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F1.3YE2 Revision Notes on Theory

1 Introduction

By a group we mean a together with some algebraic (such as or of ) 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 together with addition, the set of nonzero real numbers together with multiplication, the set of invertible n n matrices together with multiplication. × 2 Binary Operations

The formal definition of a group uses the notion of a .A binary operation on a set A is a map A A A, written (a, b) a b. Examples include most of the∗ standard operations× → on the real or7→ complex∗ numbers, such as addition (a + b), multiplication (a b), (a b). Other examples of binary operations (on suitably defined sets)× are − ab (on the set of positive reals, for example), composition of functions, 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. ∗ ∗ ∗ ∀ ∈ Addition and multiplication of numbers is commutative, as is addition of matri- ces or vectors, and intersection of sets, etc. Subtraction of numbers is not commutative, nor is . Definition A binary operation on a set A is associative if a (b c) = (a b) c a, b, c A. ∗ ∗ ∗ ∗ ∗ ∀ Addition∈ 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 for a binary operation on a set A is an e A such that e a = a = a e a A. ∗ ∈ Examples∗ are 0∗ for∀ addition∈ 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 ∈a are identities for . ∗ ∈ ∗ ∗ ∗ 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 ( 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 inverse∈ 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 (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 identity∈ 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 V is a group with respect to vector addition. The identity is the zero vector, and the inverse of v V is v. ∈ −

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 to{ multiplication∈ | | of} 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 In. × 6. Let X be a set. Then the set (X) of all 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 on X.→ In the particular case where X is the set 1, 2, . . . , n , this group is { } denoted Sn, and called the symmetric group of degree n.

7. Let n > 0 be an , and let Zn denote the set 0, 1, . . . , n 1 . Define a { − } 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 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,∈ ) 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. { } 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 { − − } 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 { } 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 = .  3 2 1   2 1 3   2 3 1   3 1 2  Its Cayley table is ι σ τ α β γ ◦ι ι σ τ α β γ σ σ τ ι γ α β τ τ ι σ β γ α α α β γ ι σ τ β β γ α τ ι σ γ γ α β σ τ ι

Note for example that α β = β α, so that S3 is not abelian. ◦ 6 ◦ 6

A of a group G is a subset H G that is also a group with respect to the same binary operation as G. Examples⊆ include Z as a subgroup of R (with respect 1 to addition), R∗ and S as subgroups of C∗ with respect to multiplication. It is important to recognise when a subset of a group G is actually a subgroup of G. The following result gives a useful criterion. Theorem (The subgroup test) Let G be a group with respect to a binary operation , and let H be a subset of G. Then H is a subgroup of G if and only if the following three∗ conditions are satisfied:

5 1. Closure: x y H x, y H. ∗ ∈ ∀ ∈ 2. Identity: eG H, where eG is the identity element of G. ∈ 1 1 3. Inverse: x− H x H, where x− is the inverse of x in G (with respect to ). ∈ ∀ ∈ ∗ Examples

1. Let n > 0 be an integer and let nZ = nx : x Z , the set of integers divisible by n. Then nZ is a subgroup of Z with{ respect∈ } to addition. For the closure property, note that nx + ny = n(x + y). The identity element is 0 = 0n Z. The inverse in Z of nx nZ is nx = n( x) nZ. ∈ ∈ − − ∈ 2. Let G be the group of n n invertible matrices with respect to matrix multipli- cation, and let H be the× set of matrices A G such that det(A) = 1. Then H is a subgroup of G. The closure property follows∈ since det(AB) = det(A) det(B); 1 In H since det(In) = 1; and finally if det(A) = 1 then det(A− ) = 1. ∈ 3. Let G = Sn and let H = σ G : σ(1) = 1 . Then H is a subgroup of G. To check the closure property,{ ∈ if σ, τ H} then σ(1) = τ(1) = 1, so (σ τ)(1) = σ(τ(1)) = σ(1) = 1, and so σ ∈τ H. Clearly the identity map ◦ ◦ ∈ 1 sends 1 to 1, so belongs to H. Finally, if σ H then σ(1) = 1, so σ− (1) = 1 1 ∈ and σ− H. ∈ Definition The order of a group G is the of elements in G (finite or infinite). It is denoted G . Note that G 1, since every group contains at least one element, namely the identity.| | The order| | ≥of an element g G is the least positive integer k such that gk is equal to the identity element of G,∈ where gk denotes g g ... g (k times). If no such positive integer k exists, then g has infinite order. ∗ ∗ ∗ Lemma Let G be a group and g G. Then the set g = gk : k Z is a subgroup of G, and its order is equal to that∈ of g. h i { ∈ } 0 1 k 1 k (Here g = eG, g− is the inverse of g in G, and if k > 0 then g− denotes (g− ) , which is the inverse of gk.) Theorem (Lagrange’s Theorem) Let G be a finite group and H a subgroup of G. Then the order of H divides that of G. Corollary Let G be a group and g G. Then the order of g divides that of G. ∈ Hence, for example, the group S5 contains no element of order 7, since S5 = 5! = | | 60. (In fact, S5 contains elements of orders 1, 2, 3, 4, 5 and 6, but no elements of order greater than 6.)

6 Definition Let H be a subgroup of G, and g G. Then the left coset of H in G represented by g is the subset gH = gh, h H∈ of G, and the right coset of H in G represented by g is the subset Hg ={ hg, h∈ H} of G. { ∈ } Exercise Show that:

1. the left cosets form a partition of G;

2. the map h gh is a bijection from H to gH. 7→ Hence prove Lagrange’s Theorem. Definition A subgroup N G is normal if each left coset is also a right coset. (That ⊂ 1 is gN = Ng for all g G.) Equivalently, gng− N for all g G and all n N. ∈ ∈ ∈ ∈ Example Let G = S5, H = σ S5, σ(1) = 1 . Then H is a subgroup of G. The { ∈ } 1 left coset gH is σ S5, σ(1) = g(1) , while the right coset Hg is σ S5, σ− (1) = 1 { ∈ } { ∈ g− (1) . If g(1) = 1, then these two subsets are different, so H is not normal. } 6 Example Let G = S5 and let A5 be the set of all even permutations in S5. Then the left coset gA5 consists of all even permutations (if g is even), or all odd permutaitons (if g is odd). In other words there are precisely two left cosets A5 and S5 A5. \ Similarly there are two right cosets A5 and S5 A5, so right and left cosets are the \ same thing, and A5 is normal.

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Definition Let (G, ) and (H, ) be groups. A map f : G H is a if ∗ † → f(x y) = f(x) f(y) x, y G. ∗ † ∀ ∈

Example The exponential map from (R, +) to (R+, ) is a homomorphism (since exp(x + y) = exp(x) exp(y)). × Lemma Let f : G H be a homomorphism. Then the image of f, → Im(f) := f(x), x G H, { ∈ } ⊂ is a subgroup of H, and the of f,

Ker(f) := x G, f(x) = eH G, { ∈ } ⊂ is a normal subgroup of G.

7 Clearly, a homomorphism f : G H is surjective iff Im(f) = H. Less obvi- → ously (but easily checked), it is injective iff Ker(f) = eG , the trivial subgroup of G. A homomorphism that is both injective and surjective{ } (ie bijective) is called an isomorphism, and two groups G, H are isomorphic (denoted G ∼= H) if there is an isomorphism G H. (Exercise: understand→ why this is the same thing as the less formal definition of iso- morphic groups given earlier.) (Another exercise: check that ∼= is an equivalence relation between groups.) Example The exponential map from (R, +) to (R+, ) is an isomorphism. It is a × homomorphism, and bijective (with inverse ln : R+ R). Hence the groups (R, +) → and (R+, ) are isomorphic. × 8 Quotient Groups

Let G be a group, and N a normal subgroup. The (or factor group) G of G by N, denoted G/N or N , is defined to be the set of left cosets gN for all g G. (Since N is normal, this is the same as the set of right cosets Ng.) The binary operation∈ on G/N is defined by

(xN)(yN) := (xy)N x, y G. ∀ ∈ Of course, one needs to check some things - firstly that the definition does not depend on the choices x, y of representatives of the two cosets, and then that the resulting binary operation on G/N satisfies all the axioms for a group. (Exercise: check these things.) Example The set 2Z of even integers is a normal subgroup of the group (Z, +). The quotient group Z/2Z has two elements: 0 + 2Z = 2Z (the set of all even integers and 1 + 2Z (the set of all odd integers). The Cayley table of this group is:

+ 0 + 2Z 1 + 2Z 0 + 2Z 0 + 2Z 1 + 2Z 1 + 2Z 1 + 2Z 0 + 2Z Z Z Z Note that this is just like addition modulo 2. In fact /2 ∼= 2. This is a special case of an important theorem. First Isomorphism Theorem Let f : G H be a homomorphism. Then → G = Im(f). Ker(f) ∼

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