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Finite Groups: Basic Structure Theory August 23, 2018 11:7 ws-book9x6 A Practitioner’s Guide to Group Theory in Physics 11081-main page 125 Chapter 3 Finite groups: Basic structure theory In this chapter we study generic properties of finite groups, whereas Chapter 4 will be concerned with linear representations of finite groups. A special attention to the permutation or symmetric group Σn is given, due to its deep relation with Young diagrams, that will turn out to be a central tool for the study of representations of classical Lie groups (see Chapter 9). As a geometrical illustration we study finite dimensional subgroups of E(2) and E(3), the two and three-dimensional Euclidean groups respectively. We further study with some details finite dimensional subgroups of SO(2) and SO(3) and their relationship to polygons or regular (also called Platonic) solids. 3.1 General properties of finite groups In this section we review the main properties and results of the theory of finite groups that will be useful for the study of linear representations. We only enumerate those results that, in one form or the other, will be used explicitly. For additional properties, the reader is referred to the literature on structural and combinatorial group theory (see e.g. Huppert (1967) and references therein). We have seen previously that in an Abelian group G two arbitrary ele- ments commute with each other. This corresponds to a particular case of an important equivalence relation, the conjugacy, that will play an important role for the structure and representations of finite groups. A group possess- ing at least one pair of elements x, y such that xy = yx is correspondingly called non-Abelian. 125 August 23, 2018 11:7 ws-book9x6 A Practitioner’s Guide to Group Theory in Physics 11081-main page 126 126 Finite groups: Basic structure theory Definition 3.1. Let G be a group. If the number |G| of elements is finite, we say that the order of G is finite and equal to |G|, otherwise that it is an infinite group.1 We observe that infinite groups are divided into two principal classes, ac- cording to their cardinality. So, a group G is called discrete if it is infinite countable, like the group of integers Z. Groups with an uncountable infi- nite number of elements are called continuous. The general linear group GL (n, R) is an example of this class. Formally, any finite group can be described through its multiplication table, constructed as follows: chose an ordering {e = g1, ··· ,gr} of the r = |G| elements, e being the neutral element, and consider a table, the ith th row and j column of which contains the element gigj. In particular, if the group is Abelian, the table is symmetric with respect to the main diagonal. The multiplication table has the following general property: in each row and column, any element of G appears only once. As an example, consider the following table: ωαβγ δ ε ω ωαβγ δ ε α αβωδ εγ β βωαεγδ (3.1) γ γεδωβα δ δγεαωβ ε εδγβαω It is easy to verify that all group axioms are satisfied for G = {ω, α, β, γ, δ, ε}.Asαγ = δ = ε = γα, we conclude that G is not Abelian. This group of order six is commonly denoted by D3 Σ3. This group can also be obtained in an apparently very different form. In GL (2, C) we consider the matrices associated to the element of the group of Table 3.1 10 01 q 0 ω → ,γ→ ,α→ , 01 10 0 q2 q2 0 0 q 0 q2 β → ,δ→ ,→ , (3.2) 0 q q2 0 q 0 where q3 − 1 = 0. These matrices form a group of order six that coincides with that given by Table 3.1. However, in this form, the group can be 1In the case of the classical groups, although they are infinite, supplementary properties will enable us to associate to them some finite characteristic scalars. August 23, 2018 11:7 ws-book9x6 A Practitioner’s Guide to Group Theory in Physics 11081-main page 127 3.1. General properties of finite groups 127 interpreted geometrically as the group of transformations in the affine plane E2 that leave invariant an equilateral triangle. As will be seen later, this ∼ actually corresponds to a linear representation of the dihedral group D3 = Σ3. Groups given in terms of its multiplication table actually contain all the required information, albeit for groups of high order, they are usually im- practical. The main inconvenience is however the fact that the associative property is not immediately derivable from the table. One of the objec- tives of the structure and representation theory of groups is to condense the information of a group to a minimal amount of data, from which all properties of the isomorphism class can be recovered. q 0 01 Let g = , h = be elements of Σ as given in (3.2). If we 0 q2 10 3 consider powers of this element, we obtain q2 0 q2 0 10 10 g2 = = ,g3 = ,h2 = . 0 q4 0 q 01 01 Clearly the powers g, g2,g3 form a group, called the group generated by an element and commonly written as g. We observe that this group is necessarily Abelian. The generalisation is straightforward. Let n ≥ 2and n n−1 "x ∈ G" be such that x = e. Define the group x = e, x, ··· ,x where " x " = n. Such a group is generated by the element x, and is Abelian by construction. Definition 3.2. AgroupG is cyclic if there exists x = e such that for any y ∈ G there is some n ∈ N with xn = y. In the following, cyclic groups of order n will be denoted by Cn either Zn. As observed before, in general the analysis of a finite group by means of its multiplication table is cumbersome, and does not constitute an optimal strategy to the analysis. Besides their representation as matrix groups, there is an elegant and concise algebraic procedure to describe finite groups that generalises naturally the previous characterisation of cyclic groups, called presentations of groups and basing on a set of generators and a set of relations satisfied by them.2 2The theory of presentations constitutes an important branch of combinatorial group theory. In this text we will only consider the most elementary properties of presentations. A good albeit demanding introduction to the subject is given in [Johnson (1976)]. August 23, 2018 11:7 ws-book9x6 A Practitioner’s Guide to Group Theory in Physics 11081-main page 128 128 Finite groups: Basic structure theory 3.1.1 Subgroups, factor groups Given a group G and a subset K,wesaythatK is a subgroup of G if k1k2 ∈ K for any k1,k2 ∈ K.If{e}= K = G,wesaythatK is a proper subgroup. Usually subgroups are denoted by K<G. As clearly not any subset of G forms a subgroup, it has sense to introduce the notion of subgroup generated by a set X ⊂ G: a a1 ··· j | ∈ ∈ N X = x1 xj xi X, ai . Obviously X is a subgroup, having the property of being the smallest subgroup of G that contains the set X. In particular, given g ∈ G, g is the smallest subgroup containing g. Cyclic groups are just a special case of this construction. Definition 3.3. The order |g| of an element g ∈ G is the order of the cyclic subgroup g. In the preceding example (3.2), we have that |g| = 3, while |h| =2. Thus any element of Σ3 distinct from the identity has order 2 or 3. Generalising the preceding property, for any two subgroups K, H of G we define the product KH = {gh | g ∈ K, h ∈ H} . Lemma 3.1. KH is a subgroup of G if and only if KH = HK. − Proof. In fact, if KH < G is a subgroup, then KH =(KH) 1 = H−1K−1 = HK, as both K and H are subgroups. Conversely, if HK = KH, (KH)(KH)=K (HK) H = K (KH) H = KH, − (KH) 1 = H−1K−1 = HK = KH , showing that the product is a subgroup. QED q 0 01 Example 3.1. Taking again g = , h = in Σ ,letK = g 0 q2 10 3 and H = h. It is easily verified that KH = HK =Σ3. Lemma 3.2. If K, H < G are finite subgroups of G such that HK = KH, then |K||H| |KH| = . |K ∩ H| August 23, 2018 11:7 ws-book9x6 A Practitioner’s Guide to Group Theory in Physics 11081-main page 129 3.1. General properties of finite groups 129 Given g ∈ G and a subgroup K not containing g, we call gK a left coset (Kg right coset, respectively). Cosets are related by means of the bijection − gK → (gK) 1 = Kg−1, showing that the number of left and right cosets is the same. It is immediate to verify that the two following properties are satisfied: (1) |gK| = |K| (thenumberofelementsisthesame) (2) gK ∩ K = ∅ (gK and K have no common elements) It follows from the latter that two cosets gK, hK either coincide or are disjoint, implying that G = K ∪ g1K ∪···∪gs−1K for suitable elements {g1, ··· ,gs−1} not belonging to K. From this partition it clearly follows that |K| divides the order of G. The number of cosets of K in G,givenby|G| / |K| = s is called the index of K in G and denoted by [G : K]. This proves the following result: Proposition 3.1 (Lagrange). Let G be a finite group and let K be a subgroup.
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