Equivalence Relations and Partitions, Normal Subgroups, Quotient Groups, and Homomorphisms Math 356 Abstract We sum up the main features of our last three class sessions, which list of topics are given above. Interspersed throughout these notes are exercises, which are due Tuesday, March 6. Any problem with an (H) behind it is required homework, anything with an (S) is suggested for yourself, and should not be handed in (although I will be happy to go over these problems with you in office hours). There are quite a few problems here, but the six problems that are to be handed in are mostly very short. 1
1 Equivalence relations and partitions Throughout this section, let A and B represent fixed sets. Whenever we refer to a symbol of the form a or a, then we are refering to an element of A (even if there is no subscript), and any symbol of the form b, or b will refer to an element of B. (In this notation, we will replace the by any index value or variable.) We will also assume that the symbols for subset, union, and intersection (,, and respectively) are understood. We can define a new set from A and B, the cartesian product or direct product of A and B, denoted A B, as follows. A B = {(a, b) a A, B B} The set A B is the set of all ordered pairs, whose first coordinates come from the set A, and whose second coordinates come from the set B. We define a relation R between A and B to be a subset of A B, that is R A B. The idea here is that we think that a is related to b by R if and only if (a, b) R. For example, a function f from A to B is a special form of relation that satisfies two further rules. 1. For all a A, there is b B so that (a, b) f. 2. For all a A, if (a, b 1 ) f and (a, b 2 ) f, then b 1 = b 2. Note that in this point of view, a function IS what we usually think of as a graph of a function! Sometimes we focus on relations from a set to itself. For example, we can define the relation < on R by the rule <= {(r 1, r 2 ) r 2 r 1 = r2 r 1 andr 2 r 1 }. There is actually a cheat here, since it is very hard to define the absolute value of something, if you cannot tell if it is greater than zero! Another sort of relation is an equivalence relation. An equivalence relation on A is a subset E of A A which satisfies three rules. 1. For all a A we have that (a, a) E (Reflexivity Rule). 2. If (a 1, a 2 ) E then (a 2, a 1 ) E (Symmetry Rule). 3. If (a 1, a 2 ), (a 2, a 3 ) E then (a 1, a 3 ) E (Transitivity Rule). Exercise 1 (H) We will say that two points in the plane are related if they are on the graph of a line of the form y = 2x + c for some constant c. Does this define an equivalence relation on the plane? What if we use the rule that two points are related if they are both in a line of the form y = cx for some constant c? We often denote the set of an equivalence relation with the symbol, and we denote the fact that (a 1, a 2 ) by the sequence a 1 a 2. Suppose that is an equivalence relation on A, and a A, then we can build the set a = {a A a a }, which is the set of all things in a which are related (equivalent) to a under. This is often called the equivalence class of a. Note that the notation a is a little non-standard, but the use of the preposition under seems quite common. 1
Exercise 2 (H) Suppose is an equivalence relation on A, and that a, b A. If a b, then a = b The set of sets, whose elements are the distinct equivalence classes of the elements of A under, is a partition of A. That is, a collection of disjoint subsets of A whose union is all of A. It will only take a moment s thought to see that given a partition of A, one can define an equivalence relation on A whose equivalence classes are the elements of the partition (note that an element of a partition of A is a subset of A). Exercise 3 (H) For each relation in the first exercise that is an equivalence relation, describe the equivalence classes. 2
2 Normal subgroups and quotient groups For this section, we will refer to a fixed group G, and its subgroup N. We will adopt the convention that symbols of the form g or g are elements of G, and n and n are elements of N. The symbol e will always denote the identity of G (and therefore of N as well). Given two subsets R and S of G, we can define a third subset RS using the binary operation of G as follows. RS = {rs r R, s S} This enables us to define a Multiplication on subsets of G. If R = {r}, then we will often denote the set RS = {rs s S} by the symbol rs instead. Likewise if S = {s}, then we may denote RS by Rs = {rs r R}. As multiplication in G is associative, we will assume the notation g 1 Ng 2 is now understood, given elements g 1, g 2 G. If H is a subgroup of G, and g is an element of G, then we call the set gh a left coset of H in G. We say that N is a normal subgroup of G if and only if, given any element g G, we have gng 1 = N. We will denote that N is a normal subgroup of G by the notation N G. Exercise 4 (H) Suppose g G, and n 1 N, where N G. Show that there is n 2 N so that gn 1 = n 2 g. Exercise 5 (H) Suppose N G, and g 1, g 2 G. Show that if g 1 N g 2 N, then g 1 g 2 N. Conclude that either g 1 N g 2 N =, or g 1 N = g 2 N. Consider the set Q = {gn g G} of left cosets of N in G. Recalling that sets do not allow duplicates, we see that Q usually has fewer elements than G. From the previous exercise, we see that Q is a partition of G. Exercise 6 (H) Show that the sets g 1 Ng 2 N and g 1 g 2 N are equal. Conclude that Q, under the binary operation g 1 Ng 2 N = (g 1 g 2 )N forms a group. (You will need to discuss closure, the existence of an identity, inverses, and associativity.) Q is called the Quotient group of G by N, or a quotient group of G, for short. It is often denoted by the notation Q = G/N, which is read as Q is G mod N or Q is the quotient of G by N, or various similar language. Exercise 7 (S) Is there a quotient group of order 3 from S 3? In other words, can you find a normal subgroup of size two in S 3? Exercise 8 (S) Give an example of the fact that if H G, where H is not normal in G, then the left cosets of H in G do not form a group under coset multiplication. 3
3 Homomorphisms We will delve much deeper into this subject as the course goes on. For now, I simply give a short explanation, definition, and a collection of exercises/facts from class. Given any class of mathematical objects, such as sets, vector spaces, or topological spaces for example, we like to be able to relate these objects to each other. For example, functions that map sets into each other, linear transformations that map vector spaces into each other, and continuous maps that map topological spaces into each other. These special types of mapping relations have one chief aspect in common; each of these operations works as a set function on the underlying sets, but they also preserve the extra structure of the object to some extent. This is not saying much in the world of sets, but for example, linear transformations preserve linear combinations, that is, they do not send lines to curvy curves! Continuous functions do not rip apart points that are close together. (Although, continuous functions may stretch a region between nearby points over a large range, in order to have points which looked close together map to places far apart. This is just a reflection of the idea that the points we started with were not Close enough together to stay together.) When the objects we are concerned with are groups, the map we use is called a homomorphism. A homomorphism between two groups G and H is a set map between the underlying sets that respects the group product operations of the two groups. More formally, we have the the following. Let (G, ) and (H, ) be groups. A function φ : G H is a homomorphism if for all g 1, g 2 G, we have that φ(g 1 g 2 ) = φ(g 1 ) φ(g 2 ). We will typically drop the explicit use of the two group operations from our notation in the future, I only include it in this definition explicitly to emphasize that the two different products are occurring in the two different groups. The homomorphism phi can be thought of as interpreting a multiplication in the group G as a multiplication in the group H. This last approach is often useful if H is simpler than G to understand; we throw away some extra information, hopefully preserving enough of the structure of G within H to answer questions about G! Where there is a homomorphism φ : G H, there is an important subset of G, called the kernel of φ, which is often denoted Ker(φ). It is defined as follows. Ker(φ) = {g G φg = e H } Here I am using e H to denote the identity element of H. The kernel of a homomorphism is very similar, in concept, to the kernel of a linear transformation from linear algebra, and just as important. Here is a list of facts about homomorphisms from class. Any fact with an (S) at the beginning was not shown in class, however, it would be good if you try to prove these facts on your own. Remark 3.1 Let φ : G H be a group homomorphism. 1. φ(g n ) = φ(g) n for any integer n (here a negative integer means the inverse taken to a positive power). 4
2. Ker(φ) is a subgroup of G. 3. (S) Ker(φ) is a normal subgroup of G. 4. (S) If R G, then φ(r) = {h H r R, φ(r) = h} is a subgroup of H. 5. (S) If S H, then φ 1 (S) = {g G φ(g) S}, is a subgroup of G. Here is one more suggested problem for fun... Let K4 = a, b a 2, b 2, a 1 b 1 ab be a group, (this group is called the Klein-four group, recall that the relation [a, b] = a 1 b 1 ab means that a and b commute). Can you find a homomorphism from D 6 onto K4? (Hint: the kernel of the homomorphism must be a normal subgroup of D 6 with three elements.) 5