Finite Group Theory David A. Craven Michaelmas Term, 2010 Contents 0 Preliminaries 1 1 The 1900s 6 1.1 PSLn(q)...................................... 6 1.2 The Transfer . 9 1.3 p-Groups and Nilpotent Groups . 14 1.4 Frobenius Groups . 18 2 The 1930s 22 2.1 The Schur{Zassenhaus Theorem . 22 2.2 Hall's Theorem on Soluble Groups . 28 2.3 The Fitting Subgroup . 30 3 The 1960s 34 3.1 Nilpotence of Frobenius Kernels . 34 3.2 Alperin's Fusion Theorem . 39 3.3 Focal Subgroup Theorem . 42 3.4 The Generalized Fitting Subgroup . 46 4 The 1990s 51 4.1 Saturated Fusion Systems . 51 4.2 Normalizers and Quotients . 55 4.3 Alperin's Fusion Theorem . 59 4.4 Thompson's Normal p-Complement Theorem . 62 5 Exercise Sheets 67 1 Sheet 1 . 67 2 Sheet 2 . 68 3 Sheet 3 . 70 i 4 Sheet 4 . 71 5 Sheet 5 . 73 6 Sheet 6 . 74 7 Sheet 7 . 76 6 Solutions to Exercises 77 1 Sheet 1 . 77 2 Sheet 2 . 80 3 Sheet 3 . 83 4 Sheet 4 . 86 5 Sheet 5 . 88 6 Sheet 6 . 91 Bibliography 96 ii Chapter 0 Preliminaries We assume that the reader is familiar with the concepts of a group, subgroup, normal subgroup, quotient, homomorphism, isomorphism, normalizer, centralizer, centre, simple group. Definition 0.1 A p-group is a group, all of whose elements have order a power of p. Proposition 0.2 let G be a finite p-group. Then jGj = pn for some n, and Z (G) 6= 1. Definition 0.3 Let G be a finite group, and let p be a prime. Suppose that pm j jGj, but pm+1 - jGj.A Sylow p-subgroup of G is a subgroup P of G of order pm. Theorem 0.4 (Sylow's theorem, 1872) Let G be a finite group, and let p be a prime. (i) Sylow p-subgroups exist, and the number of them is congruent to 1 modulo p. (ii) All Sylow p-subgroups are conjugate in G. (iii) Every p-subgroup is contained in a Sylow p-subgroup. Definition 0.5 Let G be a group. A series for G is a sequence 1 = G0 G1 G2 ··· Gr = G P P P P of subgroups of G with Gi−1 Gi for all 1 6 i 6 r. It will sometimes also be denoted (Gi). P If Gi=Gi−1 is abelian, (Gi) is an abelian series. If Gi=Gi−1 is simple, (Gi) is a composition series. The length of a series is the number of terms in it, so in the example above it has length r. If a group possesses an abelian series, we say that it is soluble. 1 Theorem 0.6 (Burnside's pαqβ-theorem, 1904) Any finite group whose order is of the form pαqβ, for primes p and q, is soluble. Definition 0.7 Let G be a group. (i) Let h and k be elements of G. The commutator of h and k, denoted [h; k], is the element h−1k−1hk. (ii) Let H and K be subgroups of G. The commutator of H and K, denoted [H; K], is the subgroup generated by all commutators [h; k] for h 2 H and k 2 K. (Note that the elements of [H; K] are products of commutators, and not necessarily commutators [h; k].) (iii) The derived subgroup of G, denoted G0, is [G; G]. Write G(1) = G0 and define G(i) = [G(i−1);G(i−1)]. (iv) By [x; y; z] we mean the left-normed commutator [[x; y]; z]. We extend this notation by induction, so that [x1; x2; x3; : : : ; xn−1; xn] = [x1; x2; : : : ; xn−1]; xn : Definition 0.8 Let G be a finite group. If G possesses a norrmal subgroup K and a sub- group H such that K \ H = 1 and G = HK, then G is the (internal) semidirect product of K by H. If K is a finite group and φ : H ! Aut(K) is a homomorphism, then we may construct a group G such that G = K o H and the elements h of H act on K by conjugation as hφ 2 Aut(K). This is to define a multiplication on the set H × K by (h0; k0)(h; k) = (h0h; (k0)hφk): This group so defined is denoted G = K o H or G = H n K. This formula is meant to mimic the conjugation formula h0k0hk = h0h(k0)hk in any finite group. This is often referred to as the external semidirect product. If G = KoH is an external semidirect product, then the natural subgroups K¯ = f(1; k)g and H¯ = f(h; 1)g make G into an internal semidirect product, and the other direction is equally easy. Hence from now on we will identify internal and external semidirect products. Theorem 0.9 Any finite abelian group is the direct product of cyclic groups. 2 Definition 0.10 Let G be a finite permutation group on a set X. (i) G is transitive if, for any two points x and y in X, there is an element g 2 G such that xg = y. (ii) G is imprimitive if there is a partition p of X that is preserved by G, and the parts of p are neither X nor singleton sets. G is primitive if it is not imprimitive. (iii) G is n-transitive if, for any two n-tuples x and y of distinct points in X there is an element g 2 G such that xg = y. If G is a permutation group on a set X, and x 2 X, denote by Gx the stabilizer of x in G, i.e., the elements g 2 G such that xg = x. Proposition 0.11 Let G be a transitive permutation group on a finite set X. (i) G is n-transitive if and only if Gx is (n − 1)-transitive, for some x 2 X. (ii) If G is 2-transitive then G is primitive. (iii) G is primitive if and only if Gx is a maximal subgroup of G, for some (and hence every) x 2 X. Proof: If G is n-transitive, then any n-tuple of distinct elements ending in x can be sent to any other n-tuple of distinct elements ending in x, so that (by removing the last term from the n-tuples) we see that Gx is (n − 1)-transitive. To see the converse, let x = (x1; : : : ; xn) and y = (y1; : : : ; yn) be any two n-tuples of distinct elements. We need to prove that there exists g 2 G such that xg = y. Since G is transitive, there exists h; k 2 G such that xnh = ynk = x. As Gx is (n − 1)-transitive, there exists l 2 Gx such that (x1h; x2h; : : : ; xn−1h)l = (y1k; y2k; : : : ; yn−1k): The element hlk−1 maps x to y, proving (i). Let p be a partition of X that is preserved by G, proving that G is imprimitive. Let x and y be elements of X lying in the same part, and let z be an element not in this part. Since G preserves p, there is no element of G fixing x and mapping y to z, so G is not 2-transitive on X. This proves (ii). Let H be the stabilizer of a point x in X, and let M be a maximal subgroup of G containing H. As X is the set of cosets of H in G, let the parts of the partition p be given by the cosets of M in G. This partition is clearly preserved by G, and hence G is not primitive. Hence if G is primitive then point stabilizers are maximal subgroups. 3 The converse is very similar: suppose that X = X1 [ X2 [··· Xn is a partition of X proving that G is imprimitive. Let x be a point in X1, and let H be the stabilizer of x. Write M for the stabilizer of X1. Notice that M < G since X1 6= X, and since X1 6= fxg we have H < M, so that H is not a maximal subgroup of G. Definition 0.12 (i) If V is a vector space, GL(V ) is the set of linear transformations of n V . If V = (Fq) we denote GL(V ) by GLn(q), and consider it as all invertible n × n matrices. (ii) SL(V ) is the subset of GL(V ) consisting of all transformations of determinant 1. (iii) The dihedral group D2n is the group with the presentation h x; y : xn = y2 = 1; xy = x−1i: (iv) By Cn we denote the cyclic group of order n. (v) The symmetric group on n letters is denoted by Sn, and the alternating group on n letters is denoted by An. n (vi) By Epn we mean the elementary abelian p-group of order p , i.e., the direct product of n copies of Cp. n−1 Proposition 0.13 (i) If G = Cpn then Aut G is abelian, of order p (p − 1). ∼ (ii) If G = Epn then Aut G = GLn(p). Theorem 0.14 For n > 5, An is simple. Proposition 0.15 If G is a simple group of order 60 then G is isomorphic with A5. Proof: We will show that G has a subgroup of order 12, since in this case the standard homomorphism from G into S5 must be injective (as G is simple) and lie inside A5 (again, since G is simple).