2 Group Theory The theory of groups is a deep mathematical subject, one which will not receive a comprehensive treatment here. The goal will be motivation and overview. 2.1 Groups The question might first arise in elementary school. We learn of the laws of addition and multiplication, and we see a pattern; there can exist “operations” which input two numbers and output a third. The question attaches itself to the curious mind: what other operations exist, or could exist? Certainly, we could invent operations to our hearts' content, simply by assigning rules for combining numbers, tantamount to a multiplication table. If we want to assign 5 ＃ 6 = 53, we are free to do so. The more nuanced question becomes: which operations are useful, or interesting? Are addition and multiplication chosen because they are the only interesting and useful operations, or are they simply the most interesting and useful? How do we define what is useful to us? Notice that the idea of composing two things to get a new thing isn't a concept devoted simply to numbers. For example, when we compose two rotations (by performing both in succession), the combination is a rotation. When we compose functions, the result is a function. When we multiply matrices, the result is a matrix. In the realm of physics, if a physical state or theory has a set of symmetries, we can compose two symmetries, and the result will be a symmetry. This question has a wide scope of answers. First, we abstract the problem to the realm of set theory. In this language, we study what is known as a law of composition. A law of composition is a map which combines two elements from a set to produce another element of the set: f : S×S S (2.1) ° = ∈ s1, s2 ⇢ s1 s2 s3 S (2.2) Thus, a law of composition has the property of closure; when we combine two elements from the set S, we remain in the set S. However, it is still a notion fairly devoid of structure. With such an unrestricted definition, it seems unlikely we have yet produced something useful. For example, let us invent a new operation, which we shall call the “turns into” operation, denoted “ti>”. The “turns into” operation will be defined as follows: a ti> b=b , for all b∈S (2.3) This is a bona fide law of composition, which is well-defined on any set S. The multiplication table would be quite simple to produce: ti> a b c ⋯ a a b c ⋯ b a b c ⋯ (2.4) c a b c ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ As you can see, this composition law is a bit stubborn; it doesn't matter at all what the first element is. Eventually, we want the objects of study to actually be “compositions”, that is, we want the composed element “a ti> b” to somehow retain some of the information about how it is composed. For the case of the “turns into” operation, this is not possible. This will not be a satisfactory operation to be placed under the category of “group”. Examples Before we seek a formal definition, it is worthwhile to provide some useful examples for context and motivation. Later, when we have developed the definition, we will show that all of these examples indeed satisfy the group axioms. We start with the real numbers, with two possible composition laws: addition or multiplication. In the case of multiplication, it turns out that we need to restrict ourselves to the real numbers with zero removed: R* = R\{0}. We can sometimes form groups out of closed subsets of these spaces. For example, the positive real numbers R⁺ also form a group under multiplication, and the integers Z form a group under addition. We can even go further, looking at the group of integers modulo n. Given an integer n, any integer m can be written in the form m=anb , where0bn (2.5) In simpler language, b is the remainder when m is divided by n. We write m≡b mod n (2.6) meaning m and b are separated by a multiple of n. The set of integers modulo n contains n elements, in that there are n possible remainders when dividing by n. Note that they can add; it works exactly like addition in the integers, except that it wraps around at n: n≡0 mod n (2.7) Because addition on this set is cyclic, this group is known as the cyclic group of order n. It is denoted Zn. Problem 2.1 Show that addition of integers is preserved when looking at integers modulo n: ab=c ⇒ ab≡c mod n (2.8) also, show that multiplication is preserved: a⋅b=c ⇒ a⋅b≡c mod n (2.9) Problem 2.2 Given the results of Problem 2.1, we look at a specific example: 10≡1 mod 3 (2.10) meaning 10⋅a≡a mod 3 (2.11) and 10m ≡1 mod 3 (2.12) Use these facts to show that a number is divisible by three if and only if the sum of its digits is divisible by three. The symmetric group, Sn, is the set of permutations of n elements. We can compose permutations in the natural way; s3 =s2 ° s1 (2.13) is the permutation found by performing the permutation s₁ followed by s₂. As an example of a permutation group, the group S₃ might represent the six possible ways of rearranging three playing cards. The symmetric group is our first example of a group whose multiplication law is not commutative. The order of permutations is important: s2° s1≠s1 ° s2 (2.14) Figure 2.1 The symmetric group is not abelian; the order of permutations matters. For example, if we have three playing cards, placed in the order J-Q-K, and we perform a cyclic permutation leading to K-J-Q, then switch the first two cards, we have J-K-Q. If we instead were to switch the first two cards first, Q-J-K, then perform the cyclic permutation, we have K-Q-J, not J-K-Q. A group whose multiplication law commutes, so that the order doesn't matter, is called “abelian”. The symmetric group is not abelian, for n > 2. Often times, we can define a group as “the set of symmetries of ____”. For example, the Klein Four group is the set of symmetries of a rectangle. There are two reflections and one 180° rotation which compose this group. The dihedral group Dn is the set of symmetries of a regular n-gon. There always exist n rotations and n reflections which map the n-gon to itself. Groups can also simply be specified by a multiplication table. For example, there is the quaternion group H given by the eight elements {±1, ±i, ±j, ±k}, with the rules i2 = j 2=k2=−1 (2.15) i⋅j=− j⋅i=k j⋅k=−k⋅j=i (2.16) k⋅i=−i⋅k= j these rules give us the following multiplication table: 1 −1 i −i j − j k −k 1 1 −1 i −i j − j k −k −1 −1 1 −i i − j j −k k i i −i −1 1 k −k − j j −i −i i 1 −1 −k k j − j (2.17) j j − j −k k −1 1 i −i − j − j j k −k 1 −1 −i i k k −k j − j −i i −1 1 −k −k k − j j i −i 1 −1 Clearly, the quaternions are another example of a nonabelian group. Matrix multiplication provides another group multiplication law. Specifically, the set of all invertible n × n matrices is a group, which we call the general linear group, GLnR. There are many subsets of GLnR which are also groups by themselves. As was hinted at previously, the set of spatial rotations and reflections (about a given point) forms a group, On, the set of rotations and reflections in n dimensions. In addition, there is Tn, the group of translations in n dimensions. It is possible to combine these groups, since the composition of a rotation with a translation is just the same rotation with its point of origin translated. This is known as the group of rigid transformations in n dimensions. It includes all spatial translations, and all rotations about all points. For an abstract example of a group, think of the set of bijective maps from a set to itself. The composition law is just given by composition of maps: hx= f ° g x= f g x (2.18) Group Axioms When we formulate the theory of groups, we will need additional structure that reflects our need to retain information about how an element is constructed out of other elements. When we compose two elements a · b = c, it should be possible to “re-separate” the element c into the combination “a · b”. For example, we can separate a translation of n- dimensional space into a series of n consecutive translations, one along each dimension. We can also separate any permutation into a series of two-element swaps. It is this concept which will motivate additional restrictions added to a set with a law of composition, making it worthy of being what we call a group. The property of being able to rewrite an element as a combination of constituent elements is primarily encapsulated in the notion of associativity. Say we take a combination of two elements, and wish to compose it with another element.