Appendix a Basics on Finite Groups

Home , Frobenius reciprocity, Integral element, Zero object (algebra)

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:

(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 assertions 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. 2017 221 M. Broué, On Characters of Finite Groups, Mathematical Lectures from Peking University, https://doi.org/10.1007/978-981-10-6878-2 222 Appendix B: Assumed Results on Galois Theory

(3) The Galois group Gal(K (ζ)/K ) is isomorphic to a subgroup of the multiplicative group (Z/nZ)×. Indeed, if σ ∈ Gal(K (ζ)/K ), then σ(ζ) is again a root of unity of order n, hence there exists a(σ) ∈ (Z/nZ)× such that g(ζ) = ζa(σ), and the map

Gal(K (ζ)/K ) → (Z/nZ)× , σ → a(σ),

is clearly an injective group morphism.

B.2 Main Theorem

∼ Theorem B.2.1 Let φ : K1 −→ K be a ﬁeld isomorphism. Let L/K1 be a ﬁnite extension, and let K /K be a Galois extension such that K contains the roots of φ(P(X)) where P(X) runs over the set of minimal polynomials over K1 of elements of L. L K

∼ K1 K φ

Then there exist exactly [L : K1] morphisms L → K extending φ. Sketch of proof This is essentially, for example, the proof of Proposition2.246 in [Bro13]. Corollary B.2.2 Let L/K be a ﬁnite extension and let K be a Galois extension of K which contains L. Let MorK (L, K ) denote the set of morphisms L → K inducing the identity on K . L K

(1) |MorK (L, K )|=[L : K ] . (2) Let x ∈ L. Then

x ∈ K ⇔∀σ ∈ MorK (L, K ), σ(x) = x .

Proof (1) is a particular case of the preceding theorem. In order to prove (2), it is enough to prove that if x ∈ L, x ∈/ K , there exists σ ∈ MorK (L, K ) such that σ(x) = x. Appendix B: Assumed Results on Galois Theory 223

So assume x ∈ L \K . The irreducible polynomial of x over L has degree at least 2, and splits into degree 1 polynomials over K , hence has a root y = x in K (since we ∼ are in characteristic zero). The map x → y induces an isomorphism K (x) −→ K (y) by Theorem B.2.1. That isomorphism extends to [L : K (x)] morphisms L → K , and, for such morphisms σ,wedohaveσ(x) = x. The following corollary is an immediate consequence of the preceding one. Corollary B.2.3 Let K /K be a Galois extension. (1) |Gal(K /K )|=[K : K ] . (2) FixGal(K /K )(K ) = K .

2πi/n Example B.2.4 Let ζn := e . Since the cyclotomic polynomial n(X) (see e.g. [Bro13, 1.1.4.3]) has degree ϕ(n) =|(Z/nZ)×|, and since it is irreducible over Q ([Bro13, Theorem 1.171]), we see that × [Q(ζn)/Q]=|(Z/nZ) | ,

× and so the morphism Gal(Q(ζn)/Q) → (Z/nZ) described in Example B.1.3,(2), is an isomorphism. := ( 2iπ ) ( ) := 4 + 3 + 2 + + Exercise B.2.5 Let ζ exp 5 , and let P X X X X X 1 (see Exercise 3.7.21). Let φ be the golden ratio, characterized by the conditions

φ2 = φ + 1 and φ > 0 .

(1) Prove that ζ2 + ζ3 =−φ and ζ + ζ4 = φ − 1 .

(2) Prove that P(X) has the following decomposition over the ﬁeld Q(φ)[X]:

P(X) = (X 2 + φX + 1)(X 2 + (1 − φ)X + 1).

(3) Prove that P(X) is irreducible over Q(21/5) and deduce that the ﬁeld Q(21/5, ζ) has degree 20 over Q. (4) Prove that Q(21/5, ζ)/Q is a Galois extension and describe its Galois group Gal(Q(21/5, ζ)/Q).

B.3 Minimal and Characteristic Polynomials, Morphisms

Let L/K be a ﬁnite extension and let x ∈ L. [ ( ) : ] ( ) The K x K elements σ x σ∈MorK (K (x),K ) (see TheoremB.2.1) are called the conjugates of x over K . 224 Appendix B: Assumed Results on Galois Theory

We denote by

• Mx (X) the minimal polynomial of x over L, so that K (x) K [X]/(Mx (X)), • m L/K,x the K -linear endomorphism of L deﬁned by the multiplication by x, and

– L/K,x (X) the characteristic polynomial of m L/K,x , –TrL/K (x) the trace of m L/K,x , – NL/K (x) the determinant of m L/K,x , called the norm of x relative to the extension L/K . The following proposition is essentially Proposition2.247 in [Bro13].

Proposition B.3.1 Let L/K be a ﬁnite extension and let K be a Galois extension of K which contains L. Let x ∈ K. We denote by Gal(K /K )x the ﬁxator of x in Gal(K /K ). We call conjugates of x the images of x under Gal(K /K ). (1) M (X) = (X − σ(x)) , x σ∈MorK (K (x),K ) (2) M (X) = (X − σ(x)) , that is, the roots of M (X) are x σ∈Gal(K /K )/Gal(K /K )x x the distinct conjugates of x. [L:K (x)] (3) L/K,x (X) = Mx (X) = ∈ ( , )(X − σ(x)) , and in particular σ MorK L K (a) Tr / (x) = σ(x), L K σ∈MorK (L,K ) (b) N / (x) = σ(x). L K σ∈MorK (L,K ) Appendix C Integral Elements

All throughout this section, A is assumed to be a subring of a commutative ring B.

C.1 Deﬁnition, Integral Closure

Deﬁnition C.1.1 Let x ∈ B. We say that x is integral over A if there exists a monic polynomial P(X) ∈ A[X] such that P(x) = 0.

Examples C.1.2 Assume A = Z and B = C. In that case, an element which is integral over Z is called an algebraic integer.

(1) The elements of Z are integral over Z√. (2) All roots√ of unity of C, all elements m n for m, n ∈ N, m, n ≥ 1, all elements (1 + √d)/2ford ∈ Z and d ≡ 1 mod 4 (thus in particular the Golden Ratio (1 + 5)/2) are algebraic integers.

Exercise C.1.3 Prove that x ∈ Q is integral over Z if and only if x ∈ Z.

Proposition C.1.4 Let x ∈ B. The following assertions are equivalent: (i) the element x is integral over A, (ii) the subring A[x] of B generated by A and x is a ﬁnitely generated A–module, (iii) there exists a subring A of B, containing x, which is a ﬁnitely generated A–module.

Proof r r−1 (i)⇒(ii): Assume that x − λr−1x −···−λ1x − λ0 = 0. We shall prove that A[x] is generated by {1, x,...,xr−1}. n r+n Since A[x] is generated by {x }n∈N, it sufﬁces to prove that, for all n ≥ 0, x is a linear combination of {1, x,...,xr−1}.Itisclearforn = 0. The proof by induction on n is easy. (ii)⇒(iii): trivial. © Springer Nature Singapore Pte Ltd. 2017 225 M. Broué, On Characters of Finite Groups, Mathematical Lectures from Peking University, https://doi.org/10.1007/978-981-10-6878-2 226 Appendix C: Integral Elements

⇒ { ,..., } ∈ (iii) (i): Assume that x1 xm is a generating system for A . Since x A , ∈ ≤ , ≤ = m . there are λi, j A (1 i j m) such that for all i, xxi j=1 λi, j x j Let be the m × m square matrix with entries λi, j ∈ A (1 ≤ i, j ≤ m). The above equations give ⎛ ⎞ ⎛ ⎞ x1 0 ⎜ . ⎟ ⎜. ⎟ (x1m − ) ⎝ . ⎠ = ⎝. ⎠ .

xm 0

t Multiplying to the left by Com(x1m − ) gives then ⎛ ⎞ ⎛ ⎞ x1 0 ⎜ . ⎟ ⎜. ⎟ det(x1m − ) ⎝ . ⎠ = ⎝. ⎠ ,

xm 0 so det(x1m − )A = 0 hence det(x1m − ) = 0 .

Now det(X1m − ) is a monic element of A[X], which proves (i).

Corollary C.1.5 The set of elements of B which are integral over A is a subring of B which contains A.

Proof We must prove that whenever x and y are two elements of B which are integral over A, then x + y and xy are integral over A. This follows from the fact that all the elements of the ring A[x, y] are integral over A, which we prove now. By PropositionC.1.4, it suffices to prove that A[x, y] is a finitely generated A-module. This follows from the following lemma (which generalizes part of a known result about fields extensions).

Lemma C.1.6 Assume that A = A0 ⊂ A1 ⊂ ··· ⊂ Am is a tower of commutative rings, where (for all i = 0,...,m − 1)Ai is a subring of Ai+1. If, for all i = 0,...,m − 1,Ai+1 is a finitely generated Ai -module, then Am is a finitely generated A-module. Proof of Lemma C.1.6 We sketch a proof in the case where m = 2 (which is sufficient). Let us set B := A1 and C := A2.Let{s1,...,sm } be a generating system of B as an A-module, and let {t1,...,tn} be a generating system of C as a B-module. Then it can be checked (do it!) that {si t j }1≤i≤m,1≤ j≤n is a generating system of C as an A-module.

Deﬁnition C.1.7 (1) The ring of elements of B which are integral over A is called the integral closure of A in B. (2) If A = Z and B = C, the integral closure of Z in C (the ring of all algebraic integers) is denoted by Z¯ . Appendix C: Integral Elements 227

C.2 A Few Properties

C.2.1 Minimal Polynomial of an Integral Element

The following property shows that if an element is integral over Z, it is the root of a monic element of Z[X] which is irreducible over Q.

Proposition C.2.1 Let x ∈ C be an algebraic integer, and let Mx (X) denote its monic minimal polynomial over Q. Then

Mx (X) ∈ Z[X] .

Sketch of proof Let P(X) ∈ Z[X] be a monic polynomial such that P(x) = 0. Whenever σ : Q(x) → K is a field morphism into a Galois extension K of Q (note that σ induces the identity on Q), then σ(x) is also a root of P(X), hence is an alge- ( ) = ( − braic integer. By PropositionB.3.1, we know that Mx X σ∈MorQ(Q(x),K ) X σ(x)) , from which it follows that the coefficients of Mx (X), which are symmetric ( ( )) polynomials evaluated at the family σ x σ∈MorQ(Q(x),K ), • arefixedbyallσ ∈ MorQ(Q(x), K ), hence (see CorollaryB.2.2, (2)) belong to Q, • are algebraic integers (by CorollaryC.1.5), hence belong to Z (by ExerciseC.1.3).

Remark C.2.2 The reader may state various generalizations of the previous proposition (for example, replacing Z by any factorial domain).

C.2.2 On Fields of Fractions and Integrality

We conclude by a lemma which is related to Proposition 7.2.6.

Lemma C.2.3 Let A be a subring of an integral domain B. Let K (resp. L) be the ﬁeld of fractions of A (resp. B). Assume that B is integral over A. (1) L is algebraic over K . (2) Each element of L may be written b/a where b ∈ B and a ∈ A. (3) Every system of generators of B as an A-module is also a system of generators of L as a K -vector space.

Proof of Lemma C.2.3 (1) Since L is generated over K by the elements of B which are all algebraic over K , L is algebraic over K . 228 Appendix C: Integral Elements

d (2) Let b ∈ B, b ∈/ A. There exist d ≥ 2 and a0,...,ad−1 ∈ A such that b + d−1 ad−1b +···+a0 = 0 . Let e be the smallest integer such that ae = 0. Since ∈/ ≤ − −1 =− −1 d−e−1 + d−e−2 +···+ , b A, e d 1, and b ae b ad−1b ae+1 showing that b−1 = b/a for some b ∈ B and a ∈ A. (3) Assume that B = Ab1 +···+Abs . Then every element b/a of L may be written b/a = (a1/a)b1 +···+(as /a)bs . Appendix D Noetherian Rings and Modules

D.1 Noetherian Modules

Theorem D.1.1 Let R be a commutative ring and let M be an R-module. The following conditions are equivalent. (i) Every nonempty family of submodules of M has a maximal element. (ii) Every increasing (ascending) sequence of submodules M0 ⊂ M1 ⊂ ··· ⊂ Mn ⊂··· of M is stationary. (iii) Every submodule of M is ﬁnitely generated.

The following deﬁnition is in honor of Emmy Noether.

Deﬁnition D.1.2 A module satisfying the above equivalent conditions is said to be Noetherian.

Proof of Theorem D.1.1 (i)⇒(iii) Let N be a submodule of M, and let F(N) be the family of all finitely generated submodules of N. Since F(N) contains the trivial module 0, it is nonempty hence it has a maximal element, say N . Let us show that N = N.Letx ∈ N.The module N + Rx is finitely generated, hence N + Rx = N , showing that x ∈ N . ⇒ ( ) (iii) (ii) Let Mn n≥0 be an increasing sequence of submodules of M. It is easy to := check that M∞ n≥0 Mn is a submodule of M. Since it is finitely generated, there exists m ∈ N such that Mm contains a set of generators, which implies M∞ = Mm, hence that the sequence (Mn)n≥0 is stationary. (ii)⇒(i) The proof relies on the following lemma. Note that this lemma uses the axiom of choice (where?).

Lemma D.1.3 Let be a (partially) ordered set. The following assertions are equivalent. (i) Every nonempty subset of has a maximal element. (ii) Every increasing sequence (ωn)n≥0 of elements of is stationary.

© Springer Nature Singapore Pte Ltd. 2017 229 M. Broué, On Characters of Finite Groups, Mathematical Lectures from Peking University, https://doi.org/10.1007/978-981-10-6878-2 230 Appendix D: Noetherian Rings and Modules

Proof of Lemma D.1.3 (i)⇒(ii): Let (ωn)n≥0 be an increasing sequence. If ωm is a maximal element, we have ωn = ωm for all n ≥ m, which shows that the sequence is stationary. (ii)⇒(i): Assume (i) is false, and let be a nonempty subset of which has no maximal element. Let ω0 ∈ . We can then build by induction a strictly increasing sequence (ωn)n≥0 in as follows: Assume ωn known. Since ωn is not maximal in , we may pick ωn+1 ∈ such that ωn+1 > ωn.

ι π Proposition D.1.4 Let 0 → M −→ M −→ M → 0 be a short exact sequence of R-modules. Then the following assertions are equivalent: (i) M is Noetherian, (ii) M and M are Noetherian.

Proof (i)⇒(ii): Any increasing sequence of submodules of M (or M) gives rise to an increasing sequence of submodules of M, hence is stationary. (ii)⇒(i): Let us prove that any submodule N of M is ﬁnitely generated. By assumption, the submodule ι−1(N) of M and the submodule π(N) of M are ﬁnitely gen- −1 erated. Let (x1,...,xm ) be the image under ι of a set of generators of ι (N), and let (y1,...,yn) be a set of preimages under π of generators of π(N). The reader will check as an exercise that (x1,...,xm , y1,...,yn) is a set of generators of N. , ,..., r . Corollary D.1.5 If M1 M2 Mr are Noetherian R-modules, so is n=1 Mn

Proof Applying PropositionD.1.4 to the short exact sequence 0 → M1 → M1 ⊕ M2 → M2 → 0 gives that M1 ⊕ M2 is Noetherian, and now an easy induction on r proves the claim.

D.2 Noetherian Rings

Deﬁnition D.2.1 AringisaNoetherian ring if it is Noetherian as a module over itself.

n ! Let k be a ﬁeld. Then the polynomial ring k[(Xn)n≥0] in a countable set of indeterminates is not Noetherian. That ring is a subring of its ﬁeld of fractions k((Xn)n≥0), hence a subring of a Noetherian ring needs not be Noetherian.

Proposition D.2.2 Let R be a commutative Noetherian ring. (1) If a is an ideal of R, the ring R/a is Noetherian. (2) Any ﬁnitely generated R-module is Noetherian. (3) If R is a subring of a ring T which is a ﬁnitely generated R-module, then T is a Noetherian ring. Appendix D: Noetherian Rings and Modules 231

Proof (1) R/a is Noetherian as an R-module by PropositionD.1.4, hence is Noetherian as an R/a-module, that is, is a Noetherian ring. (2) Assume M is an image of Rm for some integer m. Since Rm is a Noetherian R-module by CorollaryD.1.5, M is Noetherian by PropositionD.1.4. (3) T is a Noetherian R-module by (2) above, hence a fortiori a Noetherian T -module, hence a Noetherian ring.

D.3 Hilbert’s Basis Theorem

Theorem D.3.1 (Hilbert’s Basis Theorem) Let R be a Noetherian ring. Then the polynomial ring R[X1,...,Xr ] in a finite number of indeterminates is Noetherian. Proof It suffices to prove that R[X] is Noetherian. Let A be an ideal of R[X]. We shall prove that A is finitely generated. The leading coefficients of the elements of A form an ideal a in R. That ideal is finitely generated since R is Noetherian. Let a1,...,am be a set of generators of a. Let P1(X),...,Pm (X) ∈ A whose leading coefficients are respectively a1,...,am and let us denote by B the ideal of R[X] generated by P1(X),...,Pm (X). For 1 ≤ i ≤ m,wesetdi := deg Pi (X), and r := max{d1,...,dm } . Let R[X]r be the R-submodule of R[X] generated by {1, X,...,Xr−1}. In order to prove that A is finitely generated, we shall prove that

A = (A ∩ R[X]r ) + B . (D.3.2)

Since B is finitely generated by definition, and A ∩ R[X]r is finitely generated as an R-module (since it is an R-submodule of the finitely generated R-module R[X]r and R is Noetherian), that will prove the result. Let P(X) ∈ A.Letd := deg P(X). We may assume that d > r.Leta be its leading coefficient. We have a = λ1a1 +···+λmam for some λi ∈ R. Then the polynomial m d−di P(X) − λi X Pi (X) i=1 has degree strictly smaller than d. Repeating that operation, we get P(X) as the sum of an element of R[X]r and an element of B, thus proving the announced equality (D.3.2). Corollary D.3.3 If R is a Noetherian ring, any finitely generated R-algebra is a Noetherian ring. Proof Indeed, it is an immediate consequence of TheoremD.3.1 and of Proposition D.2.2,(1). Appendix E The Language of Categories and Functors

E.1 General Deﬁnitions

We brieﬂy introduce (or recall) some basic notation and deﬁnitions about categories. For more details we refer the reader to [Mac71].

E.1.1 Categories and Functors

Categories.

Deﬁnition E.1.1 A category consists of the following three mathematical entities: • AclassOb(C), whose elements are called objects (for X an object, we write X ∈ Ob(C),orevenX ∈ C), • for each pair of objects X and X ,asetMorC(X, X ) (an element f ∈ MorC(X, X ) is then called a morphism with source X and target X and denoted f : X → X ), • for each triple of objects X, X , X , a map called the composition: MorC(X , X ) × MorC(X, X ) −→ MorC(X, X ) (g, f ) → g · f ,

such that (1) (h · g) · f = h · (g · f ), (2) whenever X ∈ C, there is an element IdX ∈ MorC(X, X) such that, for every morphism f : X → X ,wehave f · IdX = IdX · f .

Let us give some deﬁnitions related to properties of morphisms. A morphism f : X → X is

© Springer Nature Singapore Pte Ltd. 2017 233 M. Broué, On Characters of Finite Groups, Mathematical Lectures from Peking University, https://doi.org/10.1007/978-981-10-6878-2 234 Appendix E: The Language of Categories and Functors

• a monomorphism (or monic) if f · g1 = f · g2 implies g1 = g2 for all morphisms g1, g2 : X → X, • an epimorphism (or epic) if g1 · f = g2 · f implies g1 = g2 for all morphisms g1, g2 : X → X , • an isomorphism if there exists a morphism g : X → X with f · g = IdX and g · f = IdX , • an endomorphism if X = X (EndC(X) denotes the set of endomorphisms of X, • an automorphism is both an endomorphism and an isomorphism. (Aut(X) denotes the group of automorphisms of X).

A full subcategory C of a category C is a category where • the objects of C are some objects of C, • for X and X objects of C ,wehaveMorC(X, X ) = MorC (X, X ).

Example E.1.2 Let k be a field and let A be a finite dimensional k-algebra. We denote by Amod the category whose objects are the finitely generated A–modules and where

( , ) := ( , ). Hom Amod X X Hom A X X

Exercise E.1.3 Prove that the monomorphisms (resp. epimorphisms) of Amod are the injective (resp. surjective) homomorphisms.

The opposite category Cop of a category C is the category where • Ob(Cop) := Ob(C), • MorCop (X, X ) := MorC(X , X), • for f ∈ MorCop (X, X ) and g ∈ MorCop (X , X ), (g. f )Cop := ( f.g)C. Functors.

Deﬁnition E.1.4 Let C and D be two categories. A (covariant) functor F : C → D • associates to each X ∈ Ob(C) an object F(X) ∈ Ob(D), • for each pair (X, X ) of objects of C it deﬁnes a map

F : MorC(X, X ) → MorD(F(X), F(X ))

such that

(1) whenever X ∈ C, F(IdX ) = IdF(X), (2) whenever f : X → X and g : X → X , then F(g. f ) = F(g).F( f ). A contravariant functor F : C → D is a (covariant) functor from Cop to D.

The essential image of F is the full subcategory of D whose objects are the objects of D isomorphic to objects of the image of F. Appendix E: The Language of Categories and Functors 235

We say that the functor F is • faithful if MorC(X, X ) → MorD(F(X), F(X )) is injective for all X, X ∈ C, • full if MorC(X, X ) → MorD(F(X), F(X )) is surjective for all X, X ∈ C, • fully faithful if it is full and faithful, • essentially surjective if the essential image of F is D. Deﬁnition E.1.5 Let F, F : C → D be two functors. A morphism ε : F → F • associates to each object X of C a morphism

εX : F(X) → F (X),

• such that, whenever f : X → X is a morphism in C, the following diagram is commutative ε F(X) X F (X)

F( f ) F( f )

ε F(X ) X F (X )

We say that a morphism ε : F → F is an isomorphism if, for all object X of C, εX : F(X) → F (X) is an isomorphism. The functors F and F are then said to be isomorphic and we write F F . Example E.1.6 Let A be a k-algebra. Whenever X is an A–module, the k–module Hom A(A, X) is endowed with a natural structure of A–module deﬁned by

(aϕ)(b) := ϕ(ba) for ϕ ∈ Hom A(A, X), a, b ∈ A . It deﬁnes a functor Hom A(A, ), which is isomorphic to the functor identity. Deﬁnition E.1.7 We say that a functor F : C → D is an equivalence of categories if there exists a functor F : D → C such that F.F IdD and F .F IdC. The proofs of the following two propositions are left to the reader. Proposition E.1.8 A functor F : C → D is an equivalence of categories if and only if it is fully faithful and essentially surjective. Proposition E.1.9 Assume that the functor F : C → D is an equivalence of categories. Then: (1) Whenever X, X ∈ C, then F induces a bijection

∼ HomC(X, X ) −→ HomD(F(X), F(X )) .

(2) The image under F of a monomorphism (resp. an epimorphism) is a monomorphism (resp. an epimorphism). 236 Appendix E: The Language of Categories and Functors

E.2 k–Linear and Abelian Categories

E.2.1 k–Linear Categories

Definition E.2.1 Let k be a field. A category A is said to be k-linear if the following conditions are satisfied. (1) • for each pair of objects X and X ,MorA(X, X ) is a k-vector space, • for each triple of objects X, X , X , the composition is k-bilinear, (2) there is a zero object 0, i.e., an object such that for all object X, both MorA(0, X) and MorA(X, 0) have a single element (“the 0 morphism”), (3) every pair of objects X, X ∈ A admits (a) a product, i.e., an object X X endowed with morphisms

: → : → prX X X X and prX X X X

such that the map

MorA(Y, X X ) −→ MorA(Y, X) × MorA(Y, X ) −→ ( . , . ) ϕ prX ϕ prX ϕ

is a bijection: Y

X X X prX

prX

(b) a coproduct, i.e., an object X X endowed with morphisms

i X : X → X X and i X : X → X X

such that the map

MorA(X, Y ) × MorA(X , Y ) −→ MorA(X X , Y )

ϕ −→ (ϕ.i X , ϕ.i X ) Appendix E: The Language of Categories and Functors 237

is a bijection: Y

X X X iX

Remark E.2.2 A k–linear category A with a single object X0 is deﬁned by the k–algebra A := EndA(X0). ( , , ) ( , , Exercise E.2.3 Given products X X prX prX (resp. coproducts X X i X i X )), ( , , ) ( , , (1) prove that there exist coproducts X X i X i X (resp. products X X prX ) prX ), and that ⎧ . + . = , ⎨⎪ i X prX i X prX IdY pr .i X = IdX , pr .i X = IdX , ⎩⎪ X X . = , . = , prX i X 0 prX i X 0

(2) there are natural isomorphisms

∼ X X −→ X X .

Deﬁnition E.2.4 For X and X two objects of a k–linear category A, we say that X is a summand of X and we write X | X if there exists an object X and an ∼ isomorphism X −→ X X .

E.2.2 Abelian k-Linear Categories

Throughout this section, objects and morphisms are those of a k–linear category A. Kernel, cokernel, image, coimage

• A kernel of a morphism f : X → Y , denoted ker( f ), is a pair (X , ι) where X ∈ A and ι : X → X, such that f ι = 0, and given another pair (Z, g : Z → X) such that f g = 0, there exists a unique h : Z → X such that the following diagram f ι commutes X X Y . The kernel is unique up to isomorphism. g h Z • A cokernel of a morphism f : X → Y , denoted coker( f ) is a pair (Y , σ) where Y ∈ A and σ : Y → Y , such that σ f = 0, and given another pair (Z, g : Y → Z) 238 Appendix E: The Language of Categories and Functors

such that g f = 0, there exists a unique h : Y → Z such that the following diagram f σ commutes X Y Y . The cokernel is unique up to isomorphism. g h Z • The image of a morphism f : X → Y , denoted im ( f ), is the kernel (if it exists) of the cokernel of f . • The coimage of a morphism f : X → Y , denoted coim( f ), is the cokernel (if it exists) of the kernel of f . The proof of next proposition is left to the reader.

Proposition E.2.5 Assume that the four objects (kernel, cokernel, image, coimage) exist for f : X → Y . There is a unique morphism f such that the following diagram (called then the canonical decomposition of a morphism) is commutative:

f X X

f ker( f ) coim( f ) im ( f ) coker( f )

Abelian k–linear categories.

Deﬁnition E.2.6 A k-linear category A is said to be abelian if the following conditions are satisﬁed. (1) Every morphism in A has a kernel and a cokernel. (2) Every monomorphism is the arrow of a kernel and every epimorphism is the arrow of a cokernel.

By slight abuse of notation, from now on we shall often forget the “arrow part” of kernels and cokernels by mentioning only the “object part”. The canonical decomposition (see above PropositionE.2.5) provides an isomorphism in an abelian category, as shown by the next proposition (whose proof is left to the reader).

Proposition E.2.7 (1) If A is abelian, then for each morphism f , the corresponding morphism fisan isomorphism. (2) In that case, the canonical decomposition of a morphism is unique up to a unique isomorphism.

Exact sequences and exact functors. Appendix E: The Language of Categories and Functors 239

Deﬁnition E.2.8 Let A be an abelian category. (1) A sequence of morphisms in A

α1 α2 αn−1 X1 −→ X2 −→ ··· −−→ Xn

is said to be exact if, for all i = 2,...,n − 1, ker αi = im αi−1 . (2) A short exact sequence is an exact sequence of the shape

ι π 0 −→ X −→ X −→ X −→ 0,

thus, we have ⎧ ⎨⎪ ι is a monomorphism, π is an epimorphism, ⎩⎪ im ι = ker π .

Deﬁnition E.2.9 A functor F : A → B is exact if whenever X → X → X is exact in A, then F(X ) → F(X) → F(X ) is exact in B.

Exercise E.2.10 Prove that a functor F : A → B is exact if and only if the image under F of any short exact sequence in A is a short exact sequence in B.

Deﬁnition E.2.11 Let A and B be abelian categories. We say A and B are equivalent as abelian categories if there exist two exact functors F : A → B and F : B → A such that F F .

Exercise E.2.12 Let vectk be the abelian k-linear category of ﬁnite dimensional k-vector spaces. Let Matk be the category such that

• Ob(Matk ) = N • For m, n ∈ N,Mor(m, n) is the space Matm,n(k) of m × n matrices with entries in k, and where the composition of morphisms is given by matrix multiplication.

Prove that vectk and Matk are equivalent as abelian categories.

Jordan–Hölder series and the Grothendieck group. A nonzero object S in an abelian category A is called irreducible if a monomorphism with goal S is either 0 or an isomorphism. If Y → X is a monomorphism, by abuse of notation we denote by X/Y its cokernel. • For X an object of A, we say that X has ﬁnite length if there exists a “ﬁltration”

0 = X0 → X1 →···→ Xn = X such that Xi / Xi−1 is irreducible for all i = 1,...,n. 240 Appendix E: The Language of Categories and Functors

Such a ﬁltration is called a Jordan–Hölder series of X. • For S an irreducible object, its multiplicity in such a Jordan–Hölder series of X is the number of values of i for which Xi / Xi−1 is isomorphic to S.

Theorem E.2.13 If X has ﬁnite length and if S is an irreducible object, the multi- plicities of S in any Jordan–Hölder series of X coincide.

Such a multiplicity as in the above theorem is denoted by m S(X).

Definition E.2.14 Let A be an abelian category all of whose objects have finite length. The Grothendieck group Gr(A) is the free abelian group with basis the set Irr(A) of isomorphism classes of irreducible objects in A. For X an object of A, its class [X] in Gr(A) is defined by the formula [X]:= m S(X)[S] . S∈Irr(A)

If 0 → X0 →···→ Xn → 0 is an exact sequence in A, then

n i (−1) [Xi ]=0 . i=0 Bibliography

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A Cl(G), 36 (A, B)-bimodule, 17 Cl(G × G)com), 181 Abelian k-linear category, 238 C(n), 72 Absolutely irreducible character, 54 Coevaluation map, 196 Absolutely irreducible module, 54 coevM , 196 Algebraic integer, 225 Cohen-Macaulay, 165 Amod, 196 Commutator, 34 A-module-B, 17 Complements of a Frobenius group, 148 Artin Theorem, 132 Completely reducible representation, 43 Aut(G), 22 Contragredient module, 197 Aut(X), 20 Contragredient representation, 32 Control of the fusion of nilpotent π- subgroups, 141 B Coxeter Group, 105 Big enough ﬁeld, 133 CS(G), 132 g, x Cl(Ω), x [G/H], x D Braided category, 201 Degree of a Set-representation, 26 Degree of a k-linear-representation, 29 deg(x), 155 C Degrees, 164 Canonical decomposition, 62 Derived Subgroup, 34 Category, 20 Dihedral group, 19 CF(G, k), 36 Disjoint KG-modules, 124 ( , G ( ), ) CF G Secπ u K , 141 Dk G, 180 CFπ(G, K ), 141 Double centralizer property, 60 ( , QG ) CF G K , 48 Double Cosets, 119 ChaK (G), 71 Doubly transitive, 59 Character of the trivial representation, 44 Drinfeld Double, 195 Character table, 75 Drinfeld element, 200 Characteristic degrees, 164 DS, 49, 52 Characterization of reﬂection groups, 175 a χM , 46 reg χG , 52 E σ χM , 46 G, 72 Class function, 36 eG , 44 © Springer Nature Singapore Pte Ltd. 2017 243 M. Broué, On Characters of Finite Groups, Mathematical Lectures from Peking University, https://doi.org/10.1007/978-981-10-6878-2 244 Index

Elementary Group, 133 Group determinant, 65 Elementary tensors, 3 G-set, 26 Equivalence of categories, 235 (G × G)com, 181 Equivalent representations, 21 G ×H Y , 115 eS, 60 G\X, 25 Essential image, 234 Gx , 22 Essentially surjective functor, 235 gY , 119 Evaluation map, 196 evM , 196 Exact sequence, 239 H Hilbert’s Basis Theorem, 231 H(K ), 91 ( , ) F HomkG M N , 29 ( , ) Faithful functor, 235 Homk X Y , 20 Faithful representation, 21 Homogeneous component, 155 ( , ) Field of quaternions, 91 Homk X Y , 20 Fixator, 22 FixG (X), 26 I Flip, 12 Indecomposable representation, 31 Fourier inversion, 61 ker(χ ), 45 Frattini argument, 220 M IndG Y , 112 Free graded, 161 H Induction, 112 Free Graded Modules, 161 Induction left adjoint to Restriction, 114 Frobenius group, 148 Induction right adjoint to Restriction, 118 Frobenius Reciprocity, 123 Inﬂation, 45 Frobenius–Schur indicator, 55 InﬂG , 45 Full functor, 235 G/H Inn(G), 23 Full subcategory, 234 Inn(g), 22, 119 Fully faithful functor, 235 Integral, 225 Integral closure, 226 Irreducible representation, 31 G IrrK (G), 45 Galois extension, 221 Isomorphism of functors, 235 ( , , ) G de e r , 104 Isomorphism of representations, 21 ( ) Gr DK G , 205 Isotypic component, 63 Generalized characters, 71 G-graded character, 204 g H, 119 J GL(X), 20 Jordan–Hölder series, 240 gπ, 218 Gπ , 141 gπ , 218 K Graded A-module morphism, 158 k, ix Graded KG-module, 167 k, L x Graded algebra, 157 K (χT ), 98 Graded character, 167 Kernel of a character, 45 Graded dimension, 156 Kernel of a Frobenius group, 148 Graded multiplicity, 168 KG , 44 Graded submodule, 158 kG-module, 37 Graded Vector Spaces, 155 k-linear category, 236 ( ) grdimk V , 156 k , 31 Grothendieck group, 240 Krdim(A), 164 Group algebra, 35 Kronecker product, 8 Index 245

Krull dimension, 164 π-component, 218 G K -conjugacy class, 73 πM , 42 π, 145, 217 π-trivial intersection subgroup of G, 146 G L ProH , 142 L\G/H, 119

Q M Q(χM ), 47 QG Möbius function, 27 K , 47 Mackey Formula for Functions, 124 Quasitriangular Hopf algebra, 198 Mackey Formula for Modules, 119 Mark, 26 m K (T ), 98 R Morphism between representations, 21 Regular representation, 32 Morphism in a category, 233 Rep(G, C), 21 Mor(X, Y ), 20 Rep(G, vectk ), 29 mS,M , 50 Representation, 20 G Multiplicity of S in M, 50 ResH , 111 Multiplicity space, 63 Restriction, 111 MultS(M), 63 Ribbon category, 202 Ribbon Hopf algebra, 202

N Nakayama’s Lemma, 160 S Nilpotent Group, 218 S, 208 NK , 46 α1, α2, 48 , NL/K (x, 224 α1 α2 G , 48 Noether parameter algebra, 165 Schur index, 98 G ( ) Noether system of parameters, 165 Secπ u , 141 Noetherian module, 229 Semisimple module, 42 Noetherian ring, 230 Set, 20 Norm (for a field extension), 224 setk , 20 Normal π-complement, 145 S(G), 26 n p, 134 Shephard–Todd Groups, 104 nπ, 217 Shift, 157 n p , 134 Short exact sequence, 239 ν2(χ), 55 Simple representation, 31 W NY,Z , 213 S-isotypic component, 63 S, 32 Splitting field for G, 56 O Splitting field for S, 56 Objects, 233 Subrepresentation (Set), 24 , 208 Summand, 237 ωS, 67 S(X), 20 Oπ(G), 145 sX,Y , 206 Outer automorphisms, 23 Symmetric algebra, 35 Out(G), 23 Symmetrizing form, 35

P T p-component, 218 Tensor Algebra, 13 p-component, 218 Tensor product, 1 p-Elementary Group, 133 θA, 131 246 Index

(G), 65 Verlinde Formula, 213 T (M), 14 |Ω|, x n Top, 20 M⊗ , 13 Transcendence degree, 164 Transitive representation, 24 Trivial representation, 29 W Tr L/K (x), 224 Weyl Group, 105

U Z (Z/ Z)× Universal R-matrix, 198 eG K , 73 Universal Property of Induction, 113 ζm , 104 ZG , 136 ZkG, 36 V Z((q)), 156 Vectk , 20 Z S, 67, 94