often denote ita ¯, or a− . If the binary operation on G is addition of some sort, it is usual to denote it by +, the identity by 0 or 0G, and the inverse of an elment a by a. − Examples 1. Z, Q, R and C are all groups with respect to addition. In each case 0 is the identity and the inverse of x is x. − 2. Any vector space V is a group with respect to vector addition. The identity is the zero vector, and the inverse of v V is v. 2 − 2 3. The set Q∗ of nonzero rational numbers is a group with respect to multiplication. a b R C The identity is 1, and the inverse of b is a . Similarly the sets ∗ and ∗ of nonzero real and complex numbers, respectively, are groups with respect to multiplication. 4. The set S1 = z C : z = 1 of complex numbers of modulus 1 is a group with respect tof multiplication2 j j ofg complex numbers. 5. The set of invertible n n matrices forms a group with respect to matrix × multiplication. The identity element is the n n identity matrix In. × 6. Let X be a set. Then the set (X) of all permutations of X, that is, bijective maps X X, forms a group withS respect to composition of maps. The identity map X !X is the identity element. This group is called the symmetric group on X.! In the particular case where X is the set 1; 2; : : : ; n , this group is f g denoted Sn, and called the symmetric group of degree n. 7. Let n > 0 be an integer, and let Zn denote the set 0; 1; : : : ; n 1 . Define a f − g binary operation on Zn by a b = a + b if a + b < n, and a b = a + b n ∗ ∗ ∗ − otherwise. Then Zn is an abelian group with respect to , with identity 0. The ∗ inverse of a > 0 in Zn is n a (the inverse of 0 is 0). This group is called the cyclic group of order n. The− binary operation is usually denoted +, and referred to as addition modulo n. ∗ 4 Cayley tables One way of describing a binary operation on a set G (provided G is not too big) is to form a grid with rows and columns labelled∗ by the elements of G, and enter the element a b in the cell in row a and column b (for all a; b G). This is called a multiplication∗ table or a Cayley table or (in the case where (G;2 ) is a group) a group table. ∗ Example + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 This is the Cayley table for Z4, the cyclic group of order 4. Example 3 e a b c ∗e e a b c a a e c b b b c e a c c b a e This describes a binary operation on the set G = e; a; b; c with respect to which G is a group. f g Two groups G and H are said to be isomorphic if they have Cayley tables which are identical, except for relabelling of the elements. For example G = 1; i; 1; i f − − g is a group with respect to multiplication of complex numbers. It is isomorphic to Z4, because its Cayley table 1 i -1 -i ×1 1 i -1 -i i i -1 -i 1 -1 -1 -i 1 i -i -i 1 i -1 is identical to that of Z4, if we relabel elements of Z4 by the rule 0 1, 1 i, 2 1, 3 i (in other words, k ik; k = 0; 1; 2; 3). 7! 7! 7! − 7! − 7! Question Is the group G = e; a; b; c in the second example above isomporphic to f g Z4? If G is a group containing only two elements, then G is isomorphic to Z2. To see this, note that one of the elements of G is the identity e. Let g be the other element 1 of G. The e g = g = g e, where is the binary operation in G. What is g− ? Since 1 ∗ ∗ 1 ∗ 1 g− g = e = g = e g, g− = e, so g− = g. Hence g g = e, and this determines the Cayley∗ table6 of G∗ as 6 ∗ e g ∗e e g g g e Clearly this is the same as that of Z2, using the relabelling 0 e, 1 g. 7! 7! Exercise Show that any group containing exactly three elements is isomorphic to Z3. 4 5 Calculations with permutations Symmetric groups are common examples of finite groups. They are not in general abelian, as we shall see below. The symmetric group Sn contains n! elements. A 1 2 : : : n common way to denote permutations is in the form σ = , where a1 a2 : : : an 1 2 : : : n 1 2 : : : n ak = σ(k). If τ = , then σ τ = , where b1 b2 : : : bn ◦ c1 c2 : : : cn ck = abk . Example 1 2 3 1 2 3 1 2 3 = : 1 3 2 ◦ 3 2 1 2 3 1 Example 1 2 3 1 2 3 The group S has precisely six elements: ι = , α = , 3 1 2 3 1 3 2 1 2 3 1 2 3 1 2 3 1 2 3 β = , γ = , σ = and τ = .