Class group relations from Burnside ring idempotents

Robert Boltje Institut für Mathematik Universität Augsburg 86135 Augsburg Germany email: boltje@math.uni-augsburg.de

November 6, 1996

(revised: February 24, 1996)

Abstract

We show that for each finite cohomological Mackey functor on a finite group G there exist explicit relations in the category of finite abelian groups between the evaluations of the Mackey functor at all the subgroups, one for each conjugacy class of non-hypo-elementary subgroups of G. Furthermore we show that the class groups of the intermediate fields of a Galois extension of number fields form such a Mackey functor on the Galois group, thereby obtaining class group relations by using the presence of the structure of a cohomological Mackey functor.

Introduction

The aim of this paper is to introduce explicit relations between the class groups cl(E) of intermediate fields E of a finite Galois extension L/K of number fields with Galois group G.

More precisely, we obtain for each subgroup H of G which is not p-hypo-elementary (i.e. has no normal Sylow p-subgroup with cyclic factor group) an explicit relation between the p-parts of all class groups of intermediate fields E with Gal(L/E) ≤ H. This relation comes from a primitive idempotent of the Burnside algebra ℚ ⊗ Ω(G) which is associated to H. In particular, the relation does not depend on the specific realization L/K of G as a Galois group, but only on G itself.

The methods we use are more group theoretic than number theoretic. In fact, the only thing we use about the class groups cl(E), K ⊆ E ⊆ L, is that they carry a rigid structure which has not been used systematically in the past, namely the structure of a cohomological Mackey functor. A Mackey functor is a structure which axiomatices the features of the character rings R(H), H ≤ G, namely the existence of conjugation, restriction and induction maps between them satisfying a list of natural properties.

In Section 1 we recall the definition of a Mackey functor and derive a general result about relations between the evaluations of a finite cohomological Mackey functor on the subgroups H of G, cf. Theorem 1.8. Crucial is the use of a result due to Yoshida (Theorem 1.3) which allows to consider cohomological Mackey functors as functors on the category of permutation modules and to use relations between permutation modules.

In Section 2 we show in an elementary way that the class groups form a cohomological Mackey functor and apply the results of Section 1 to this case,

Section 3 starts with a comparison of the results of Section 2 with classical results of Dirichlet in [8] and Brauer in [4]. We also give examples for the explicit relations of Section 2 in the case of small groups. The reader should have no problems to apply the same method to any group G she/he is interested in. Moreover, it is pointed out that if G is not p-hypo-elementary for any prime p, and if the Möbius coefficient μ(1, G) with respect to the partially ordered set (poset) of subgroups of G is non-zero, then the class group cl(L) is uniquely determined by the class groups of smaller fields in L/K. Interestingly, this is a topological property, since μ(1, G) is also the reduced Euler characteristic of the geometric realization of the poset of all non-trivial proper subgroups of G.

I am grateful to P. Webb for bringing the article [10] of Roggenkamp and Scott to my attention. This article uses similar ideas to obtain relations between class groups from relations between permutation modules, but without determining these relations explicitly and systematically. I would also like to mention P. Webb’s article [16] which applies similar methods to group cohomology. Finally I’m indebted to the referee for bringing the results listed in Remark 3.5 to my attention.

1 Cohomological Mackey functors

Throughout this section let G denote a finite group and k a commutative ring. For the reader’s convenience we recall the definition of a Mackey functor, see for instance [13].

1.1. Definition A k-Mackey functor M = (M, c, res, ind) on G is a quadruple consisting of

• a family of k-modules M(H), where H ≤ G,

• a family of k-module homomorphisms c_{g,H}: M(H) → M(^{g}H), the conjugation maps, where g ∈ G, H ≤ G, and ^{g}H = gHg^{−1},

• a family of k-module homomorphisms res_{K}^{H}: M(H) → M(K), the restriction maps, where K ≤ H ≤ G, and • a family of k-module homomorphisms ind_{K}^{H}: M(K) → M(H), the induc-

tion maps, where K ≤ H ≤ G,

such that the following axioms are satisfied:

(M1) (Triviality) c_{h,H} = res_{H}^{H} = ind_{H}^{H} = id_{M(H)} for all H ≤ G and h ∈ H.

(M2) (Transitivity) c_{g′g,H} = c_{g′,^{g}H} ∘ c_{g,H}, res_{L}^{K} ∘ res_{K}^{H}, and ind_{K}^{H} ∘ ind_{L}^{K} for all L ≤ K ≤ H ≤ G and g, g′ ∈ G.

(M3) (G-equivariance) c_{g,K} ∘res_{K}^{H} = res _{^{g}K}^{^{g}H} ∘c_{g,H} and c_{g,H} ∘ind_{K}^{H} = ind _{^{g}K}^{^{g}H} ∘c_{g,K} for all K ≤ H ≤ G and g ∈ G.

(M4) (Mackey formula) For all H ≤ G and U, K ≤ H one has

res_{U}^{H} ∘ ind_{K}^{H} = ∑_{h∈U\H/K} ind_{U∩^{h}K}^{U} ∘ res_{U∩^{h}K}^{^{h}K} ∘ c_{h,K} ,

where h ∈ H runs through a set of representation for the double cosets U\H/K

A morphism f: M → N between k-Mackey functors on G is a family of kmodule homomorphisms f_{H}: M(H) → N(H), H ≤ G, which commute in the obvious way with conjugation, restriction, and induction maps.

A k-Mackey functor M on G is called cohomological, if the axiom

(M5) ind_{K}^{H} ∘ res_{K}^{H} = [H : K] · id_{M(H)}, for all K ≤ H ≤ G,

holds.

The k-Mackey functors on G and the morphisms between them form a category k-Mack(G) which has the structure of a k-category, i.e. it is abelian and the morphism sets are k-modules such that the composition of morphisms is k-bilinear. In fact, we can define direct sums, kernels, cokernels, etc. in the usual way for each subgroup H ≤ G separately. The cohomological k-Mackey functors on G form a full k-subcategory k-Mack^{coh}(G) of k-Mack(G). In the sequel we will often write ^{g}x instead of c_{g,H}(x) for x ∈ M(H), H ≤ G, and M ∈ k-Mack(G). Note that by the axioms (M1) and (M2) the conjugation maps are isomorphisms and provide M(H), H ≤ G, with a module structure for the group ring k[N_{G}(H)/H]. If k = ℤ we speak of a Mackey functor instead of a ℤ-Mackey functor on G.

1.2 Example The standard example of a Mackey functor is the Burnside ring functor Ω(H), H ≤ G, with the following structure maps:

c_{g,H} : Ω(H) → Ω(^{g}H) , [H/U] ↦ [^{g}H/^{g}U] ,
res_{K}^{H} : Ω(H) → Ω(K) , [H/U] ↦ ∑_{h∈K\H/U}[K/K ∩ ^{h}U] ,
ind_{K}^{H} : Ω(K) → Ω(H) , [K/V] ↦ [H/V] ,

for K ≤ H ≤ G, g ∈ G, U ≤ H, and V ≤ K. The group Ω(G) is a commutative ring with unity [G/G] under the multiplication [G/H] · [G/K] = ∑_{g∈H\G/K}[G/H ∩ ^{g}K] for H, K ≤ G. We refer to [7, § 80] for the notation and basic results concerning the Burnside ring.

Another example is the character ring functor R(H), H ≤ G, with the usual structure maps.

For a kG-module V, the cohomology groups H^{n}(H, V), H ≤ G, with the usual conjugation and restriction maps together with the corestrictions, form a cohomological k-Mackey functor on G.

We denote by kG-per the category of all finitely generated kG-permutation modules, i.e. kG-modules which have a finite G-stable k-basis. For a finite G-set S we denote the associated kG-module with k-basis S by k[S].

By k-Funct(kG-per, k-Mod) we denote the category of k-functors from kG-per to k-Mod, i.e. functors which are k-linear on morphisms. The morphisms in this category are as usually the natural transformations between such functors. The following theorem is due to Yoshida, cf. [18, Thm. 4.3] or [12, Sect. 1].

1.3 Theorem There is an equivalence of k-categories

Ψ: k-Mack^{coh}(G) ↦ k-Funct(kG-per, k-Mod)

with the property that

(Ψ(M))(k[G/H]) ≅ M(H)  (1.3.a)

in k-Mod for each M ∈ k-Mack^{coh}(G) and each H ≤ G.

In what follows we don’t need the fact that Ψ is an equivalence, but only that there exists a functor with the property (1.3.a). Since a functor maps isomorphic objects to isomorphic objects, we have the following immediate corollary.

1.4 Corollary Let H_{1}, …, H_{m}, U_{1}, …, U_{n} be subgroups of G such that

⨁_{i=1}^{m} k[G/H_{i}] ≅ ⨁_{i=1}^{n} k[G/U_{i}]  (1.4.a)

in kG-per. Then

⨁_{i=1}^{m} M(H_{i}) ≅ ⨁_{i=1}^{n} M(U_{i})

in k-Mod for each M ∈ k-Mack^{coh}(G).

1.5 The hypothesis (1.4.a) in the above corollary suggests to study the functor

k[−]: G-set → kG-per , S ↦ k[S]

on the category of finite G-sets. If we denote by P_{k}(G) the Grothendieck group of kG-per with respect to direct sums, this functor induces a surjective ring homomorphism

i_{G}: Ω(G) ↦ P_{k}(G) , [G/H] ↦ [k[G/H]] ,  (1.5.a)

where the ring structure on P_{k}(G) comes from taking the tensor product M ⊗_{k}N for M, N ∈ kG-per with the diagonal G-action.

Note that for k = ℤ_{p}, the p-adic completion of ℤ, we know by the Krull-Schmidt-Azumaya theorem, cf. [6, Thm. 12], that M ≅ N in kG-per, if and only if [M] = [N] in P_{k}(G) . In fact, this becomes apparent, if one passes further to the category ℤ_{p}G-lat of ℤ_{p}G-lattices which contains ℤ_{p}G-per as a full subcategory, and to the Grothendieck ring A_{ℤ_{p}}(G) of ℤ_{p}G-lat with respect to direct sums, for which we have a canonical injective ring homomorphism P_{ℤ_{p}}(G) → A_{ℤ_{p}}(G).

Therefore, for k = ℤ_{p}, it suffices to determine the kernel of i_{G} in (1.5.a) in order to obtain all relations of the form (1.4.a). Actually it suffices to determine the kernel of

i_{G}: ℚ ⊗ Ω(G) → ℚ ⊗ P_{ℤ_{p}}(G)  (1.5.b)

which is easier, since the ring structure of ℚ ⊗ Ω(G) is more convenient than the one of Ω(G).

1.6 Let S_{G} denote the poset of subgroups of G, ordered by inclusion and considered as a G-set by the conjugation action. Let R_{G} ⊆ S_{G} be a set of representatives for the G-orbits, i.e. conjugacy classes of subgroups of G. Then for each H ∈ R_{G} there is a ring homomorphism ϕ_{H}^{(G)} : Ω(G) → ℤ which maps the class [S] of a finite G-set S to the number |S^{H}| of H-fixed points. The collection of these maps ϕ_{H}^{(G)}, H ∈ R_{G}, induces a ℚ-algebra isomorphism

ρ_{G}: ℚ ⊗ Ω(G) → ∏_{H∈R_{G}} ℚ ,

cf. [7, 80.13], and we have a set of mutually orthogonal primitive idempotents ε_{H}^{(G)} ∈ ℚ ⊗ Ω(G), H ∈ R_{G}, whose sum is 1, and which are defined by ϕ_{K}^{(G)}(ε_{H}^{(G)}) = δ_{K,H} for K, H ∈ R_{G}. There is an explicit formula, due to Gluck (cf. [9]),

ε_{H}^{(G)} = \frac{1}{|N_{G}(H)|} ∑_{K≤H} |K|μ(K, H)[G/K] ,  (1.6.a)

where μ(K, H) denotes the Möbius function on S_{G}, i.e.

μ(K, H) = ∑_{K=H_{0}<···<H_{n}=H}(−1)^{n} ,  (1.6.b)

cf. [11], where the sum runs over all strictly increasing chains in ρ_{G} connecting K to H.

A finite group H is called p-hypo-elementary for a prime p, if H/O_{p}(H) is cyclic, where O_{p}(H) denotes the biggest normal p-subgroup of H.

1.7 Proposition Let p be a prime. Then the kernel of the map

i_{G}: ℚ ⊗ Ω(G) → ℚ ⊗ P_{ℤ_{p}}(G) , [G/H] ↦ [ℤ_{p}[G/H]] ,

is the ℚ-subspace with basis ε_{H}^{(G)}, where H runs through all H ∈ R_{G} which are not p-hypo-elementary.

Proof There are full embeddings of categories

ℤ_{p}G-per ⊆ ℤ_{p}G-triv ⊆ ℤ_{p}G-lat ,

where the category ℤ_{p}G-triv consists of those ℤ_{p}G-lattices which are permutation modules when restricted to a Sylow p-subgroup of G. These modules are called p-permutation modules and also trivial-source modules, cf. [7, § 81B] and [5]. If P P_{ℤ_{p}}(G) denotes the Grothendieck ring of ℤ_{p}G-triv with respect to direct sums we obtain injective natural subring inclusions

P_{ℤ_{p}}(G) ⊆ P P_{ℤ_{p}}(G) ⊆ A_{ℤ_{p}}(G) .

Since i_{G}: ℚ ⊗ Ω(G) → ℚ ⊗ P_{ℤ_{p}}(G) is a ℚ-algebra homomorphism, its kernel is an ideal, and therefore spanned over ℚ by those primitive idempotents ε_{H}^{(G)}, H ∈ R_{G}, with i_{G}(ε_{H}^{(G)}) = 0. Now Lemma 81.28 in [7] states that ε_{H}^{(G)}, H ∈ R_{G}, is in the kernel of the map

ℂ ⊗ Ω(G) →^{i_{G}} ℂ ⊗ P_{ℤ_{p}}(G) ⊆ ℂ ⊗ P P_{ℤ_{p}}(G) ,

if and only if H is not p-hypo-elementary. This completes the proof.

Let K_{0}(ab) denote the Grothendieck group of the category ab of finite abelian groups with respect to direct sums. For A ∈ ab let [A] denote the associated element in K_{0}(ab). Note that two finite abelian groups A and B are isomorphic, if and only if [A] = [B] in K_{0}(ab), since each finite abelian group has a unique decomposition (up to isomorphism and ordering) into a direct sum of indecomposable groups (i.e. cyclic groups of prime power order). For A ∈ ab and a prime p we write A_{p} for the Sylow p-subgroup of A. For A ∈ ab and n ∈ N we write A^{n} for the n-fold direct sum A ⊕ … ⊕ A.

We call a k-Mackey functor on G finite, if M(H) is finite for all H ≤ G. A finite group H is called hypo-elementary, if it is p-hypo-elementary for

some prime p.

1.8 Theorem Let M be a finite cohomological Mackey functor on G.

(i) Let p be a prime and let H ≤ G be not p-hypo-elementary. Then the relation

∑_{K≤H} |K|μ(K, H) [M(K)_{p}] = 0  (1.8.a)

holds in K_{0}(ab). More precisely,

⨁_{{H_{0}<···<Hn=H \atop n even}} M(H_{0})_{p}^{|H_{0}|} ≅ ⨁_{{H_{0}<···<Hn=H \atop n odd}} M(H_{0})_{p}^{|H_{0}|}  (1.8.b)

in ab, where the sums run over all strictly increasing chains in S_{G} which end in H and which have even (resp. odd) length.

(ii) Let H ≤ G be not hypo-elementary. Then the relation

∑_{K≤H} |K|μ(K, H)[M(K)] = 0  (1.8.c)

holds in K_{0}(ab). More precisely we have

⨁_{{H_{0}<···<Hn=H \atop n even}} M(H_{0})^{|H_{0}|} ≅ ⨁_{{H_{0}<···<Hn=H \atop n odd}} M(H_{0})^{|H_{0}|}  (1.8.d)

in ab.

Proof (i) Note that M_{p}(H): = M(H)_{p} for H ≤ G defines a cohomological ℤ_{p}-Mackey functor M_{p} on G. Hence, the isomorphism (1.8.b) in ab (or equivalently in ℤ_{p}-Mod) follows from Corollary 1.4 and Proposition 1.7 by taking into account that two objects in ℤ_{p}G-per are isomorphic, if and only if their classes in P_{ℤ_{p}}(G) coincide, and by using the explicit formulae (1.6.a) and (1.6.b). Equation (1.8.a) then follows immediately.

(ii) This follows from part (a), since A ≅ ⨁_{p} A_{p} for any finite abelian group A, where the sum runs over all primes dividing the order of A.

2 Class group relations

Throughout this section let L/K denote a Galois extension of number fields with Galois group G. For H ≤ G we denote the associated fixed field by L^{H}.

For any number field F, we denote by O_{F} the ring of integers of F, by I_{F} the group of fractional ideals of F and by P_{F} the subgroup of principal fractional ideals of F. Then the class group cl(F) of F is the factor group I_{F} /P_{F}.

2.1 For U ≤ H ≤ G and g ∈ G we have group homomorphisms

c_{g,H}: I_{L^{H}} → I_{gL^{H}} , 𝔞 ↦ g(𝔞) ,
res_{U}^{H}: I_{L^{H}} → I_{L^{U}} , 𝔞 ↦ 𝔞O_{L^{U}} ,  (2.1.a)
ind_{U}^{H}: I_{L^{U}} → I_{L^{H}} , 𝔟 ↦ N_{L^{U} /L^{H}} (𝔟) ,

for 𝔞 ∈ I_{L^{H}} and 𝔟 ∈ I_{L^{U}}, where N_{L^{U} /L^{H}} (𝔟) ∈ I_{L^{H}} is the O_{L^{H}} -module generated by the elements N_{L^{U} /L^{H}} (b) ∈ L^{H}, b ∈ 𝔟. For a non-trivial prime ideal 𝔓 of O_{L^{U}} one has N_{L^{U} /L^{H}} (𝔓) = 𝔭^{f_{𝔓/𝔭}} with 𝔭 = O_{L^{H}} ∩ 𝔓 and f_{𝔓/𝔭} the residue degree of 𝔓/𝔭. Note that res_{U}^{H} is injective. It is clear that the subgroups P_{L^{H}}, H ≤ G, of I_{L^{H}} are stable under these three types of maps (N_{L^{U} /L^{H}} (xO_{L^{U}}) = N_{L^{U} /L^{H}} (x)O_{L^{H}} for x ∈ L^{U}), and therefore, we obtain induced maps

c_{g,H}: cl(L^{H}) → cl(L^{^{g}H)}) ,
res_{U}^{H}: cl(L^{H}) → cl(L^{U}) ,  (2.1.b)
ind_{U}^{H}: cl(L^{U}) → cl(L^{H}) ,

for U ≤ H ≤ G and g ∈ G.

2.2 Proposition The groups I_{L^{H}}, H ≤ G, together with the maps in (2.1.a) form a cohomological Mackey functor on G. In particular, the class groups cl(L^{H}), H ≤ G, together with the maps in (2.1.b) form a finite cohomological Mackey functor on G.

Proof The axioms (M1), (M2), (M3), and (M5) are clearly satisfied for I_{L^{H}}, H ≤ G. We only show that the Mackey axiom (M4), namely

res_{U}^{H}(ind_{V}^{H}(𝔓)) = ∑_{h∈U\H/V} ind_{U∩^{h}V}^{U} (res_{U∩^{h}V}^{^{h}V} (h(𝔓))) ,  (2.2.a)

holds for U, V ≤ H ≤ G and all prime ideal 𝔓 ≠ 0 of O_{L^{V}}.

Step 1. We show (2.2.a) in the case V = U = 1:

res_{1}^{H}(ind_{1}^{H}(𝔓)) = res_{1}^{H}(𝔭^{f}) = (𝔓_{1} ⋯ 𝔓_{r})^{ef} = ∏_{h∈H} h(𝔓) ,

where 𝔭 = O_{L^{H}} ∩ 𝔓, f and e are the residue degree and the ramification index of 𝔓/𝔭 respectively, and 𝔭O_{L} = (𝔓_{1} ⋯ 𝔓_{r})^{e} with pairwise distinct prime ideals 𝔓_{1}, ⋯, 𝔓_{r} of O_{L}.

Step 2. We show (2.2.a) in the case U = 1. Using Step 1, (M3), and (M5) we have

res_{1}^{H}(ind_{V}^{H}(𝔓))^{|V |} = (res_{1}^{H} ∘ ind_{V}^{H} ∘ ind_{1}^{V} ∘ res_{1}^{V})(𝔓)

= (res_{1}^{H} ∘ ind_{1}^{H})(res_{1}^{V} (𝔓))

= ∏_{h∈H} ^{h}(res_{1}^{V} (𝔓))

= ∏_{h∈H/V} (res_{1}^{^{h}V} (h(𝔓)))^{|V |} .

Since I_{L} is torsion free, (2.2.a) follows in this case.

Step 3. We show (2.2.a) in general. Since res_{1}^{U} is injective, it suffices to show that

(res_{1}^{H} ∘ ind_{V}^{H})(𝔓) = ∑_{h∈U\H/V} (res_{1}^{U} ∘ ind_{U_{∩} ^{h}V}^{U})(res_{U∩^{h}V}^{^{h}V} (h(𝔓))) . (2.2.b)

By Step 2 we have,

(res_{1}^{H} ∘ ind_{V}^{H})(𝔓) = ∑_{h∈H/V} res_{1}^{^{h}V} (h(𝔓)) ,

and for arbitrary h ∈ H we have by Step 2,

(res_{1}^{U} ∘ ind_{U∩^{h}V}^{U})(res_{U∩^{h}V}^{^{h}V} (h(𝔓))) =

= ∑_{u∈U/U∩^{h}V} ^{u}[res_{1}^{U∩^{h}V} (res_{U∩^{h}V}^{^{h}V} (h(𝔓)))]

= ∑_{u∈U/U∩^{h}V} res_{1}^{^{uh}V} (uh(𝔓)) .

Now Equation (2.2.b) follows, since if h runs through a set of representatives for U\H/V and for each such h, u_{h} runs through a set of representatives for U/U ∩ ^{h}V, then u_{h} · h runs through a set of representatives for H/V. This completes the proof.

2.3 Theorem (i) If H ≤ G is not p-hypo-elementary for a prime p, then

⨁_{{H_{0}<···<Hn=H \atop n even}} cl(L^{H_{0}})_{p}^{|H_{0}|} ≅ ⨁_{{H_{0}<···<Hn=H \atop n odd}} cl(L^{H_{0}})_{p}^{|H_{0}|} .

(ii) If H ≤ G is not hypo-elementary, then

⨁_{{H_{0}<···<Hn=H \atop n even}} cl(L^{H_{0}})^{|H_{0}|} ≅ ⨁_{{H_{0}<···<Hn=H \atop n odd}} cl(L^{H_{0}})^{|H_{0}|} .

Proof This follows immediately from Theorem 1.8 and Proposition 2.2.

2.4 Corollary (i) If p is a prime and H ≤ G is not p-hypo-elementary, then

∑_{U≤H} |U|μ(U, H)[cl(L^{U})_{p}] = 0

in K_{0}(ab). In particular, cl(L^{H})_{p} is uniquely determined by the groups cl(L^{U})_{p}, U < H.

(ii) If H ≤ G is not hypo-elementary, then

∑_{U≤H} |U|μ(U, H)[cl(L^{U})] = 0

in K_{0}(ab). In particular, cl(L^{H}) is uniquely determined by the groups cl(L^{U}), U < H.

Proof This follows from Theorem 2.3 by taking the associated elements in the Grothendieck group K_{0}(ab) and using Equation (1.6.b).

3 Remarks and Examples

We first compare Theorem 2.3 with the following two classical results. For a number field K we denote by h_{K} = |cl(K)| its class number.

3.1 Theorem (Dirichlet 1842, [8]) Let 1 < d ∈ ℕ be square-free, and let L = ℚ(\sqrt{d}, \sqrt{−d}). Then

2h_{L} = Q · h_{ℚ(\sqrt{d})} · h_{ℚ(\sqrt{−d})} ,

where Q ∈ {1, 2} depends on d.

3.2 Theorem (Brauer 1951, [4]) Let G be a finite group and let

∑_{H≤G} a_{H} · ind_{H}^{G}(1_{H}) = 0

be a relation in the character ring R(G) of G, with integers a_{H}, H ≤ G. Then the product

∏_{H≤G} h_{L^{H}}^{a_{H}}

assumes only finitely many values, as L runs over all Galois extensions of K = ℚ with Galois group G.

3.3 Remark (i) Theorem 3.2 can be considered as a generalization of Theorem 3.1, since there is a relation

ind_{1}^{G}(1) − ind_{H_{1}}^{G}(1) − ind_{H_{2}}^{G}(1) − ind_{H_{3}}^{G}(1) + 2 · ind_{G}^{G}(1) = 0  (3.3.a)

for the group G = ℤ/2ℤ × ℤ/2ℤ, where H_{1}, H_{2}, H_{3} denote the three subgroups of order 2. Both theorems make use of number theoretic results. Brauer for instance uses the theory of Artin L-functions and the analytic class number formula. In contrast to this we only used very little knowledge from number theory to verify that class groups carry the structure of a cohomological Mackey functor. The rest followed from the general theory of cohomological Mackey functors. Note that the character relations used in Theorem 3.2 come from the idempotents ε_{H}^{(G)} ∈ ℚ⊗Ω(G) with non-cyclic H ≤ G, cf. [17, Lemma 3.7]. More precisely, the kernel of the natural map i_{G}: ℚ ⊗ Ω(G) → ℚ ⊗ R(G) , [G/H] ↦ ind_{H}^{G}(1), has as a ℚ-basis the set of idempotents ε_{H}^{(G)}, where H runs through a set of representatives for the conjugacy classes of non-cyclic subgroups of G, cf. Proposition 1.7. Therefore, Brauer obtains more relations than we do in Theorem 2.3. But his relations are only exact up to a factor which may assume finitely many values. For the relations coming from idempotents ε_{H}^{(G)} with H non-cyclic but hypo-elementary, this finiteness result can’t be obtained by our general approach.

For the group G = ℤ/2ℤ × ℤ/2ℤ in Dirichlet’s Theorem 3.1 we obtain only one relation, coming from the idempotent

ε_{G}^{(G)} = \frac{1}{2}(2[G/G] − [G/H_{1}] − [G/H_{2}] − [G/H_{3}] + [G/1]) ,  (3.3.b)

since G is not p-hypo-elementary for odd primes p. In particular we obtain

cl(ℚ(\sqrt{d}))_{p} ⊕ cl(ℚ(\sqrt{−d}))_{p} ≅ cl(ℚ(\sqrt{d}, \sqrt{−d}))_{p}

for odd primes (since cl(ℚ) and cl(ℚ(i)) are trivial), showing that the factor Q in Theorem 3.1 has to be a power of 2 or 1/2.

But note that for all primes which do not divide the order of the Galois group G we obtain exact relation, even for the class groups not only for the class numbers. Moreover, we do not need the restriction that the base field is ℚ, as in Theorem 3.1 and 3.2.

(ii) There have been improvements of Brauer’s Theorem 3.2. Walter proved in [15, Thm. 2.1] that each relation

∑_{H≤G} a_{H} · ind_{H}^{G}(1) = ∑_{H≤G} b_{H} · ind_{H}^{G}(1)

in the character ring R(G) with a_{H}, b_{H} ∈ ℕ_{0}, induces a relation

⨁_{H≤G} cl(L^{H})_{π}^{a_{H}} ≅ ⨁_{H≤G} cl(L^{H})_{π}^{b_{H}}

in the category ab, where π is the set of all primes not dividing |G|. This is covered by our results, since each relation in the character ring comes from idempotents ε_{H}^{(G)} ∈ ℚ ⊗ Ω(G), H ≤ G non-cyclic, and since each non-cyclic H ≤ G is non-p-hypo-elementary if p does not divide |G|.

(iii) Roggenkamp and Scott have an approach in [10] which is very similar to the one we outlined here. What they call ‘Hecke actions’ is basically a functor in k-Funct(kG-per, k-Mod). They showed that Picard groups (in particular class groups) give rise to a Hecke action. In particular, they have a result analogous to Corollary 1.4. But they didn’t determine all possible relations in ℤ_{p}G-per explicitly.

If L/K is a Galois extension of number fields with Galois group G, for practical uses one would rather like to determine the class group of L from the class groups of all smaller intermediate fields, than the class group of K by all the other ones, as Corollary 2.4 does.

3.4 Proposition If L/K is a Galois extension of number fields with Galois group G and p a prime such that there is a non p-hypo-elementary subgroup H ≤ G with μ(1, H) ≠ 0, then cl(L)_{p} is uniquely determined by the p-parts of the class groups of all smaller intermediate fields, and this in an explicit way which is independent of the specific Galois extension L/K.

Proof This follows immediately from Corollary 2.4 (a) and the fact that the isomorphisms type of an abelian group is uniquely determined by its class in K_{0}(ab).

3.5 Remark Unfortunately not too much is known about the number μ(1, G) for a finite group G. The following results were brought to my attention by the referee.

S. Bouc showed in [1, Cor. to Prop. 6] that μ(1, G) = m(G)|G′| for an integer m(G) where G′ is the derived subgrouup of G.

In [2, Prop. 3] it is shown that, if G has a normal subgroup N with 1 ≠ N ≠ G which has no complement in G (i.e. there is no subgroup H ≤ G with HN = G and H ∩ N = 1), then m(G)=0. Moreover, in [2, Prop. 5] there are stated several equivalent conditions for a finite solvable group G to satisfy m(G) = 0. One of them is the existence of a normal subgroup N as above. The same Proposition also gives a nice formula for μ(1, G) in the solvable case.

If G_{1} and G_{2} are two finite groups without common factor group except 1, then μ(1, G_{1} × G_{2}) = μ(1, G_{1})μ(1, G_{2}), cf. [1, Prop. 7(ii)] or [3, Lemma 22].

The methods in [1] also allow to compute m(G) for non-solvable groups, as for example for the series G = SL_{2}(p), p an odd prime:

m(G) = \begin{cases}2, & if p^{2} ≡ 1 mod 5 and p^{2} ≡ 1 mod 16, \\ 1, & if p^{2} ≡ 1 mod 5 and p^{2} ≡ 9 mod 16, \\ 0, & if p^{2} ≡ 4 mod 5 and p^{2} ≡ 1 mod 16, \\ −1, & if p^{2} ≡ 4 mod 5 and p^{2} ≡ 9 mod 16. \end{cases}

If G is a p-group, then μ(1, G) ≠ 0, if and only if G is elementary abelian (in the other case, the Frattini subgroup Φ(G), i.e. the intersection of all maximal subgroups of G, is a non-trivial proper normal subgroup of G without complement in G), and in the elementary abelian case one has m(G) = (−1)^{r}p^{\binom{r}{2}}, if |G| = p^{r}, cf. [11, 5, Ex. 2].

A check with the computer program package GAP for computational group theory yields that among the 168 non-hypo-elementary groups G of order between 1 and 95 there are exactly 50 with μ(1, G) ≠ 0.

3.6 Example Let L/K be a Galois extension of number fields with Galois group G.

(i) Let G = ℤ/2ℤ × ℤ/2ℤ. Then G is not p-hypo-elementary for all odd primes p and we obtain from the idempotent ε_{G}^{(G)} in (3.3.b) the relation

cl(K)_{2′} ⊕ cl(K)_{2′} ⊕ cl(L)_{2′} ≅ cl(K_{1})_{2′} ⊕ cl(K_{2})_{2′} ⊕ cl(K_{3})_{2′} ,

if K_{1}, K_{2}, K_{3} are the proper intermediate fields of L/K.

(ii) Let G = S_{3}, the symmetric group on 3 letters. Then G is not p-hypo-elementary for all primes p except p = 3, and we obtain from the idempotent

ε_{G}^{(G)} = \frac{1}{2}([G/1] − 2[G/C_{2}] − [G/C_{3}] + 2[G/G]) ∈ ℚ ⊗ Ω(G) ,

with C_{2} and C_{3} the Sylow subgroups for the primes 2 and 3 respectively, the relation

cl(K)_{3′} ⊕ cl(K)_{3′} ⊕ cl(L)_{3′} ≅ cl(E)_{3′} ⊕ cl(E)_{3′} ⊕ cl(F)_{3′} ,

with [E: K] = 3 and [F: K] = 2.

(iii) The smallest non-hypo-elementary group is the dihedral group D_{12} of order 12. Since μ(1, D_{12}) = −6 ≠ 0, Proposition 3.4 applies to G = D_{12} and all prime p. Also G = S_{4}, the symmetric group on 4 letters, is not hypoelementary with μ(1, S_{4}) = −12. Unfortunately there are, for fixed p, minimal non-p-hypo-elementary groups G with μ(1, G) = 0, as for example the group G =< a, b|a^{6} = 1, b^{4} = 1, bab^{−1} = a^{5}, b^{2} = a^{3} > of order 12 for the prime 2. For this G we obtain by our approach no information about the 2-part of cl(L) from the 2-parts of the class groups of all proper subfields containing K.

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