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Appendix a Basics on Finite Groups Appendix A Basics on Finite Groups A.1 π-Parts of Elements Let G be a group. Proposition A.1.1 Let g ∈ G be an element of order n. Assume n = ab where a and b are relatively prime integers. (1) There exist elements ga and gb of G, uniquely defined by the two conditions: (c) g = gagb = gbga , (o) the order of ga divides a, and the order of gb divides b. (2) Assume ga and gb as above. vb ua (a) If ua + vb = 1, we have ga = g and gb = g . (b) ga has order a and gb has order b. Proof Assume ua + vb = 1. We have g = gvbgua = guagvb; moreover, the order of gvb divides a and the order ua vb va of g divides b, which shows the existence of elements ga := g and gb := g satisfying (c) and (o). Let us now prove the uniqueness. Thus assume that there exist ga and gb in G = ua vb = vb vb = vb vb = satisfying (c) and (o). We have ga ga ga ga ,sog ga gb ga. Similarly, ua g = gb, which establishes the uniqueness. Finally, the order of g is the lcm of the orders of ga (a divisor of a) and of gb (a divisor of b), and since that order is ab it follows that the order of ga is a and the order of gb is b. Notation A.1.2 Let π be a set of prime numbers. We denote by π the complement of π, that is, the set of all prime numbers which do not belong to π. (1) For n ∈ N, we write n = nπnπ where all the prime divisors of nπ belong to π while all the prime divisors of nπ belong to π . © Springer Nature Singapore Pte Ltd. 2017 217 M. Broué, On Characters of Finite Groups, Mathematical Lectures from Peking University, https://doi.org/10.1007/978-981-10-6878-2 218 Appendix A: Basics on Finite Groups (2) For g ∈ G of order n,weset := := , gπ gnπ and gπ gnπ and we call gπ the π-component of g (hence gπ is the π -component of g). (3) If π ={p} is a singleton, we set gp := g{p} and gp := g{p} . Then gp and gp are called respectively the p-component and the p-component of g. A.2 Nilpotent Groups Definition A.2.1 A group G is said to be nilpotent if there exists a finite chain of normal subgroups 1 = G0 ⊂ G1 ⊂···⊂ Gn = G such that for i = 1,...,n, Gi /Gi−1 = Z(G/Gi−1). The sequence described in the above definition is called the ascending central chain. Notice that for i = 1,...,n, [G, Gi ]⊂Gi−1. Examples A.2.2 • An abelian group is nilpotent, but the group S3 is not nilpotent. • A direct product of nilpotent groups is nilpotent. • If p is a prime, any finite p-group is nilpotent. Indeed, this follows from the following lemma. Lemma A.2.3 Let p be a prime, and let G be a finite p-group. Then Z(G) = 1. Proof We let G act on itself by conjugation. The decomposition into the disjoint union of orbits gives G = Z(G) C C where C runs over the set of non central conjugacy classes. Thus |G|=|Z(G)|+ | | | ( )| ( ) = C C and p divides Z G , hence Z G 1. Let us give a few properties of nilpotent groups. Proposition A.2.4 Let G be a nilpotent finite group. (1) If H is a proper subgroup of G, then H is a proper subgroup of its normalizer. (2) If G is nonabelian, there exists an abelian normal subgroup of G not contained in Z(G). Appendix A: Basics on Finite Groups 219 Proof (1) Let 1 = G0 ⊂ ··· ⊂ Gn = G be a chain as in DefinitionA.2.1. Since H = HG0 ⊂ HG1 ⊂ ··· ⊂ HGn = G, there exists i < n such that Gi ⊂ H and H ⊂ Gi+1. Then H HGi+1, and H HGi+1 since [H, Gi+1]⊂Gi ⊂ H. (2) Since G is nonabelian, G1 is a proper subgroup of G, hence G1 G2. Choose g ∈ G2 \ G1. Then the group A := G1g is abelian and not contained in G1.Itis normal since [G, A]⊂G1. Theorem A.2.5 Let G be a finite group. The following assertions are equivalent. (i) G is nilpotent. (ii) There exists a (finite) family of primes p and of p-subgroups G p of G such that G is isomorphic to the direct product of the G p’s. Proof (ii)⇒(i) is trivial (see Example A.2.2). Let us prove (i)⇒(ii). 1. We first prove that, for every prime p, a Sylow p-subgroup P of G is normal in G. This follows from PropositionA.2.4, (1) above, and from the following general lemma. Lemma A.2.6 Whenever G is a finite group, p a prime number, and P a Sylow p-subgroup of G, then NG (NG (P)) = NG (P). Proof of Lemma A.2.6 Indeed, since P is the unique Sylow p-subgroup of NG (P), P is a characteristic subgroup of NG (P) (that is, a subgroup which is stable under any automorphism of NG (P)). 2. Let p1,...,pm be the different prime numbers dividing |G| and let P1,...,Pm be the corresponding Sylow subgroups of G, all normal in G. Thus for all i = j we have [Pi , Pj ]⊂Pi ∩ Pj , hence [Pi , Pj ]=1. It follows that the map P1 ×···× Pm → P1 ···Pm ⊂ G ,(x1,...,xm ) → x1 ···xm , is a surjective group morphism. Since the order of its image P1 ···Pm is divisible by the order of each Sylow subgroup of G,wehaveP1 ···Pm = G, and the above map is an isomorphism. A.3 Complements on Sylow Subgroups The next proposition is known as the “local characterization” of “Sylow subgroups”. Proposition A.3.1 Let P be a p-subgroup of a finite group G. The following asser- tions are equivalent. 220 Appendix A: Basics on Finite Groups (i) P is a Sylow p-subgroup of G. (ii) P is a Sylow p-subgroup of its normalizer NG (P). Proof (i)⇒(ii) is clear. Let us prove (ii)⇒(i). For this purpose, we prove that if there exists a p-subgroup Q of G such that P Q, then P NQ(P). Indeed, since Q is nilpotent, this follows from item (1) of Proposition A.2.4. The next proposition is known as the “Frattini argument”. Proposition A.3.2 (Frattini argument) Assume that H is a normal subgroup of G. Let p be a prime and let P be a Sylow p-subgroup of H. Then G = HNG (P). Proof For g ∈ G, gPg−1 is still a Sylow p-subgroup of H. Hence there exists h ∈ H −1 −1 −1 such that gPg = hPh . Thus n := h g ∈ NG (P) and g = hn. A.4 Schur–Zassenhaus Theorem A proof of the following statement may be found in [Gor80], Chap. 6, Theorem 2.1. Theorem A.4.1 (Schur–Zassenhaus) Assume that N is a normal subgroup of a finite group G such that |N| and |G/N| are relatively prime. (1) There exists a subgroup H of G such that G = N H and N ∩ H = 1, i.e., G N H. (2) If either N or G/N is solvable, then the subgroups H of G such that G = NH and N ∩ H = 1 form a single conjugacy class of subgroups of G. Remark A.4.2 A celebrated (very deep and difficult) theorem of Feit and Thompson (W. Feit and J.G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029) proves that any group of odd order is solvable. Hence, by this theorem, we see that the assumption of solvability can be dropped in the second assertion of the Schur–Zassenhaus Theorem. Appendix B Assumed Results on Galois Theory For the contents of this appendix, see for example [Ste03]. All fields considered here are commutative and have characteristic zero. B.1 Galois Extensions For the following fundamental property, the reader may also look at [Bro13], Proposition1.78. Proposition B.1.1 Let K be a finite extension of a field K . The following assertions are equivalent. (i) There exists P(X) ∈ K [X] such that K is the splitting field of P(X) over K . (ii) Whenever Q(X) is an irreducible element of K [X] which has at least one root in K , then Q(X) splits into a product of degree one factors over K . Definition B.1.2 (1) A finite field extension K /K is called a Galois extension if it satisfies properties (i) and (ii) as above. (2) The Galois group Gal(K /K ) of such an extension is the set of automorphisms of K which induce the identity on K . Examples B.1.3 √ (1) Let α := 3 2 ∈ R. Then the extension Q(α)/Q is not Galois. Indeed, two of the roots of X 3 − 2 (which is irreducible in Q[X]) do not belong to Q(α). (2) Let n ∈ N, n ≥ 1, and let ζ be a root of unity of order n in an extension of K . Then the extension K (ζ)/K is Galois. Indeed, since any n-th root of 1 is a power of ζ, the field K (ζ) is the splitting field of X n − 1 over K . © Springer Nature Singapore Pte Ltd.
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