Burnside’s Theorem Julie Linman, Oregon State University Advisor: Holly Swisher April 2010 Contents 1 Introduction 2 2 Group Theory Background 4 2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Important Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Lagrange’s Theorem and Consequences . . . . . . . . . . . . . . . . . 7 2.2.2 The Isomorphism Theorems . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Introduction to the Representation Theory of Finite Groups 14 3.1 Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Modules and Tensor Products . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Representations and Complete Reducibility . . . . . . . . . . . . . . . . . . . 19 3.4 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4 Proofs of Burnside’s Theorem 33 4.1 A Representation Theoretic Proof of Burnside’s Theorem . . . . . . . . . . . 34 4.2 A Group Theoretic Proof of Burnside’s Theorem . . . . . . . . . . . . . . . . 36 5 Conclusion 38 1 Chapter 1 Introduction Just as prime numbers can be thought of as the building blocks of the natural numbers, in a similar fashion, simple groups may be considered the building blocks of finite groups. More precisely, every group G has a composition series, which is a series of subgroups 1 = G (cid:69)G (cid:69)···(cid:69)G = G, 0 1 k such that G /G is simple for all 0 ≤ i ≤ k − 1, and the Jordan-H¨older Theorem states i+1 i that any two composition series of a group G are equivalent. Therefore, the following goals naturally emerged in finite group theory: 1. Classify all finite simple groups. 2. Find all ways to construct other groups out of simple groups. Toward the end of the 19th century, much of the research in finite group theory was related to the search for simple groups. Following the work of German mathematician Otto Ho¨lder [15] and American mathematician Frank Nelson Cole [7], in 1895 English mathematician William Burnside found all simple groups of order less than or equal to 1092 [3]. However, it was Ho¨lder’s result, stating that a group whose order is the product of two or three primes is solvable [15], that prompted Burnside to consider the following questions: 1. Do there exist non-abelian simple groups of odd order? 2. Do there exist non-abelian simple groups whose orders are divisible by fewer than three distinct primes? In 1904, Burnside answered question 2 when he used representation theory to prove that groups whose orders have exactly two prime divisors are solvable[4]. His proof is a clever application of representation theory, and while purely group-theoretic proofs do exist, they are longer and more difficult than Burnside’s original proof. For more information on Burn- side’s work see [19]. The goal of this paper is to present a representation theoretic proof of Burnside’s Theorem, providing sufficient background information in group theory and the 2 representation theory of finite groups first, and then give a brief outline of a group theoretic proof. In this paper we begin by reviewing some definitions and theorems from group theory in Chapter 2. In particular, we prove Lagrange’s Theorem, the Class Equation, and the Isomorphism Theorems in Sections 2.2.1 and 2.2.2, which are necessary for the proofs of the results concerning solvable groups given in Section 2.2.3. In Chapter 3 we give an in- troduction to the representation theory of finite groups, beginning with a brief discussion of linear algebra and modules in Sections 3.1 and 3.2. Next, in Section 3.3 we define and give examples of representations of finite groups and prove that every representation can be decomposed uniquely into a direct sum of irreducible representations. In Section 3.4 we introduce characters and prove some results about characters that allow us to determine the specific irreducible representations of a group and the decomposition of a general represen- tation into a direct sum of these irreducible representations. Moreover, we prove that the set of characters of irreducible representations of a finite group G form an orthonormal basis for the set of class functions on G. We will then present two proofs of Burnside’s Theorem in Chapter 4. In Section 4.1 we give a proof that relies on results stated in Section 3.4, and in Section 4.2 we outline a purely group theoretic proof. We finish with some consequences of Burnside’s Theorem in Chapter 5. 3 Chapter 2 Group Theory Background 2.1 Definitions In this section we begin by reviewing some definitions and results from group theory. See [8] and [9] for an in-depth introduction to abstract algebra. Recall that a group is a nonempty set G with a binary operation · : G×G → G such that 1. (g ·h)·k = g ·(h·k), for all g,h,k ∈ G. 2. There exists an element 1 ∈ G, called the identity of G, which satisfies g·1 = 1·g = g, for all g ∈ G. 3. For each g ∈ G there exists an element g−1 ∈ G, called the inverse of g, such that g ·g−1 = g−1 ·g = 1. Moreover, a group G is called abelian if g ·h = h·g, for all g,h ∈ G. The order of a group G, denoted |G|, is the cardinality of the set G, and a group is called finite is it has finite order. Note that when discussing an abstract group G, it is common to use juxtaposition to indicate the group’s operation. Furthermore, if a group G has the operation +, we use the conventional notation of 0 for the identity element and −g for the inverse of g ∈ G. You are probably already familiar with several groups. For example, Z, Q,R, and C are all groups under addition. These sets, however, are not groups under multiplication because the identity 0 does not have a multiplicative inverse. Yet, all nonzero elements in Q,R, and C have multiplicative inverses, so we can conclude that Q \ {0},R \ {0}, and C \ {0} are groups under multiplication. In fact, all of the above examples of groups are abelian. Example 2.1.1. For an example of a non-abelian group consider the set of all bijections of {1,...,n} to itself, where the cycle (a ...a ) of length m ≤ n denotes the permutation 1 m that takes a to a for all 1 ≤ i ≤ m−1, takes a to a , and fixes a for all m+1 ≤ j ≤ n. i i+1 m 1 j This set, denoted S , is a group of order n! under function composition and is called the n symmetric group of degree n. ToseethatS isnotanabeliangroupforalln > 2, considerthe n permutations (12),(23) ∈ S . We see that (12)(13) = (132), but (13)(12) = (123) (cid:54)= (132). n 4 A cycle of length 2 is called a transposition, and it turns out that every element of S n can be written as a product of transpositions. Although this product might not be unique, the parity of the number of transpositions is, and therefore we say that g ∈ S is an odd n permutation if g is a product of an odd number of transpositions and an even permutation if g is a product of an even number of transpositions. Moreover, we define the sign of a permutation g ∈ S to be n (cid:40) −1 if g is an odd permutation sgn(g) = 1 if g is an even permutation. A subgroup of a group G is a nonempty subset H ⊆ G such that for all h,k ∈ H, hk ∈ H and h−1 ∈ H, often denoted H ≤ G. In fact, to determine whether a nonempty subset H ⊆ G is a subgroup of G, it suffices to show that gh−1 ∈ H for all g,h ∈ H. This is referred to as the Subgroup Criterion and it is an easy exercise to show that these two definitions of a subgroup are equivalent. Example 2.1.2. For any n ∈ N, consider the subset nZ = {nk | k ∈ Z} ⊆ Z. Take any elements x,y ∈ nZ. There exist a,b ∈ Z such that x = na and y = nb. Therefore, x−y = na−nb = n(a−b) ∈ nZ. Hence, by the Subgroup Criterion, nZ ≤ Z. A proper subgroup M of G is called maximal if the only subgroups of G containing M are M and G. A subgroup N of G is called normal, denoted N (cid:69)G, if for all n ∈ N, g ∈ G, we have that gng−1 ∈ N. Equivalently, N (cid:69)G if gNg−1 = N, for all g ∈ G. Note that if G is abelian, then each subgroup of G is normal. However, while every group G has both itself and {1} as normal subgroups, which are referred to as trivial subgroups, it is not necessarily true that G has any other normal subgroups. A group which has no nontrivial normal subgroups is called simple. Example 2.1.3. The alternating group of degree n, denoted A , is the set of all even per- n mutations in S . It turns out that A is a simple group for all n ≥ 5 and A is the smallest n n 5 non-abelian simple group. Several important subgroups can be generated given a nonempty subset A of a group G. 1. The normalizer of A in G is the subgroup of G defined by N (A) = {g ∈ G | gag−1 ∈ A, for all a ∈ A}. G 2. The centralizer of a A in G is the subgroup C (A) = {g ∈ G | ga = ag, for all a ∈ A}. G 5 3. Similarly, the center of G is the subgroup Z(G) = {g ∈ G | gx = xg, for all x ∈ G}. Notice that Z(G) = C (G) and Z(G)(cid:69)G. G 4. Moreover, for every subset A ⊆ G, there exists a unique smallest subgroup of G containing A, namely the subgroup of G generated by A, defined by (cid:92) (cid:104)A(cid:105) = H, A⊆H H≤G or equivalently, (cid:104)A(cid:105) = {a(cid:15)1a(cid:15)2···a(cid:15)n | n ∈ N,a ∈ A,(cid:15) = ±1}. 1 2 n i i A subgroup H ≤ G which is generated by a single element g ∈ G is called cyclic and is written H = (cid:104)g(cid:105) = {gn | n ∈ Z}. The order of an element g ∈ G, denoted |g|, is the smallest positive integer n such that gn = 1. If no such integer exists, then we say g has infinite order. As a consequence of the division algorithm, it is easy to see that if H is a cyclic group generated by g ∈ G, then |H| = |g|. A group of order pn, where p ∈ N is a prime and n ∈ N, is called a p-group. Similarly, a subgroup of a group G which is itself a p-group is called a p-subgroup. Suppose G is a group of order pam, where p is a prime that does not divide m and a ∈ N. A Sylow p-subgroup of G is a subgroup of order pa. In fact, there exists at least one Sylow p-subgroup of G for each prime p ∈ N dividing |G|. A proof of this will be given in the following section. Example 2.1.4. Consider the group S , which has order 6. S has three Sylow 2-subgroups, 3 3 (cid:104)(12)(cid:105),(cid:104)(23)(cid:105), and (cid:104)(13)(cid:105), and one Sylow 3-subgroup, (cid:104)(123)(cid:105). For any subgroup N of G and any element g ∈ G, the set gN = {gn | n ∈ N} is called a left coset of N in G. We use G/N to denote the set of all left cosets of N in G, and the index of N in G, denoted |G : N|, is equal to the cardinality of G/N. We will see in the next section that the set of left cosets of N in G partition G and two cosets gN, hN ∈ G/N are equal if and only if h−1g ∈ N. If N is a normal subgroup of G, then G/N forms a group called the quotient group, with the operation gN ·hN = (gh)N. Example 2.1.5. Consider the group (Z,+). Since this is an abelian group, all of its sub- groups are normal. Therefore, for each n ∈ N,Z/nZ is a group. Let g +nZ and h+nZ be cosets in Z/nZ such that g+nZ = h+nZ. Then it must hold that g−h ∈ nZ, or in other words, g ≡ h (mod n). Hence, Z/nZ is the group of integers modulo n containing n distinct left cosets. In fact, Z/nZ is cyclic and equal to (cid:104)1+nZ(cid:105). 6 Let G and H be groups. A (group) homomorphism is a map φ : G → H such that for all g,h ∈ G, φ(gh) = φ(g)φ(h). The kernel of the map φ is the set ker(φ) = {g ∈ G | φ(g) = 1}, and the image of φ is the set φ(G) = {h ∈ H | h = φ(g) for some g ∈ G}. A bijective homomorphism between two groups is called a (group) isomorphism. If the map ∼ φ : G → H is an isomorphism, then G and H are said to be isomorphic, denoted G = H. In other words, G and H are the same group, up to relabeling of elements. Example 2.1.6. Recall the groups Z and nZ under addition. For a fixed n ∈ N, the map φ : Z → nZ defined by φ(k) = nk is an isomorphism. The map ψ : Z → Z/nZ defined by ψ(k) = k +nZ is also a homomorphism. However, ψ is not an isomorphism, since it is clearly not injective. Note that it is not necessary for two isomorphic groups to have the same operation. For example, the map f : (R,+) → (R+,×) defined by f(x) = ex is an isomorphism, since for all x,y ∈ R, ex+y = exey. 2.2 Important Results Here we prove some important results from group theory, which are necessary for the proofs of Burnside’s Theorem presented in Chapter 4. 2.2.1 Lagrange’s Theorem and Consequences We begin by proving Lagrange’s Theorem, Cauchy’s Theorem, and the Class Equation, which give us information about the order of a group and its subgroups. We will then use these results to prove the existence of Sylow p-subgroups, as mentioned in the previous section, and show that Z(P) is nontrivial for all p-groups P. These results are all needed for the representation theoretic proof of Burnside’s Theorem given in Section 4.1. Theorem 2.2.1. (Lagrange’s Theorem) Let G be a finite group and H ≤ G. Then |H| divides |G| and |G| |G : H| = . |H| Proof. Let G be a finite group and let H ≤ G such that |H| = m and |G : H| = k. For any g ∈ G, define a map φ : H → gH by φ (h) = gh. This map is clearly surjective. g g Furthermore, for any distinct h (cid:54)= h ∈ H, we have that gh (cid:54)= gh . Thus φ is a bijection, 1 2 1 2 g so |gH| = |H| = m. 7 It is easy to see that the set of left cosets of H in G form a partition of G. First note (cid:83) (cid:83) that for any g ∈ G, g ∈ gH. Thus, G ⊆ gH, and clearly gH ⊆ G. Therefore, g∈G g∈G (cid:91) G = gH. g∈G To show that left cosets are disjoint, suppose that for distinct cosets g H (cid:54)= g H ∈ G/H, 1 2 there exists an element x ∈ g H ∩ g H. Then there exist elements h ,h ∈ H such that 1 2 1 2 x = g h = g h . Therefore, g = g h h−1, so for any g h ∈ g H, 1 1 2 2 1 2 2 1 1 1 g h = (g h h−1)h = g (h h−1h) ∈ g H. 1 2 2 1 2 2 1 2 Hence, g H ⊆ g H. But we have seen that |g H| = |g H|, so g H = g H, which is a 1 2 1 2 1 2 contradiction. Thus, the k left cosets of H in G are in fact disjoint and, hence, partition G. Since each has cardinality m, it follows that |G| = km. Therefore |H| divides |G| and |G| = k. |H| Corollary 2.2.2. (Cauchy’s Theorem) Let G be a finite abelian group, and let p ∈ N be a prime dividing |G|. Then G has an element of order p. Proof. We will perform induction of |G|. Take any nonidentity element g ∈ G. If |G| = p, then by Lagrange’s Theorem, g has order p. Now assume that |G| > p and that all subgroups of order less than |G| whose orders are divisible by p have an element of order p. First consider the case where p divides the order of g. Then we can write |g| = np, for some n ∈ N. Thus, 1 = gnp = (gn)p, so the order of gn must divide p. But p is prime, so |gn| = p. So we now consider the case where p does not divide the order of g. Let H = (cid:104)g(cid:105), which is a normal subgroup of G, since G is abelian. By Lagrange’s Theorem, |G/H| < |G|, since H is nontrivial. Moreover, it must hold that p divides |G/H|, because p divides |G|, but does not divide |H|. Hence, by induction, G/H contains an element of order p, say xH. We have that xp ∈ H, but x ∈/ H, so (cid:104)xp(cid:105) (cid:54)= (cid:104)x(cid:105), giving that |xp| < |x|. By Lagrange’s Theorem, we find that |xp| divides |x|, so we have that p divides |x|. This brings us back to the previous case. So, by induction, G has an element of order p. Let G be a group and X a set. A group action of G on X is a map · : G×X → X such that 1. g ·(h·x) = (gh)·x, for all g,h ∈ G, x ∈ X, 2. 1·x = x, for all x ∈ X. In fact, the action of a group G on a set X induces an equivalence relation ∼ on X, where we say that x ∼ y if there exists g ∈ G such that x = g · y. For x ∈ X, the orbit of G containing x is the equivalence class [x] = {g ·x | g ∈ G}. 8 We can define a group action of G on itself by conjugation, i.e. for all g,x ∈ G, g ·x = gxg−1. It is easy to check that this action satisfies the conditions above to be a group action. Moreover, the orbits of G with respect to this action are called the conjugacy classes of G. Theorem 2.2.3. (The Class Equation) Let G be a finite group, and let g ,g ,...,g be 1 2 n representatives of the distinct conjugacy classes of G that are not contained in Z(G). Then, n (cid:88) |G| = |Z(G)|+ |G : C (g )|. G i i=1 Proof. Let G be a finite group. By definition, if z ∈ Z(G), then gzg−1 = z for all g ∈ G, so {z} is a conjugacy class of G containing a single element. Write Z(G) = {1,z ,...,z }, 2 k and let K ,...,K denote the distinct conjugacy classes of G not contained in Z(G) with 1 n respective representatives g ,g ,...,g . Since conjugation is a group action of G on itself, 1 2 n and thus induces an equivalence relation on G, the conjugacy classes partition G. Hence we have k n n (cid:88) (cid:88) (cid:88) |G| = 1+ |K | = |Z(G)|+ |K |. i i i=1 i=1 i=1 For each 1 ≤ i ≤ n, define a map φ : K → G/C (g ) by i i G i φ (gg g−1) = gC (g ). i i G i Suppose that for some g,h ∈ G, gg g−1 = hg h−1. Then h−1gg g−1h = g , so we have that i i i i h−1g ∈ C (g ), which implies that gC (g ) = hC (g ). Therefore, φ is well-defined. Now G i G i G i i take any distinct elements gg g−1 (cid:54)= hg h−1 ∈ K . Then h−1gg (cid:54)= g h−1g, so h−1g ∈/ C (g ). i i i i i G i Thus, gC (g ) (cid:54)= hC (g ), so we have that φ is injective. We can easily see that φ is also G i G i i i surjective, so |K | = |G/C (g )| = |G : C (g )|. Hence, i G i G i n (cid:88) |G| = |Z(G)|+ |G : C (g )|. G i i=1 Corollary 2.2.4. (Sylow’s Theorem) Let G be a group of order pam, where a ∈ N and p ∈ N is a prime that does not divide m. Then there exists a Sylow p-subgroup of G. Proof. We will perform induction on |G|. If |G| = 1, the result is trivial. Assume that Sylow p-subgroups exist for all groups of order less than |G|. First consider the case where p divides |Z(G)|. Since Z(G) is an abelian group, then by Cauchy’s Theorem, Z(G) has a cyclic subgroup, N, of order p. So by Lagrange’s Theorem, |G/N| = pa−1m. Therefore, by induction, G/N has a Sylow p-subgroup, P, of order pa−1. By the Fourth Isomorphism 9