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MT300: Groups of a Square-Free Order Iordan Ganev Royal Holloway, University of London Supervisor: Dr. Benjamin Klopsch Spring 2009 Updated January 2010∗ Abstract Hölder’s formula for groups of a square-free order is an early advance in the enumeration of finite groups. We aim to elucidate this classical result through a structural approach, emphasizing topics such as nilpotency, the Fitting subgroup, and extensions. These topics, which are usually not covered in undergraduate group theory, feature in the proof of Hölder’s result and have wide applicability in group theory. Finally, we remark on further results and conjectures in the enumeration of finite groups. Contents 1 Introduction 2 2 Preliminaries 2 3 Commutators 3 4 Nilpotent Groups 6 5 The Fitting Subgroup 7 6 Split Extensions 9 7 The Transfer Homomorphism 10 8 Groups of a Square-Free Order 14 9 Further Results and Conjectures 18 References 20 ∗ An abbreviated version of this report has been accepted for publication in the Spring 2010 issue of the RoseHulman Undergraduate Mathematics Journal. 1 Introduction How many non-isomorphic groups are there of order n? This is one of the simplest yet most mysterious questions in group theory. Groups of order 16 or less were classified in the late nineteenth century as part of early advances in group theory. It has been clear from even earlier that for any prime p, there is only one group of order p. In general, however, the tabulation of the nonisomorphic groups of order n requires careful consideration of the prime-power factorization of n, and the constraints on group structure imposed by the relationships between the divisors of n. Up to date, the groups of order less than 2048 have been tabulated [5]. Throughout this project, f (n) denotes the number of groups, up to isomorphism, of order n. Group theorists agree that there is no hope for a precise formula for the group number function f (n) in general. Nonetheless, there have been several remarkable asymptotic estimates due to Pyber which use the classification of finite simple groups, Hall systems, and combinatorial estimates. Moreover, for certain types of orders it is possible to determine explicit formulas, precise estimates, or other characterizations of f (n). Much of the current research in the enumeration of finite groups attempts to extend the known results to more types of orders [3, Chapter 22]. One of the first mathematicians to make advances in the enumeration of finite groups was Otto Hölder. In 1893, he described groups of order p3 and p4 [14]. Shortly thereafter, he derived a remarkable formula for the number of groups of order n where n is square-free [15]: f (n) = X Y pc(p) − 1 m|n p p−1 where p is a prime divisor of n/m and c(p) is the number of prime divisors q of m that satisfy q ≡ 1 (mod p). The aim of this project is to elucidate this classical result through a structural approach, demonstrating that what makes the formula possible is that a group of square-free order has restricted structure as a Sylow tower group. In order to render the result more accessible and to illustrate my progress through the term, we include introductory explanations of concepts used in this project that are beyond a standard undergraduate group theory course. Hence we devote much attention to topics such as nilpotency, the Fitting subgroup, and extensions. In addition, our approach emphasizes notions that have wide application in other areas of group theory and relate to open research problems. The presentation of the advanced concepts in sections 3 to 7 relies heavily on Rotman [21]. For the derivation of Hölder’s formula (section 8), Blackburn et al. [3] give an argument that we follow, but their comments are not a complete proof and warrant more explanation in our view. We also examine briefly Hölder’s original proof and an argument given in Conway et al. [4]. First, let us remind ourselves of basic terminology from group theory. 2 Preliminaries If xy = yx for all elements x, y in a group G, then G is said to be abelian. The center Z(G) of a group G is the set of elements that commute with any element of G, that is, Z(G) = {x ∈ G | xg = gx for all g ∈ G}. Clearly Z(G) = G if and only if G is abelian. For an element a in G, the centralizer CG (a) of a in G is the set of all elements that commute with a; in notation, CG (a) = {g ∈ G | ga = ag}. The centralizer of a subgroup H of G is defined analogously: CG (H) = {g ∈ G | gh = hg for all h ∈ H}. 2 For elements x, y in any group G, the conjugate of x under y is conventionally written as := y −1 xy. The normalizer NG (H) of a subgroup H in G is the set {g ∈ G | g −1 Hg = H}, where −1 g Hg = {ghg −1 | h ∈ H}. A subgroup N of G is normal if g −1 N g = N for all g ∈ G, that is, if NG (N ) = G; in this case we write N E G. It is trivial to verify that Z(G) is a normal subgroup1 in G and that, more generally, any subgroup of the center is normal. One can also verify that for any normal subgroup N of G, CG (N ) E G and Z(N ) E G. A simple group has exactly two normal subgroups: the identity subgroup {1} and the group itself. We will also make use of factor groups, products of groups, the Chinese Remainder Theorem, the Fundamental Theorem of Finite Abelian Groups (also known as the Basis Theorem), and the Sylow Theorems. Please refer to [11] and [21] for more details. A homomorphism of groups is a map f from a group G to a group H with the property that for any x, y ∈ G, f (xy) = f (x)f (y). The kernel of f is the pre-image of the identity of H: ker(f ) = f −1 (1H ). Note that the identity of G is always mapped to the identity of H, so 1G ∈ ker(H). Also, ker(f ) is a normal subgroup of G. The image im(f ) of f in H is a subgroup of H. An isomorphism of groups f : G → H is a bijective homomorphism. When an isomorphism exists from G to H, we write G ∼ = H to indicate that the groups are isomorphic. An isomorphism from a group G to itself is called an automorphism. The set of automorphisms of G, denoted Aut(G), forms a group under the operation of composition of functions. Conjugation by an element g ∈ G is an example of an automorphism. A subgroup H of G is a characteristic subgroup if φ(H) = H for all automorphisms φ of G. Hence, all characteristic subgroups of G are normal in G. For a homomorphism f : G → H, the First Isomorphism Theorem states that G/ker(f ) ∼ = ∼ im(f ). If f is surjective, then its image is H and G/ker(f ) = H. A cyclic group of finite order n consists of all powers of an element x, with the defining condition that xn = 1. The generic cyclic group of order n is denoted here as Cn . The integers modulo n, denoted Z/nZ, form a cyclic group of order n under addition. The subset U (n) of Z/nZ consisting of elements relatively prime to n forms a group under multiplication modulo n. It is well-known that Aut(Cn ) ∼ = U (n). We prove the following proposition (known as the N/C Lemma) before completing this revision. xy Proposition 1. If H is a subgroup of G, then CG (H) E NG (H) and the quotient NG (H)/CG (H) is isomorphic to a subgroup of Aut(H). Proof. Let γg |H denote conjugation by g ∈ G restricted to H, that is γg |H : H → G, x 7→ g −1 xg. If g ∈ NG (H), then γg |H is an automorphism of H, from the definition of the normalizer. Hence let φ : NG (H) → Aut(H) be the homomorphism which maps g to γg |H . Then, g ∈ ker(φ) ⇔ γg = id|H ⇔ g −1 xg = x for all x ∈ H ⇔ g ∈ CG (H). Since the kernel of a homomorphism in normal in the domain, CG (H) E NG (H). By the First Isomorphism Theorem, NG (H)/CG (H) = NG (H)/ker(φ) ∼ = im(φ) ≤ Aut(H). 3 Commutators Let G be a group. For any two elements x and y in G, define the commutator of x and y to be [x, y] := x−1 y −1 xy = x−1 xy . Similarly, if H and K are subgroups of G then [H, K] denotes the 1 In fact, Z(G) is characteristic in G. 3 subgroup of G generated by all commutators [h, k] with h ∈ H and k ∈ K. Since [h, k] = [k, h]−1 , [H, K] = [K, H]. The commutator subgroup G0 := [G, G] of G is generated by the set {[x, y] | x, y ∈ G}. This subgroup is also known as the derived subgroup of G. Proposition 2. The derived subgroup G0 of a group G is characteristic (hence normal) in G. Furthermore, for any normal subgroup N in G, G/N abelian ⇔ G0 ⊆ N. In other words, G0 is the smallest normal subgroup in G with an abelian factor group. Proof. To prove that G0 is characteristic in G, we show that any automorphism φ of G maps elements of the generating set for G0 , {[x, y] | x, y ∈ G}, to other elements in the generating set: φ([x, y]) = φ(x−1 y −1 xy) = φ(x)−1 φ(y)−1 φ(x)φ(y) = [φ(x), φ(y)]. Taking φ to be conjugation by an element of G verifies that G0 is normal. Let N E G. For any x, y ∈ G, G/N abelian ⇔ xN · yN = yN · xN ⇔ xyN = yxN ⇔ x−1 y −1 xyN = N ⇔ [x, y] = x−1 y −1 xy ∈ N ⇔ G0 ⊆ N. A nonidentity group with the property G = G0 is called perfect. For any group G, the derived series {G(i) }i∈N is a descending sequence defined as G(0) := G, G(1) := G0 , G(2) := [G0 , G0 ], . . . , G(i) := [G(i−1) , G(i−1) ], . . . of successive commutators. If G is finite, then the orders of the groups in this sequence are finite and non-increasing. Hence there exists an integer d such that G(d) = G(d+1) , and consequently G(d) = G(s) for all s ≥ d. Observe that such a G(d) is a perfect group when it contains more than the identity. If G(d) is only the identity {1}, then the original group G is called solvable. The smallest positive integer d for which G(d) = {1}, is the derived length of G. Let G be non-trivial and solvable with derived length d. Then it must be that G(d−1) is abelian since G(d−1) ∼ = G(d−1) /{1} = (d−1) (d) G /G is abelian (Proposition 2). Note that any subgroup of a solvable group is solvable. Example 1. The commutator subgroup of any abelian group is trivial, so abelian groups are solvable. Example 2. Consider D4 , the dihedral group of order 8. What is D40 = [D4 , D4 ]? Recall that D4 has presentation ha, b | a4 = b2 = (ab)2 = 1i. Rather than looking for • an explicit list of elements of D40 , or • an explicit list of all commutators [x, y] = x−1 y −1 xy, or • an explicit shorter list of generators for D40 , we look for an explicit list of generators of D40 as a normal subgroup. This list will have to imply that the quotient D4 /D40 is abelian. Since the generators for D4 are a and b, the generators of 4 D4 /D40 are the cosets a and b. Furthermore, a and b must commute, so we deduce the relation [a, b] = 1. As a consequence, [a, b] = 1 ⇒ a−1 b−1 ab ∈ D40 a−2 ∈ D40 ⇒ ⇒ a2 ∈ D40 . The second implication follows from the relations in our presentation of D4 : b−1 = b and bab = a−1 . The single relation [a, b] = 1 is sufficient because if the generators of D4 /D40 commute, then it is abelian. Thus, D40 is equal to the smallest subgroup of D4 generated by a2 and its conjugates. To compute the conjugates it is enough to consider repeated conjugation of a2 by the generators of D4 : (a2 )a = a2 and (a2 )b = ba2 b = a−2 = a2 , using the relations cited above. Finally we have that D40 = ha2 i = {1, a2 }. Observe that the quotient here is the abelian group C2 × C2 . Since D40 is abelian, D400 = {1}, so D4 is solvable. A natural generalization is to ask: for which n is Dn solvable? The argument above, with a few modifications, shows that for any n, Dn0 = ha2 i and Dn00 = {1}, so dihedral groups are solvable in general. If n is odd, Dn0 = ha2 i = hai, so [Dn : Dn0 ] = 2n/n = 2 and Dn /Dn0 ∼ = C2 . If n is even, then Dn /Dn0 ∼ = C2 × C2 . Indeed, [Dn : Dn0 ] = 2n/(n/2) = 4, and the three nonidentity elements have order 2. The order of a and b is 2 in Dn /Dn0 , and from the abelian property of the quotient, the order of their product ab is the least common multiple of the orders of a and b, which is again 2. Non-abelian simple groups are perfect, but not all perfect groups are simple. A simple counterexample is Alt(5) × Alt(5) which has three proper normal subgroups: Alt(5) ×{1}, {1}× Alt(5), and the identity {1}. In each case the quotient is nonabelian, and so (Alt(5) × Alt(5))0 = Alt(5) × Alt(5). Proposition 3. Let N be a normal subgroup of a group G. If N is solvable and G/N is solvable, then G is solvable. Proof. For any subgroup K of G write K of the subgroup K/N in G/N . Let c and d be the derived lengths of G/N = G and N , respectively, so that: 0 00 G ⊇ G ⊇ G ⊇ ··· ⊇ G (c) = {1} and N ⊇ N 0 ⊇ N 00 ⊇ · · · ⊇ N (d) = {1}. First we show by induction that G (i) = G(i) for each i. Indeed, Base step, i = 0 : G (0) = G = G(0) Induction step, i ≥ 1 : G (i) = [G (i−1) ,G (i−1) ] = [G(i−1) , G(i−1) ] = [G(i−1) , G(i−1) ] = G(i) . Therefore, the first series above implies that the derived series for G enters N (the pre-image of the identity in G under the natural homomorphism) after at most c steps. Combining this conclusion with the second series reveals that the derived series of G includes the identity: G, G0 , G00 , . . . , G(c) ⊆ N, G(c+1) ⊆ N 0 , . . . , G(c+d) ⊆ N (d) = {1}. Hence G is solvable with derived length at most c + d. The solvable radical of a finite group G is the largest normal solvable subgroup of G. It was recently proved that the solvable radical of a finite group G is equal to the set of all elements g ∈ G such that for any x ∈ G, the subgroup generated by g and x is solvable [13]. The solvable radical also occurs and has an important role in the theory of linear groups. 5 4 Nilpotent Groups Another important descending sequence for a group G is the lower central series: G = γ1 (G) ⊇ γ2 (G) ⊇ · · · ⊇ γi (G) . . . where γi+1 (G) := [γi (G), G] = hx−1 y −1 xy | x ∈ γi (G), y ∈ Gi. Note that γ2 (G) = G0 . Proposition 4. For each i, γi (G) is a characteristic subgroup of G. Proof. We argue by induction on i. Clearly γ1 (G) = G is characteristic in G. Suppose γi (G) is characteristic in G for some i ≥ 1. The generating set for γi+1 (G) is {[x, g] | x ∈ γi (G), g ∈ G}. Similar to the proof of Proposition 2, we show that any automophism φ of G maps elements of this set to other generators. Let φ ∈ Aut(G), x ∈ γi (G), and g ∈ G. Then φ([x, g]) = φ(x−1 g −1 xg) = φ(x)−1 φ(g)−1 φ(x)φ(g) = [φ(x), φ(g)]. Since γi (G) is characteristic, φ(x) ∈ γi (G), so φ([x, g]) is in the generating set for γi+1 (G). Proposition 5. The group γi (G)/γi+1 (G) is central in G/γi+1 (G), i.e. all of its elements commute with all other elements of the factor group. Proof. For any x ∈ γi (G) and g ∈ G, [x, g] ∈ γi+1 (G) implies equality between the cosets (xg)γi+1 (G) and (gx)γi+1 (G) in G/γi+1 (G). Lemma 5 explains the name “lower central series”: each member is central in G modulo its successor. If G is finite, then the orders of the groups γi (G) are finite and non-increasing. If there exists a d such that γd+1 (G) = {1}, then G said to be nilpotent. The nilpotency class of a nilpotent group G is the smallest such d, and we write nc(G) = d. Note that nilpotent groups are solvable, and that subgroups of nilpotent groups are also nilpotent. Example 3. We calculate the lower central series for D4 and show that D4 is nilpotent. Indeed, γ1 (D4 ) = D4 ; γ2 (D4 ) = D40 = ha2 i = {1, a2 }; γ3 (D4 ) = [{1, a2 }, D4 ] = {1}; since, using the relations from Example 2, a−2 a−1 a2 a = 1 and a−2 b−1 a2 b = a2 a−2 = 1. The nilpotency class of D4 is 2. In general, γ2 (Dn ) = Dn0 = ha2 i, and to calculate γ3 (Dn ) = [ha2 i, Dn ] we use the fact that 2 ha i/γ3 (Dn ) is central in Dn /γ3 (Dn ). The first group is generated by a2 and the second by a and b, hence we realize that [a2 , a] = 1 and −1 2 [a2 , b] = a−2 b a b = a−4 are trivial in Dn /γ3 (Dn ). The second condition implies a4 ∈ γ3 (Dn ). As before, conjugation of a4 by a and b gives a4 and a−4 , respectively. Thus, γ3 (Dn ) = ha4 i. k−1 Inductively we can show that γk (Dn ) = ha2 i. Indeed, the inductive hypothesis implies that k−1 k−1 k−1 k [a2 , b] = 1 in Dn /γk+1 (Dn ), but this means that a−2 a−2 = a−2 ∈ γk+1 (Dn ), so also k k a2 ∈ γk+1 (Dn ). The conjugates remain in ha2 i, so this is γk+1 (Dn ). k Now, the order of a in Dn is n, and ha2 i = {1} for some k if and only if the order of a is a power of 2. Hence, the nilpotent dihedral groups are precisely the groups Dn where n = 2d . In this case, the nilpotency class is d. 6 A characterization of finite nilpotent groups is given in the following theorem, whose proof is omitted here. See Rotman [21, Theorem 5.39]. Theorem 6. A finite group G is nilpotent if and only if it is the direct product of its Sylow subgroups. That is (1) G nilpotent ⇔ G = Sp1 × Sp2 × · · · × Spr , where p1 , p2 , . . . , pr are the prime divisors of |G| and Spi are Sylow subgroups of G. Finite nilpotent groups can also be characterized in terms of two-variable Engel words [20, Theorem 12.3.4], and there is an analogous description of finite solvable groups [1]. Unlike the Sylow subgroups in the theorem above, Engel words can also be used to study infinite groups, though the characterizations mentioned above for finite nilpotent and finite solvable groups do not carry over to the infinite case [20, § 12.3] . 5 The Fitting Subgroup Nilpotent subgroups have several important properties given in the Lemmas 7 and 8 that will allow us to construct a group’s maximal nilpotent subgroup, called the Fitting subgroup. This subgroup is the nilpotent analogue of the solvable radical and is important in the structural decomposition of a group. It relates, for example, to research questions about infinite polycyclic groups [12, 22, 2], though here it features in the proof of Hölder’s formula for finite groups of a square-free order. Lemma 7. If G is nilpotent and N E G, then G/N is nilpotent. Proof. Let G := G/N . We will first show by induction that γi (G) = γi (G). The base step is clear: γ1 (G) = G = γ1 (G). For the induction step, if i ≥ 2 then γi (G) = [γi−1 (G), G] = h[x, y] | x ∈ γi−1 (G), y ∈ Gi = h[x, y] | x ∈ γi−1 (G), y ∈ Gi = h[x, y] | x ∈ γi−1 (G), y ∈ Gi = h[a, b] | a ∈ γi−1 (G), b ∈ Gi = h[a, b] | a ∈ γi−1 (G), b ∈ Gi = γi (G). Hence, if γd+1 (G) is trivial for some d, then γd+1 (G) = γd+1 (G) is also trivial. In addition, this shows that nc(G) ≤ nc(G). Now, the product of two subgroups N and M of G is defined to be N M = hab | a ∈ N, b ∈ M i = hN ∪ M i. If N E G and M is another subgroup of G, then N M = {xy | x ∈ N, y ∈ M }. To y −1 see why, let x1 , x2 ∈ N and y1 , y2 ∈ M . Then x1 y1 x2 y2 = (x1 x21 )(y1 y2 ) ∈ {xy | x ∈ N, y ∈ M } since N E G. Moreover, if both N and M are normal in G, then N M is also normal in G. Indeed, if g ∈ G, x ∈ N , and y ∈ M , then (xy)g = g −1 xyg = (g −1 xg)(g −1 yg) = xg y g ∈ N M since both subgroups are normal. Lemma 8. For normal subgroups A, B, and C, of a group G, (i) [AB, C] = [A, C][B, C] (ii) [A, BC] = [A, B][A, C]. 7 Proof. Let a, b, and c be elements of A, B, and C, respectively. Then [ab, c] = b−1 a−1 c−1 abc = b−1 a−1 c−1 acbb−1 c−1 bc = [a, c]b [b, c] ∈ [A, C]b [B, C]. Now, both A and C are normal, so [A, C]b = [A, C], giving the inclusion [AB, C] ⊆ [A, C][B, C]. Conversely, [A, C] and [B, C] are both contained in [AB, C], so the product [A, C][B, C] is contained in [AB, C] as well, giving the first result. The second identity follows from equation (3) and (i): [A, BC] = [BC, A] = [B, A][C, A] = [A, B][A, C]. This operation of taking products of groups preserves normality and nilpotency, as verified in the next theorem. In proving the result, we will use “left-normed commutators”: [X1 , X2 , X3 , . . . , Xn ] := [. . . [[X1 , X2 ], X3 ], . . . , Xn ]. In this notation, γi (G) = [G, G, . . . , G] (i times). Theorem 9. If N and M are nilpotent normal subgroups of a group G, then N M is nilpotent and normal in G. Moreover, nc(N M ) ≤ nc(N ) + nc(M ). Proof. We already showed that N M E G. Let c and d be the nilpotency classes of M and N , respectively, and let r = c + d. Then, applying Lemma 8, γr+1 (M N ) = [M N, M N, . . . , M N ] Y = [X1 , X2 , . . . , Xr+1 ], where the product includes all tuples (X1 , X2 , . . . , Xr+1 ) ∈ {M, N }r+1 = {M, N }c+d+1 . In each term, either at least c+1 of the Xi ’s are equal to M or at least d+1 of them are equal to N . In the first case, the corresponding group is contained in γc+1 (M ) = {1}; in the second case, it is contained in γd+1 (N ) = {1}. Therefore, nc(N M ) ≤ c + d = nc(N ) + nc(M ). G N MH HH HH HH HH vv vv v vv vv NH HH HH HH HH v vv vv v v vv N ∩M {1} 8 M Using this result, we can find a unique maximal nilpotent normal subgroup F (G) in a finite group G, which is referred to as the Fitting subgroup of G. Equivalently, F (G) is the subgroup generated by the maximal normal p-subgroups of a finite group G, where p runs over all prime divisors of |G|. Theorem 10. If G is a finite solvable group, then CG (F (G)) = Z(F (G)). Proof. For convenience, let F = F (G), C = CG (F (G)), and Z = Z(F (G)). Now, Z E C and Z E G. The former follows since Z ≤ F and elements in C commute with those in F . To see why Z E G, let −1 z ∈ Z, f ∈ F , and g ∈ G. Because F is normal in G, z g and f1 := f g belong to F . Meanwhile, z g ∈ C since z g f = g −1 zgf g −1 g = g −1 zf1 g = g −1 f1 zg = g −1 f1 gg −1 zg = f z g . Thus, z g ∈ C ∩ F = Z. Suppose, for a contradiction, that Z is strictly contained in C and let M/Z be a minimal nontrivial normal subgroup of G/Z that is contained in C/Z. Since G is solvable, M/Z is solvable. We show that (M/Z)0 E G/Z. Since M/Z is normal in G/Z, conjugation by an element g ∈ G/Z is an automorphism of M/Z. Also, (M/Z)0 is characteristic in M/Z (Proposition 2) , so conjugation by g maps (M/Z)0 to itself. Therefore, (M/Z)0 E G/Z. By the minimality of M/Z, (M/Z)0 must equal either 1 or M/Z. But M/Z is solvable, so (M/Z)0 = 1 equivalently M 0 ≤ Z. Since M ⊆ C and M 0 ⊆ Z ⊆ F , we have that γ3 (M ) = [M 0 , M ] ⊆ [C, F ] = 1.Therefore M nilpotent and normal in G, which implies that M ⊆ F from the definition of the Fitting subgroup. But M ≤ C, so M ⊆ C ∩ F = Z. Then M/Z is trivial, contradicting the choice of M . This means that there are no nontrivial normal subgroups between Z and C, giving the result Z = C. 6 Split Extensions Let H and K be groups. The direct product H × K of H and K is the set of ordered pairs {(h, k) | h ∈ H, k ∈ K} with operation (h1 , k1 )·(h2 , k2 ) = (h1 ·h2 , k1 ·k2 ). In terms of presentations, if H = hh1 , h2 , . . . | r1 , r2 , . . . i and K = hk1 , k2 , . . . | s1 , s2 , . . . i, then a presentation for H × K is hh1 , h2 , . . . , k1 , k2 , . . . | r1 , r2 , . . . , s1 , s2 , . . . , hi kj = kj hi for all i, ji. More generally, let φ : H → Aut(K) be a homomorphism2 . The semidirect product H n K with respect to φ is defined as the set {(h, k) | h ∈ H, k ∈ K} with operation (h1 , k1 ) · (h2 , k2 ) = φ(h ) (h1 · h2 , k1 2 · k2 ). If G = H n K, then G is also known as a split extension of K by H and sometimes the notation G = H : K appears. By means of the natural embeddings K → G, k 7→ (1, k), and H → G, h 7→ (h, 1) we may regard K and H as subgroups of G. Then K is a normal subgroup of G, H is a subgroup of G disjoint from K except for the identity, and H and K generate the entire group G. The subgroup H is called a complement of K in G, while the normal subgroup K is called a normal complement of H in G. 2 A remark on notation: here φ is written on the left as φ(h), while elements in Aut(K) are written on the right, so if σ ∈ Aut(K), then we write kσ for the image of k under σ. 9 Note that extensions of groups need not be split. If K is a normal subgroup of group G, then K may or may not admit a complement3 in G. Theorem 11 (Schur-Zassenhaus Lemma, 1973). Let K be a normal subgroup in G. If [G : K] and |K| are coprime, then K has a complement H in G. That is, G is a split extension of K by H. We will not prove or use this result, only mention it briefly later. See [21, Theorem 7.41] for a proof. Example 4. The alternating group of degree 4, Alt(4), is isomorphic to C3 n (C2 × C2 ). Indeed, we may take H = {(1), (123), (132)} ∼ = C3 , K = {(1), (12)(34), (13)(24), (14)(23)} ∼ = C2 × C2 . In this case, K is isomorphic to the Klein four group. For convenience, we will write K ∼ = {e, i, j, k} so that Aut(K) = {(1), (ij), (ik), (jk), (ijk), (ikj)} ≤ Sym(K). Since elements of order 3 in H must be mapped to elements of order 3 or the identity in Aut(K), there are three possibilities for φ : H → Aut(K), namely • The trivial homomorphism φ1 ((1)) = φ1 ((123)) = φ1 ((132)) = (1) which leads to C3 × (C2 × C2 ) ∼ = C6 × C2 . • φ2 ((1)) = (1), φ2 ((123)) = (ijk), φ2 ((132)) = (ikj). • φ3 ((1)) = (1), φ3 ((123)) = (ikj), φ3 ((132)) = (ijk). Relabeling of i as j and j as i reveals that φ2 and φ3 lead to isomorphic groups. In fact, φ2 is the correct one for our choice of K since iφ((123)) = (123)−1 [(12)(34)](123) = (13)(24) = k = iφ3 ((123)) . Example 5. The dihedral group Dn of order 2n is the semidirect product C2 n Cn of cyclic groups. The homomorphism φ from C2 = {1, σ} to Aut(Cn ) ∼ = U (n) sends σ to multiplication by −1. For example, if n = 5, then D5 is the group of symmetries of a regular pentagon. The normal ◦ subgroup isomorphic to C5 is the set of rotations {R0 , Rα , R2α , R3α , R4α }, where α = 2π 5 = 72 . The image of σ under homomorphism φ is the automorphism that sends Rα to R−α = R4α . The geometric interpretation of multiplication by −1 is reflection across an axis between an vertex and the midpoint of the opposite side. 7 The Transfer Homomorphism Suppose we want to know whether a group G is solvable. Clearly we must search for a proper, nontrivial normal subgroup in G (unless G ∼ = Cp ). If G can be written as the split extension of a normal subgroup K by a complement Q, then according to Proposition 3, we can reduce to studying K and Q ∼ = G/K. Therefore, a common strategy to prove that a group is solvable is to begin with an accessible subgroup Q and, if possible, construct a homomorphism from G to Q, whose kernel will be a normal complement of Q in G. This homomorphism is known as the transfer. For the next few results, let Q be a subgroup of G with finite index n. 3 In infinite polycyclic groups one finds almost complements and almost split extensions 10 Lemma 12. Let {l1 , . . . , ln } and {h1 , . . . , hn } be two left coset representatives of Q in G. For any fixed g ∈ G and each i ∈ {1, ..., n}, there is a unique σ(i) ∈ {1, .., n} and a unique xi ∈ Q such that ghi = lσ(i) xi . Moreover, σ is a permutation of {1, ..., n} (i.e. σ ∈ Sym(n)). Proof. The left cosets partition G, so ghi is contained in exactly one left coset lj Q. Hence σ(i) = j and xi = lj−1 ghi ∈ Q are unique such that ghi = lσ(i) xi . To prove that σ is a permutation, note that it is a map from the finite set {1, . . . , n} to itself, so it is enough to prove that σ is injective. Suppose σ(i) = σ(k) = j. Then ghi = lj xi and ghk = lj xk , which implies −1 ghi x−1 ⇒ hi−1 hk = x−1 i = ghk xk i xk ∈ Q ⇒ hi Q = hk Q ⇒ i = k. In the case where the two coset representatives are the same (li = hi for all i ∈ {1, ..., n}), the −1 previous lemma guarantees a unique xi such that xi = lσ(i) gli for each g ∈ G and i ∈ {1, ..., n}. 0 With this xi in mind, define the function V : G → Q/Q where V (g) = n Y x i Q0 . i=1 This function is known as the transfer 4 . Proposition 13. The transfer function V is a homomorphism whose definition does not depend on the choice of a left transversal of Q in G. Proof. Let {l1 , ..., ln } and {h1 , ..., hn } be two left coset representatives of Q in G. By Lemma 12, for each g ∈ G we have xi , yi ∈ Q and σ, τ ∈ Sym(n) such that gli = lσ(i) xi ghi = hτ (i) yi To prove that the transfer is independent of the choice of left transversal, we shall show that n Y 0 xi Q = i=1 n Y yi Q0 . i=1 Setting g = 1 in Lemma 12, we have α ∈ Sym(n) and zi ∈ Q such that hi = lα(i) zi forall i. Fix i. Define j := α−1 σα(i) so hj = lσα(i) zj . Then ghi = glα(i) zi = lσα(i) xi zi = hj zj−1 xi zi By the uniqueness assertion in Lemma 12 and the definition of j, j = τi Consider and yi = zj−1 xi zi = zα−1 −1 σα(i) xi zi . n Y i=1 4 0 yi Q = n Y 0 zα−1 −1 σα(i) xi zi Q . i=1 The letter V abbreviates the original German term V erlagerung. 11 The order of multiplication is irrelevant since Q/Q0 is abelian. Furthermore, α−1 σα is a permutation of {1, . . . , n}, so the inverse of each zi appears and cancels zi . Hence, n Y 0 yi Q = i=1 n Y xi Q0 i=1 as desired. Hence the definition of V does not depend on the choice of left transversal. To prove that V is a homomorphism, let g, g 0 ∈ G and let xi and yi be the unique elements in −1 0 Q such that xi = lσ(i) gli and yi = lτ−1 (i) g li . Then gg 0 li = glτ (i) yi = lστ (i) xτ (i) yi . Using the facts that Q/Q0 is abelian and that τ ∈ Sym(n), ! n ! n n Y Y Y V (gg 0 ) = xτ (i) yi Q0 = xτ (i) Q0 yi Q0 = i=1 i=1 i=1 n Y ! xi Q0 i=1 n Y ! yi Q0 . i=1 Thus V is a homomorphism. Lemma 14. Let Q be a subgroup of G with finite index n and left coset representatives {l1 , . . . , ln }. For any fixed g ∈ G, there exist m ∈ N; h1 , . . . , hm ∈ G; and positive integers n1 , . . . , nm with (i) hi ∈ {l1 , . . . , ln } for all i; ni (ii) h−1 i g hi belongs to Q; Pm (iii) i=1 ni = n; and Q −1 ni 0 (iii) V (g) = m i=1 (hi g hi )Q . −1 Proof. Lemma 12 provides unique σ ∈ Sym(n) and unique xs ∈ Q such that xs = lσ(s) gls for all s ∈ {1, . . . , n}. Factorize σ as the product of disjoint cycles, including a 1-cycle for each fixed point: σ = α1 . . . αm . Then for a fixed i ∈ {1, . . . , m}, the cycle αi is of the form5 (j1 , . . . , jr ) and −1 xj1 = lσ(j glj1 , 1) ⇒ xj1 = lj−1 glj1 , 2 −1 xj2 = lσ(j glj2 , 2) xj2 = lj−1 glj2 , 3 ..., ..., −1 xjr = lσ(j gljr r) xjr = lj−1 gljr 1 gljr ) . . . (lj−1 glj2 )(lj−1 glj1 ) = lj−1 g r lj1 , ⇒ xjr . . . xj2 xj1 = (lj−1 1 3 2 1 which is in Q. Define hi := lj1 and ni := r. Then (i) and (ii) follow immediately. The sum of the ni ’s is the sum of the length of the cycles of σ – a permutation of n elements. Hence, (iii) holds. Finally, to show (iv), m m m Y Y Y −1 r ni 0 0 (h−1 g h )Q = (l g l )Q = xjr . . . xj2 xj1 Q0 i j 1 i j1 i=1 = i=1 ni m Y Y xjk Q0 = i=1 n Y xs Q0 = V (g), s=1 i=1 k=1 Pm where each s corresponds to one of i=1 ni = n combinations of i and k, i.e. one for each coset of Q. The final equality holds because the product includes all the xs ’s of the necessary form. 5 The j’s and r depend on i, but additional notation is omitted. 12 Corollary 15. Let Q be a subgroup of G of finite index n. If Q ≤ Z(G), then V (g) = g n for all g ∈ G. Proof. Any subgroup of the center is normal (see Section 2), so Q is a normal subgroup of G. Also, Q0 is trivial because Q is abelian, and we may regard the transfer as a homomorphism V : G → Q. If g, h ∈ G satisfy h−1 g r h ∈ Q for some integer r, then g r = h(h−1 g r h)h−1 ∈ Q since Q is normal. But Q is in the center, so h−1 g r h = g r . By parts (iii) and (iv) of Lemma 14, V (g) = m Y ni h−1 i g hi = i=1 m Y g ni = g Pm i=1 ni = gn. i=1 We now turn our attention to Sylow subgroups. Proposition 16. Let Q be a Sylow subgroup of a finite group G, and let h and k be elements of CG (Q). If h and k are conjugate in G, then they are conjugate in NG (Q). Proof. Suppose k = g −1 hg for some g ∈ G. Then k ∈ g −1 CG (Q)g. Now, g −1 CG (Q)g = {g −1 ag ∈ G | a ∈ G and ax = xa for all x ∈ Q} = {b ∈ G |gbg −1 x = xgbg −1 for all x ∈ Q} = {b ∈ G | bg −1 xg = g −1 xgb for all x ∈ Q} = {b ∈ G | by = yb for all y ∈ g −1 Qg} = CG (g −1 Qg). Hence, k ∈ CG (g −1 Qg). By hypothesis, Q ⊆ CG (k) and now we have shown that g −1 Qg ⊆ CG (k). Since conjugation is an automorphism, |g −1 Qg| = |Q|; thus, both groups are Sylow subgroups of CG (k). By the third Sylow Theorem (see [11, Theorem 24.5]), there exists a c ∈ CG (k) such that Q = c−1 g −1 Qgc. Then gc ∈ NG (Q) and c−1 g −1 hgc = c−1 kc = k. Theorem 17 (Burnside, 1900). Let G be a finite group and let Q be an abelian Sylow subgroup with the property that Q ≤ Z(NG (Q)), i.e. Q is contained in the center of its normalizer. Then Q has a normal complement K in G. Proof. Since Q is abelian, Q0 = {1} and we may regard the transfer as a homomorphism V : G → Q. We will show that V is surjective and that K = ker(V ) is the desired complement. Let g ∈ Q. Then, using Lemma 14, V (g) = m Y ni h−1 i g hi i=1 with ni h−1 i g hi ∈ Q ni for all i ∈ {1, . . . , m}. For any i, g ni and h−1 i g hi are elements of Q which are conjugate in Q. −1 ni n i Note that g and hi g hi belong to CG (Q). This is because Q is abelian, and hence Q ≤ CG (Q). ni By Proposition 16, g ni and h−1 i g hi are already conjugate in NG (Q), i.e. there exists ci ∈ NG (Q) 13 −1 ni ni ni commutes with c and, combining several with h−1 i i g hi = ci g ci . But Q ≤ Z(NG (Q)), so g steps, V (g) = m Y ni h−1 i g hi = i=1 m Y ni c−1 i g ci = i=1 m Y g ni = g Pm i=1 ni = gn, i=1 where n = [G : Q], as before. Let |Q| = q. Then gcd(n, q) = 1, since Q is a Sylow subgroup. There are integers a and b such that an + bq = 1. V is surjective because for any g ∈ Q, we have that g = g an+bq = g an g bq = (g a )n = V (g a ). G @ @ @ Z(NG (Q)) K Q S S S S S {1} By the First Isomorphism Theorem, G/K ∼ = Q. It remains to show that K ∩ Q is trivial. Indeed, as seen above, V restricted to Q is exponentiation by n. Since n is relatively prime to q = |Q|, this shows that V restricted to Q is injective, hence K ∩ Q = 1. We conclude that G = Q n K and K is the desired complement of Q in G. Theorem 18. Let G be a finite group, and let p be the smallest prime divisor of |G|. Let Q be a Sylow p-subgroup of G. If Q is cyclic, then Q has a normal complement in G. Proof. Let N = NG (Q) and C = CG (Q). From Proposition 1, C E N and N/C is isomorphic to a subgroup of Aut(Q). What can we say about |N/C|? Say |Q| = pm for some m. Then Aut(Q) ∼ = U (pm ) because Q is cyclic (see Section 2), and it is easy to show that |U (pm )| = pm−1 (p − 1). Thus, |N/C| divides pm−1 (p − 1). Since Q is abelian, Q ≤ C and Q is the Sylow p-subgroup of C. Hence p does not divide |N/C| and so |N/C| divides p − 1. Finally, N ≤ G, and therefore |N/C| = |N |/|C| divides |G|. But p is the smallest prime divisor of |G|; therefore |N/C| = 1, which means N = C. Because Q is abelian, Q ≤ Z(C), which implies Q ≤ Z(N ). By Theorem 17, Q has a normal complement in G. Corollary 19. Let G be a finite group, and let p be the smallest prime divisor of |G|. Let Q be a Sylow p-subgroup of G. If Q is cyclic, then G is a split extension of a normal subgroup by Q. 8 Groups of a Square-Free Order As mentioned in the Introduction, it is a difficult task in general to determine f (n), the number of groups of finite order n. In this section, we restrict our attention to cases when n is square-free. In 14 other words, n = p1 p2 · · · pr where the pi ’s are distinct primes. This is a very special situation of the broader problem, and lends itself to relatively straightforward results, most notably Hölder’s formula. We begin with an algebraic classification of groups whose order is the product of two primes, then prove two general results for groups of square-free order before considering Hölder’s formula. Example 6: Classification of groups with order pq . For the case when r = 2, denote p := p1 and q := p2 with p < q. By the third Sylow Theorem, the number of Sylow q-subgroups of G has the form qa + 1 for some integer a with qa + 1 dividing the order of the group. Thus qa + 1 must equal one of 1, p, q, or pq, the divisors of pq. But since q > p, it follows that there is precisely one (normal) Sylow q-subgroup. (We will see below by a different argument that in a group G of square-free order, the largest prime divisor of the order of the group always has a unique Sylow subgroup.) Similarly, the number of p-subgroups of G has the form pa + 1 for some integer a with pa + 1 dividing the order of the group. Thus pa + 1 must equal one of 1, p, q, or pq, the divisors of pq. Here we have two cases: (i) p divides q − 1 and (ii) p does not divide q − 1. In the second case, a = 0 is the only possibility, and G has only one Sylow p-subgroup. Both Sylow subgroups are normal as a consequence of the third Sylow Theorem, and both are cyclic, so let x and y be a their respective generators. Also note that their intersection hxi ∩ hyi is trivial. But [x, y] = x−1 y −1 xy = x−1 (xy ) ∈ hxi, [x, y] = x−1 y −1 xy = (y x )y ∈ hyi. So [x, y] = {1}. Therefore, xy = yx so the order of xy is pq, the product of the orders of x and y. Hence G is cyclic and f (pq) = 1 when p < q and p does not divide q − 1. In the other case, if p divides q − 1 then f (pq) = 2. This is because there is either 1 Sylow p-subgroup, which leads to the cyclic group of order pq, as above; or there are q Sylow p-subgroups. Extensions give another approach to classify groups with order pq. Since there is a unique normal Sylow q-subgroup Sq , we may regard G as the extension of Sq by a Sylow p-subgroup Sp . To determine the isomorphism class, we must specify a corresponding map φ : Sp → Aut(Sq ). One option is that φ maps Sp to the identity, and this possibility gives the cyclic group Cpq . The other possibility is for the image of Sp in Aut(Sq ) to be a subgroup of size p. Since Aut(Sq ) ∼ = Aut(Cq ) ∼ = U (q) is cyclic of order q − 1, it has a subgroup of order p if and only if p divides q − 1. This subgroup of order p, if it exists, is unique in Aut(Sq ). Since it has several generators, the group Sp can be mapped to it in different ways, but up to composition with an automorphism of Sp there is only one choice. Therefore, if p divides q − 1, there is a second non-abelian isomorphism class for groups of order pq in addition to the cyclic one. Let G be a group of square-free order n. All Sylow subgroups have prime order and are therefore cyclic. Also, if G is abelian, it must be the product of cyclic groups of prime order by the Fundamental Theorem of Finite Abelian Groups [11, Theorem 11.1]. Hence G is abelian if and only if it is cyclic. Before continuing the classification of groups of square-free order, we now prove that such groups are solvable (a result due to Frobenius [10]) and that they are so-called Sylow tower groups. Proposition 20. Every group of square-free order is solvable. Proof. Let p be the smallest prime divisor of |G| with Sp a Sylow p-subgroup of G. Then Sp is cyclic, solvable, and has a normal complement K in G (Theorem 18). Then |K| is square-fee, and, 15 by induction on the order of the group, K is solvable. Also, G/K ∼ = Sp is solvable, so by Proposition 3, G is solvable. We remark that this proposition is consistent with the Classification of Finite Simple Groups. In particular, from the classification one can deduce that 4 divides the order of any non-abelian finite simple group6 . Let G be a group of square-free order. Clearly 4 cannot divide |G|. So if G is simple, G must be abelian and hence solvable. If G is not simple, it has a proper normal subgroup N such that |G/N | and |N | are square-free, and it follows by induction that G is solvable. If G is abelian with square-free order it may be simple (e.g. G ∼ = Cp for a prime p) but it is solvable (Example 1). Clearly 4 cannot divide the square-free order of G, so a nonabelian group of square-free order isn’t simple. Thus it has a normal subgroup whose factor group has square-free order and the result follows by induction. A finite group G is a Sylow tower group if there exists a series of normal subgroups of G G = G0 ⊇ G1 ⊇ G2 ⊇ · · · ⊇ Gr = {1}, such that Gi−1 /Gi is a Sylow pi -subgroup of G/Gi where pi is the largest prime divisor of |G/Gi |. Theorem 21. Groups of square free order are Sylow tower groups. Proof. Let |G| = p1 p2 · · · pr with pi < pi+1 . By Theorem 18 a Sylow p1 -subgroup Sp1 has a normal complement G1 in G = G0 , with |G1 | = p2 · · · pr . Similarly, the Sylow p2 -subgroup of G1 has a normal complement G2 in G1 , with |G2 | = p3 p4 · · · pr . For i = 1, . . . , r − 1 define Gi to be the normal complement of the Sylow pi -subgroup of Gi−1 , where G0 := G. Then Gi E Gi−1 and in particular G1 E G0 = G. Also, |Gi | = pi+1 pi+2 . . . pr . We prove by induction that Gi E G. Suppose Gi−1 is normal in G for some i ≥ 1. Then conjugation by g ∈ G is an automorphism of Gi−1 . Therefore, the image of Gi under conjugation by g ∈ G, denoted γg (Gi ), is a subgroup of Gi−1 with order pi+1 pi+2 . . . pr . Now consider γg (Gi )Gi /Gi , the corresponding subgroup of Gi−1 /Gi . The order of the subgroup γg (Gi )Gi /Gi must divide pi+1 pi+2 . . . pr (the order of γg (Gi ) in Gi−1 ) as well as pi (the order of the group Gi−1 /Gi ). But gcd(pi+1 pi+2 . . . pr , pi ) = 1 implies that γg (Gi )Gi /Gi is trivial, hence γg (Gi ) ⊆ Gi . Since this is true for all g ∈ G, Gi is normal in G for i ∈ {1, 2, . . . r − 1}. For each i, 1 ≤ i ≤ r, pi is the largest prime divisor of |G/Gi | = p1 p2 · · · pi , while |Gi−1 /Gi | = pi . Hence Gi−1 /Gi is the Sylow pi -subgroup of G/Gi and the result follows. The proof of the theorem shows that for the largest prime factor p of |G|, the Sylow p-subgroup Sp is unique in G. Also, one can algebraically build groups of square-free order as iterated split extensions by cyclic groups of prime order. Finally, we turn our attention to Hölder’s classical result [15]: Theorem 22 (Hölder, 1895). The number of groups of order n, where n is square-free is given by f (n) = X Y pc(p) − 1 m|n p p−1 where p runs over all prime divisors of n/m and c(p) is the number of prime divisors q of m that satisfy q ≡ 1 (mod p). 6 By the Feit-Thompson Theorem, 2 divides the order of any non-abelian finite simple group, so groups of odd order are not simple. Suppose G is a non-abelian group such that 2 divides its order but 4 does not. Then its Sylow 2-subgroup is cyclic and has a normal complement in G by Theorem 18. Hence G is not simple. Another way to arrive at this result is to show that the set of elements of G with odd order form a normal subgroup. 16 We explain the derivation of this formula expanding on the comments given by Blackburn et al. [3]. Let G be a group of square-free order n and let m be the order of the Fitting subgroup F (G). Since F (G) is nilpotent, it is the product of its Sylow subgroups by Theorem 6. But these are all cyclic and their orders are distinct primes, so F (G) is cyclic by the Chinese Remainder Theorem. Moreover, gcd(F (G), [G : F (G)]) = 1 since |G| is square-free. By Theorem 11, F (G) has a complement H ∼ = G/F (G) and G can be regarded as a split extension F (G) by H. This shows that G is a split metacyclic group. We will give a direct proof which also shows that H is cyclic. Lemma 23. In a group G of square-free order, F (G) admits a cyclic complement H. Proof. By the above comments, F (G) is cyclic and, in particular, Aut(F (G)) is abelian. Let γg : F (G) → F (G) denote conjugation by g ∈ G — an automorphism since F (G) is a normal subgroup of G. Consider the map φ : G → Aut(F (G)) with g 7→ γg . The kernel of φ is {g ∈ G | x = g −1 xg for all x ∈ F (G)} = CG (F (G)) = Z(F (G)) = F (G), the penultimate equality following from Theorem 10 (G is solvable). Hence G/F (G) is isomorphic to a subgroup of the abelian group Aut(F (G)). This makes G/F (G) abelian, and it is also cyclic since it is of square-free order (see above). Choose an element g ∈ G such that G/F (G) = hgi. As G/F (G) and F (G) have coprime orders we may replace g by a suitable power of g so that hgi ∩ F (G) = 1. Then H = hgi is the desired complement of F (G) in G. Since F (G) is cyclic, CG (F (G)) = F (G) (Theorem 10) and Aut(F (G)) is abelian. Consider a map φ from H to Aut(F (G)) in which h is mapped to conjugation by h, denoted γh : F (G) → F (G). The map φ is injective since its kernel is trivial: h ∈ ker φ ⇔ γh = id ⇔ h−1 xh = x for all x ∈ F (G) ⇔ h ∈ CG (F (G)) = F (G) ⇔ h ∈ F (G) ∩ H = {1}. Hence G embeds into the holomorph Aut(F (G)) n F (G). We need to understand the isomorphism classes of certain subgroups of this holomorph. Proposition 24. Suppose K is a finite cyclic group (equivalently K is abelian with Aut(K) abelian). Let H1 and H2 be subgroups of Aut(K), and consider H1 n K and H2 n K as subgroups of Aut(K) n K. Then an isomorphism ∼ = φ : H1 n K → H2 n K with φ(K) = K exists if and only if H1 = H2 . Proof. “⇐” : trivial. “⇒” : Conversely, suppose there exists an isomorphism ∼ = φ : H1 n K → H2 n K with φ(K) = K. Step 1: We may assume that φ|K = idK . Subproof. Write α := φ|K ∈ Aut(K), and consider φ ψ φ̃ : H1 n K → H2 n K → αH2 α−1 n K 17 −1 where ψ(h, k) = (αhα−1 , k α ). As Aut(K) is abelian, we have αH2 α−1 = H2 . ∼ (−1) = φ̃ : H1 n K → H2 n K with φ̃|K = idK , for φ̃(k) = ψ(φ(k)) = (k α )α = k. Moreover, Step 2: For h ∈ H1 we have φ(h) = ψ(h) · xh where φ proj ψ : H1 ,→ H1 n K → H2 n K → H2 and xh ∈ K. Since K is abelian we deduce that k h = φ(k h ) = (φ(k))φ(h) = k ψ(h)xh = k ψ(h) for all k ∈ K. Thus h = ψ(h), and consequently H1 = H2 . Therefore the isomorphism class of the extension G = H n F (G) is determined by the image of H in Aut(F (G)). In other words, the number of nonisomorphic groups of square-free order n whose Fitting subgroup has order m is the number of distinct groups of size n/m in Aut(F (G)) ∼ = U (m). Now, write m = q1 q2 . . . qk where each distinct prime qj equals pi for some i. Then, using the Chinese Remainder Theorem, Aut(F (G)) ∼ = U (m) ∼ = U (q1 ) × U (q2 ) × · · · × U (qk ) ∼ = Cq −1 × Cq −1 × · · · × Cq −1 . 1 2 k Suppose p divides n/m. How many subgroups of size p does Aut(F (G)) have? A factor Cqj −1 has a subgroup of size p if and only if p divides qj − 1, i.e. if and only if q ≡ 1 (mod p) and these subgroups are unique. For each prime divisor p of n/m, let c(p) denote the number of primes q dividing m such that q ≡ 1 (mod p). Hence Aut(F (G)) has a subgroup isomorphic to Cp × Cp × · · · × Cp = Cpc(p) and all subgroups of Aut(F (G)) with order p are contained in this subgroup. This subgroup has pc(p) −1 c(p) −1 nonzero elements in C c(p) generates p p−1 subgroups of order p. This is because any of the p a subgroup of order p, but each such subgroup has p − 1 generators. Hence we obtain Hölder’s formula. Hölder’s original proof is similar in some ways. He uses the maximal normal cyclic subgroup H of G, which turns out in this case to coincide with the Fitting subgroup F (G), and establishes that G/H is cyclic. From there he determines the possible relations for generators of G. The explanation given in Conway et al. [4] also focuses on the generators and relations of a group. 9 Further Results and Conjectures A natural question that arises from Hölder’s formula is: for n square-free, can we relate f (n) to n more explicitly? McIver and Neumann [16] determined that f (n) ≤ n4 for n square-free. A better bound, given in [17], is f (n) ≤ φ(n), where φ is Euler’s function. For square-free n = p1 p2 · · · pr and greater than 1, this last result implies that f (n) ≤ φ(n) = (p1 − 1)(p2 − 1) · · · (pr − 1) < n. Furthermore, if n is even and square-free, then p1 = 2 and f (n) ≤ φ(n) = 1(p2 −1) · · · (pr −1) < n/2. Murty and Srinivasan [18] proved a more detailed result: there exist real numbers A, B > 0 such that n f (n) ≤ O (log n)A log log log n 18 for all square-free n and f (n) > n (log n)B log log log n for infinitely many square-free n. Such results are based on our understanding of the distribution of prime numbers. In this sense they are more number theoretic than the arguments presented in this report. A positive integer n is cube-free if no cube divides n. Similarly, a positive integer n is (k+1)-free if it is not divisible by any (k + 1)-th power greater than 1. It is tempting to think that Hölder’s result generalizes in some way to groups of cube-free order, or even to groups whose order is a (k + 1)-free integer. A general explicit formula is very unlikely, however, since such groups are not necessarily solvable, and if solvable need not be Sylow tower groups. A more promising direction, therefore, is to understand the asymptotic behavior of f (n) when n is (k + 1)-free. In this light, define log f (n) M (k) := lim n→∞ log n where the limit superior ranges just over (k + 1)-free integers n. For the square-free case (k = 1), Erdös, Murty, and Murty [8] have shown that M (1) = 1; their proof uses Dirichlet’s Theorem on primes in arithmetic progressions, among other techniques. The following conjecture for the cube-free case (k = 2) is attributed to Peter Neumann. Conjecture 25. With M (k) defined as above, M (2) = 2. A proof of Conjecture 25 may also use Dirichlet’s Theorem, and will likely invoke several known properties of groups of cube-free order. For a start, McIver and Neumann have shown that f (n) ≤ n8 for n cube-free [16]. In any group, the solvable residual is the smallest normal subgroup with solvable factor group, and recall that the solvable radical is that largest normal subgroup. Although groups of cube-free order are not solvable in general, they are the product of the solvable radical and the solvable residual [3, Proposition 21.15], so their structure can be studied through these two subgroups. It may be feasible to determine estimates for the number of groups of cube-free order since a large part of any such group is normal with a Sylow tower structure. A natural extension of Conjecture 25 is to find the value of M (k) for other small k. Such results may provide insight into another problem: what are good bounds for M (k)? For general k, one can verify that M (k) ≤ k 2 + k + 2 using results of McIver and Neumann [16]. Also, Pyber’s 2 2 theorem [19] gives M (k) ≤ 27 k + O(k 3/2 ), which may turn out to be the best asymptotic bound. Another, more significant conjecture in the enumeration of finite groups is Graham Higman’s PORC conjecture. In order to describe it, let us first define a polynomial on residue classes (PORC). The residue class of k with respect to N is RN (k) = {n ∈ Z | n ≡ k (mod N )}. Let f be a function defined on a set of integers. Suppose that for some integer N and for each k, there is a polynomial fk such that whenever n ∈ RN (k) ∩ dom(f ), we have fk (n) = f (n). Then the function f is said to be PORC. Now we state the conjecture: Conjecture 26 (Higman’s PORC conjecture). Let gn (p) = f (pn ) where p is a prime and f is the group number function. For a fixed n, the function gn is PORC as a function of p. The result has been verified in some limited cases. Combining the work of several researchers, it can be shown by means of sophisticated computation that gn is PORC for n less than or equal to 7. Using cohomology theory and algebraic representations of algebraic groups, Evseev has verified the result for a related function φn (p) which counts the number of groups of order pn whose Frattini group is central [9]. Marcus du Sautoy has applied zeta functions of finitiely generated torsion-free 19 nilpotent groups to this problem [7]. His approach features algebraic groups, p-adic Lie groups, p-adic integration, among others. Based on these advances, it is clear that the techniques necessary to prove the PORC conjecture would have a profound impact on several areas of mathematics. If f (pk ) is confirmed PORC, there will likely be relevant implications for f (n). Maybe there are other classes of orders for which f (n) is PORC? It has also been suggested that the function that counts metabelian or even metacyclic groups of a given order is PORC. We mention one final, curious conjecture in the enumeration of finite groups: Conjecture 27. The group enumeration function is surjective. That is, for every positive integer m, the conjecture asserts that there exists n such that f (n) = m. This conjecture may well be resolved through consideration of square-free n, largely because of Hölder’s formula. Indeed, it has been verified that every m less than 10,000,000 is equal to f (n) for some square-free n, and a forthcoming paper by R. Keith Dennis promises to shed more light on the topic [5, 6]. Hölder’s formula was a ground-breaking result in the early development of group theory, and it continues to influence research today. In this project, we have seen how a structural approach to the formula can be used to re-interpret the classical result using more recent ideas. In addition, understanding the formula can serve as an introduction to topics in graduate-level group theory and to current research topics. References [1] T. Bandman, G.-M. Greuei, F. Grunewald, B. Kunyavski, G. Pfister, and E. Plotkin. Identities for finite solvable groups and equations in finite simple groups. Compositio Mathematica, 142(3):734–764, 2006. [2] O. Baues and F. J. Grunewald. Automorphism groups of polycyclic-by-finite groups and arithmetic groups. Publications Mathématiques de l’IHÉS, 104:213–268, 2006. [3] S. R. Blackburn, P. M. Neuman, and G. Venkatarmaman. Enumeration of Finite Groups. Cambridge Tract in Mathematics, 173. Cambridge University Press, Cambridge, UK, 2007. [4] J. Conway, H. Burgiel, and C. Goodman-Strauss. The Symmetries of Things. A K Peters, Ltd., Wellesley, Massachusetts, 2008. [5] J. Conway, H. Dietrich, and E. A. O’Brien. Counting groups: gnus, moas and other exotica. Mathematical Intelligencer, 30(2):6–15, 2008. [6] R. K. Dennis. The number of groups of order n. In preparation. [7] M. du Sautoy. Counting subgroups in nilpotent groups and points on elliptic curves. Journal für die reine und angewandte Mathematik (Crelle’s Journal), 549(1-21), 2002. [8] P. Erdös, M. R. Murty, and V. K. Murty. On the enumeration of finite groups. Journal of Number Theory, 25:360–378, 1987. [9] A. Evseev. On Higman’s PORC Conjecture. Preprint, Mathematical Institute, Oxford, 2005. [10] Frobenius. Über auflösbare Gruppen. Sitzungsberichte der Königlichen Preussischen Akademie der Wissenschaften zu Berlin, pages 337–345, 1893. 20 [11] J. A. Gallian. Contemporary Abstract Algebra. Houghton Mifflin, Boston, Massachusetts, 6th edition, 2006. [12] F. J. Grunewald, P. R. Pickel, and D. Segal. Finiteness theorems for polycycic groups. Bulletin of the American Mathematical Society (New Series), 1(3):575–578, 1979. [13] R. Guralnick, B. Kunyavski, E. Plotkin, and A. Shalev. Thompson-like characterizations of the solvable radical. Journal of Algebra, 300:363–375, 2006. [14] O. Hölder. Die Gruppen der Ordnungen p3 , pq 2 , pqr, p4 . Mathematische Annalen, 43:301–412, 1893. [15] O. Hölder. Die Gruppen mit quadratfreier Ordnungszahl. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, pages 211–219, 1895. [16] A. McIver and P. M. Neuman. Enumerating finite groups. Quarterly Journal of Mathematics Oxford, 38(2):473–488, 1987. [17] M. R. Murty and V. K. Murty. On the number of groups of a given order. Journal of Number Theory, 18(178-191), 1984. [18] M. R. Murty and S. Srinivasan. On the number of groups of a squarefree order. Canadian Mathematics Bulletin, 30:412–420, 1987. [19] L. Pyber. Enumerating finite groups of a given order. Annals of Mathematics, 137:203–220, 1993. [20] D. J. S. Robinson. A Course in the Theory of Groups. Springer, New York, second edition, 1996. [21] J. J. Rotman. An Introduction to the Theory of Groups. Springer, New York, fourth edition, 1995. [22] D. Segal. Polycyclic Groups. Cambridge University Press, Cambridge, UK, 2005. 21