Lecture 1: Groups

Lecture 1: Groups 1 Groups 1.1 Introduction The lecturer is Davesh Maulik. These notes are taken by Jakin Ng, Sanjana Das, and Ethan Yang, and the note-taking is supervised by Ashay Athalye. Here is some basic information about the class: • The text used in this class will be the 3rd edition of Algebra, by Artin. • The course website is found on Canvas, and the problem sets will be submitted on Gradescope. • The problem sets will be due every Tuesday at midnight. Throughout this semester, we will discuss the fundamentals of linear algebra and group theory, which is the study of symmetries. In this class, we will mostly study groups derived from geometric objects or vector spaces, but in the next course, 18.7021 , more exotic groups will be studied. As a review of basic linear algebra, let’s review invertible matrices. Defnition 1.1 An n×n matrixa A is invertible if there exists some other matrix A−1 such that AA−1 = A−1 A = I, the n×n identity matrix. Equivalently, A is invertible if and only if the determinant det(A) ̸= 0. a An array of numbers (or some other type of object) with n rows and n columns Example 1.2 (n  = 2) a b Let A = be a 2×2 matrix. Then its inverse A−1 is c d 1 ad−bc  d −c  −b . a Example 1.3 (GLn (R)) A main example that will guide our discussion of groupsa is the general linear group, GLn (R), which is the group of n×n invertible real matrices. a The concept of a group will be feshed out later in this lecture Throughout the course, we will be returning to this example to illustrate various concepts that we learn about. 1.2 Laws of Composition With our example in mind, let’s start. Guiding Question How can we generalize the nice properties of matrices and matrix multiplication in a useful way? Given two matrices A, B ∈ GLn (R), there is an operation combining them, in particular matrix multiplication, which returns a matrix AB ∈ GLn (R).2 The matrices under matrix multiplication satisfy several properties: • Noncommutativity. Matrix multiplication is noncommutative, which means that AB is not necessarily the same matrix as BA. So the order that they are listed in does matter. • Associativity. This means that (AB)C = A(BC), which means that the matrices to be multiplied can be grouped together in diferent confgurations. As a result, we can omit parentheses when writing the product of more than two matrices. • Inverse. The product of two invertible matrices is also invertible. In particular, (AB)−1 = B −1 A−1 . 1 Algebra 2 Since 2 the determinant is multiplicative, det(AB) = det(A) det(B), which is nonzero. Lecture 1: Groups Another way to think of matrices is as an operation on a diferent space. Given a matrix A ∈ GLn (R), a function or transformation on Rn3 can be associated to it, namely TA : Rn −→ Rn #» v = (x1 , · · · , xn ) 7−→ A #v»4 . Since A #v» is the matrix product, we notice that TAB ( #v») = TA (TB ( #v»)), and so matrix multiplication is the same as function composition. With this motivation, we can defne the notion of a group. Defnition 1.4 (Group) A group is a set G with a composition (or product) law G×G −→ G (a, b) 7−→ a · b5 fulflling the following conditions: • Identity. There exists some element e ∈ G such that a · e = e · a = a • Inverse. For all a ∈ G, there exists b ∈ G, denoted a−1 , such that a · b = b · a = e. • Associative. For a, b, c ∈ G, (ab)c = a(bc). Also denoted ab In the defnition, both the frst and second conditions automatically give us a unique inverse and identity. For example, if e and e′ both satisfy property 1, then e · e′ = e = e′ , so they must be the same element. A similar argument holds for inverses. Why does associativity matter? It allows us to defne the product g1 · g2 · · · · · gn without the parentheses indicating which groupings they’re multiplied in. Defnition 1.5 Let g taken to the power n be the element g n = g · · · · · g for n > 0, g n = g −1 · · · · · g −1 for n < 0, and e for | {z } | {z } n times n times n = 0. Example 1.6 Some common groups include: Group GLn (R)a Zb × C = C \ {0}c Composition Law matrix multiplication + × Identity In 0 1 Inverse A 7→ A−1 n 7→ −n z 7→ z1 a The general linear group integers under addition c The complex numbers (except 0) under multiplication b The For the last two groups, there is additional structure: the composition law is commutative. This motivates the following defnition. Defnition 1.7 A group G is abelian if a · b = b · a for all a, b ∈ G. Otherwise, G is called nonabelian. 3 Vectors 4 The with n entries which are real numbers. notation A #v» refers to the matrix product of A and #» v , considered as n×n and n×1 matrices. Lecture 1: Groups Often, the composition law in an abelian group is denoted + instead of ·. 1.3 Permutation and Symmetric Groups Now, we will look at an extended example of another family of nonabelian groups. Defnition 1.8 Given a set S, a permutation of S is a bijection a p : S −→ S. a A function f : A −→ B is a bijection if for all y ∈ B, there exists a unique x ∈ A such that f (x) = y. Equivalently, it must be one-to-one and onto. Defnition 1.9 Let Perm(S) be the set of permutations of S. In fact, Perm(S) is a group, where the product rule is function composition.6 • Identity. The identity function e : x 7−→ x is is the identity element of the group. • Inverse. Because p is a bijection, it is invertible. Let p−1 (x) be the unique y ∈ S such that p(y) = x. • Associativity. Function composition is always associative. Like groups of matrices, Perm(S) is a group coming from a set of transformations acting on some object; in this case, S. Defnition 1.10 When S = {1, 2, · · · , n}, the permutation group Perm(S) is called the symmetric group, denoted Sn . Defnition 1.11 For a group G, the number of elements in the set G, |G|, is called the order of the group G, denoted |G| or ord(G). The order of the symmetric group is |Sn | = n!7 so the symmetric group Sn is a fnite group. For n = 6, consider the two permutations p and q i p(i) 1 2 2 4 3 5 4 1 5 3 6 6 i q(i) 1 3 2 4 3 5 4 6 5 1 6 , 2 where the upper number is mapped to the lower number. We can also write these in cycle notation, which is a shorthand way of describing a permutation that does not afect what the permutation actually is. In cycle notation, each group of parentheses describes a cycle, where the number is mapped to the following number, and it wraps around. Example 1.12 (Cycle notation) In cycle notation, p is written as (124)(35), where the 6 is omitted. In the frst cycle, 1 maps to 2, 2 maps to 4, and 4 maps to 1, and in the second cycle, 3 maps to 5 and 5 maps back to 3.a a In 6 We fact, we say that p has cycle type (3, 2), which is the lengths of each cycle. can check that the composition of two bijections p ◦ q is also a bijection. number of permutations of the numbers 1 through n is n! — there are n possibilities for where 1 maps to, and then n − 1 for where 2 maps to, and so on to get n(n − 1) · · · (2)(1) = n! 7 The Lecture 1: Groups Similarly, q is written as (135)(246).8 In cycle notation, it is clear that there are multiple ways to write or represent the same permutation. For example, p could have been written as (241)(53) instead, but it represents the same element p ∈ S6 . Cycle notation allows us to more easily invert or compose two permutations; we simply have to follow where each number maps to. Example 1.13 (Inversion) The inverse p−1 fips the rows of the table: 2 1 i p(i) 4 2 5 3 1 4 3 5 6 6 In cycle notation, it reverses the cycles, since each number should be mapped under p−1 to the number that maps to it under p: p−1 = (421)(53) = (142)(35). Example 1.14 (Composition) The composition is q ◦ p = (143)(26). Under p, 1 maps to 2, which maps to 4 under q, and so 1 maps to 4 under q ◦ p.a Similarly, 4 maps to 3 and 3 maps back to 1, which gives us the frst cycle. The second cycle is similar. a Remember that the rightmost permutation is applied frst, and then the leftmost, and not the other way around, due to the notation used for function composition. Example 1.15 (Conjugation) Another example of composition is p−1 ◦ q ◦ p = (126)(345). This is also known as conjugation of q by p.a a Notice that under conjugation, q retains its cycle type (3, 3). In fact, this is true for conjugation of any element by any other element! 1.4 Examples of Symmetric Groups For n ≥ 3, Sn is always non-abelian. Let’s consider Sn for small n ≤ 3. Example 1.16 (S1 ) In this case, S1 only has one element, the identity element, and so it is {e}, the trivial group. Example 1.17 (S2 ) For n = 2, the only possibilities are the identity permutation e and the transposition (12). Then S2 = {e, (12)}; it has order 2. Once n gets larger, the symmetric group becomes more interesting. 8 It has cycle type (3, 3). Lecture 1: Groups Example 1.18 (S3 ) The symmetric group on three elements is of order 3! = 6. It must contain the identity e. It can also contain x = (123). Then we also get the element x2 = (132), but x3 = e. Higher powers are just x4 = x, x5 = x2 , and so on. Now, we can introduce y = (12), which is its own inverse, and so y2 = e . Taking products gives xy = (13) and x2 y = (23). So we have all six elements of S3 : S3 = {e, (123), (132), (12), (13), (23)}. In fact, yx = (23) = x2 y, so taking products in the other order does not provide any new elements. The relation yx = x2 y holds. In particular, using the boxed relations, we can compute any crazy combination of x and y and reduce it to one of the elements we listed. For example, xyx−1 y = xyx2 y = xyyx = xy 2 x = x2 .