Journal of Pure and Applied Algebra 104 (1995) 235-241

Induction for Finite Groups Revisited

ANTHONY BAK Department of Mathematics University of Bielefeld 33501 Bielefeld, Germany

Abstract Let G be a finite group and I(G) a nonempty family of subgroups of G, which is closed under conjugation and taking subgroups of its members. Let F be a Green ring functor on G. Let Ω be the Burnside ring functor on G and let Ω_{F} denote the image of the canonical natural transformation Ω ⟶ F. Suppose that for each subgroup H ⊂ G, every ℤ-torsion element of Ω_{F} (H) is nilpotent. Then the following are equivalent: (1) F is I(G)-hypercomputable. (2) Ω_{F} is I(G)-hypercomputable. (3) The restriction homomorphism Res_{I(G)}^{G}(ℚ ⊗ F) : ℚ ⊗_{ℤ} F (G) ⟶ ∏_{H∈I(G)}ℚ ⊗_{ℤ} F (H) is injec-

tive. (4) the induction homomorphism Ind_{I(G)}^{G}(ℚ⊗F) : ∐_{H∈I(G)}ℚ⊗_{ℤ}F (H) ⟶

ℚ ⊗_{ℤ} F (G) is surjective.

1. Introduction

We recall the notions of Mackey functor, Green ring functor, Green module, and hypercomputability for Mackey functors, as well as the universal properties of the Burnside ring functor. A detailed exposition of these notions is included in [BM, §7].

Let G be a finite group. Let Sub(G) denote the category whose objects are all subgroups H ⊂ G and whose morphism are all symbols i_{H,g,K} : H ⟶ K where H and K are subgroups of G and g ∈ G such that gHg^{−1} ⊂ K. Composition is defined by i_{K,g′,K′}i_{H,g,K} = i_{H,g′g,K′}. Each morphism i_{H,g,K} has a natural realization as a group homomorphism H ⟶ K, h ⟼ ghg^{−1}, but distinct morphisms can have the same realization. A Mackey functor on G is a bifunctor M : Sub(G) ⟶ ((abelian groups)) [G, (1.3) axioms G1 and G2] satisfying the Mackey subgroup property [G, (1.3) axioms G3 and G4]. If M is a Mackey functor on G, one often lets Ind_{g,H}^{K}M = M_{⋆}(i_{H,g,K}) and Res_{g,H}^{K}M = M^{⋆}(i_{H,g,K}) and sets Ind_{H}^{K} = Ind_{1,H}^{K} and Res_{H}^{K} = Res_{1,H}^{K}. The map Ind_{g,H}^{K}M (resp. Res_{g,H}^{K}M) is called an induction map (resp. restriction map). If I(G) is a family of subgroups of G, let Ind_{I(G)}^{G}M denote

the canonical map ∐_{H∈I(G)}M(H) ⟶ M(G), ∐_{H∈I(G)}x_{H} ⟼ ∑_{H∈I(G)} Ind_{H}^{G}M(x_{H}),

and let Res_{I(G)}^{G}M denote the canonical map M(G) ⟶ ∏_{H∈I(G)}M(H), x ⟼

∏_{H∈I(G)}Res_{H}^{G}M(x).

Let I(G) be a nonempty family of subgroups of G closed under conjugation and taking subgroups of its members. A Mackey functor M on G is called I(G)-computable if the canonical maps colim_{I(G)} M_{⋆} ⟶ M(G)

and M(G) ⟶ lim_{I(G)}M^{⋆} induced by induction and restriction, respectively, are

bijective. Let S be a set consisting of the natural number 1 and some natural primes p, possibly none, such that p divides |G| := order of G. Let I(G)^{S} = {H ⊂ G | ∃N ⊴ H, N ∈ I(G), | H/N | = power of s, s ∈ S}. Let S′ denote the multiplicative subset of ℤ generated by 1 and all natural primes p such that p ∉ S and p divides |G|. A Mackey functor M on G is called I(G)-hypercomputable if for all sets S as above, the functor S′^{−1}M := S′^{−1}ℤ ⊗_{ℤ} M is I(G)^{S}-computable.

Let ((associative rings with identity))^{w} denote the category whose objects are all associative rings with identity, whose morphisms are all identity preserving ring homomorphisms between associative rings with identity, and whose weak morphisms (accounting for the superscript w above) are all group homomorphisms between associative rings with identity. A Green ring functor on G is a bifunctor F : Sub(G) ⟶ ((associative rings with identity))^{w} such that for any morphism i_{H,g,K} ∈ Sub(G), F^{⋆}(i_{H,g,K}) is a morphism in ((associative rings with identity))^{w} and F_{⋆}(i_{H,g,K}) is a weak morphism in ((associative rings with identity))^{w}, F satisfies the Mackey subgroup property (in particular, F is a Mackey functor), and F satisfies the Frobenius reciprocity law [G, (1.3) axiom G5].

A Mackey functor M on G is called a Green module over a Green ring functor F on G if M is a Frobenius module [L, Chap. III §1] over F.

Let

Ω : Sub(G) ⟶ ((commutative associative rings with identity))^{w}

denote the Burnside ring functor. It is well known, cf. [t.D, §1], that Ω is a Green ring functor, that every Mackey functor on G is canonically a Green module over Ω, and that for any Green ring functor F on G, there is a canonical natural transformation Ω ⟶ F of Green ring functors such that for each subgroup H ⊂ G, image (Ω(H) ⟶ F (H)) ⊂ center (F (H)). Let

Ω_{F} := image (Ω ⟶ F).

The following theorem is dual to A. Dress’ induction theorems [D, (1.2) and (1.7)]. A formulation of Dress’ results, in the language of the present article is given in [B, (12.13)].

Theorem 1.1. Let G be a finite group and I(G) a nonempty family of subgroups of G, which is closed under conjugation and taking subgroups of its members. Let F be a Green ring functor on G such that for each subgroup H ⊂ G, every ℤ-torsion element of F (H) is nilpotent. Then the restriction homomorphism Res_{I(G)}^{G}(ℚ⊗F) : ℚ⊗_{ℤ}F (G) ⟶ ∏_{H∈I(G)}ℚ⊗_{ℤ}F (H) is injective

⟺ F is I(G)-hypercomputable.

The theorem above allows one to prove a result which could be called the

Fundamental Theorem 1.2. Let G be a finite group and I(G) a nonempty family of subgroups of G, which is closed under conjugation and taking subgroups of its members. Let F be a Green ring functor on G such that for each subgroup H ⊂ G, every ℤ-torsion element of Ω_{F} (H) is nilpotent. Then the following are equivalent:

(1) F is I(G)-hypercomputable.

(2) Ω_{F} is I(G)-hypercomputable.

(3) The homomorphism Res_{I(G)}^{G}(ℚ ⊗ F) : ℚ ⊗_{ℤ} F (G) ⟶ ∏_{H∈I(G)}ℚ ⊗_{ℤ} F (H)

is injective.

(4) The homomorphism Ind_{I(G)}^{G}(ℚ ⊗ F) : ∐_{H∈I(G)}ℚ ⊗_{ℤ} F (H) ⟶ ℚ ⊗_{ℤ} F (G)

is surjective.

Proof (1) ⇒ (2). (1) implies that the restriction homomorphism Res_{I(G)}^{G}(ℚ⊗ F) : ℚ ⊗_{ℤ}F (G) ⟶ ∏_{H∈I(G)}ℚ ⊗_{ℤ}F (H) is injective. It follows that the restric-

tion homomorphism Res_{I(G)}^{G}(ℚ⊗Ω_{F}) : ℚ⊗_{ℤ}Ω_{F} (G) ⟶ ∏_{H∈I(H)}ℚ⊗_{ℤ}Ω_{F} (H) is

injective. One checks easily that every subbifunctor of a Green ring functor is a Green ring functor. Hence, Ω_{F} is a Green ring functor and by Theorem (1.1), Ω_{F} is I(G)-hypercomputable.

(2) ⇒ (3). Since F is a Green module over Ω_{F} and Ω_{F} is I(G)-hypercomputable, it follows from Dress’ result [D, (1.2)] (cf. [B, (12.13)(a)]) that F is I(G)- computable. In particular, the restriction homomorphism Res_{I(G)}^{G}(ℚ ⊗ F) :

ℚ ⊗_{ℤ} F (G) ⟶ ∏_{H∈I(G)}ℚ ⊗_{ℤ} F (H) is injective.

(3) ⇒ (4). (3) implies, as in the proof of (1) ⇒ (2), that the restriction

homomorphism Res_{I(G)}^{G}(ℚ ⊗_{ℤ} Ω_{F}) : ℚ ⊗_{ℤ} Ω_{F} (G) ⟶ ∏_{H∈I(G)}ℚ ⊗_{ℤ} Ω_{F} (H) is

injective. Thus, by Theorem (1.1), Ω_{F} is I(G)-hypercomputable. Hence, as in the proof of (2) ⇒ (3), F is I(G)-hypercomputable. In particular, the induction homomorphism Ind_{I(G)}^{G}(ℚ ⊗ F) : ∐_{H∈I(G)}ℚ ⊗_{ℤ} F (H) ⟶ ℚ ⊗_{ℤ} F (G)

is surjective.

(4) ⇒ (1). This follows from Dress induction [D, (1.2) and (1.7)]. Q.E.D.

The following corollaries of the fundamental theorem have applications in K-theory [BM, (12.22) and (12.23)], surgery theory [BM, (13.20)], and transformation groups [BM, §14 ].

Corollary 1.3. Let G be a finite group and I(G) a nonempty family of subgroups of G, which is closed under conjugation and taking subgroups of its members. Let F be a Green ring functor on G such that for each subgroup H ⊂ G, every ℤ-torsion element of Ω_{F} (H) is nilpotent. Then if F is I(G)-hypercomputable, every submackey functor F′′ ⊂ F and every quotient Mackey functor F′ of F is I(G)-hypercomputable.

Proof The important observation to make is that F′ and F′′ are Green modules over Ω_{F} (but not necessarily over F). By the Fundamental Theorem (1.2), Ω_{F} is I(G)-hypercomputable. Thus, by [D, (1.2)], F′ and F′′ are I(G)- hypercomputable.

Corollary 1.4. Let G be a finite group and I(G) a nonempty family of subgroups of G, which is closed under conjugation and taking subgroups of its members. Let F_{0}, F_{1}, …, F_{m} be a sequence of Green ring functors on G such that for each i > 0, one of the following holds: (1) There is a natural transformation F_{i−1} ⟶ F_{i} of Green ring functors. (2) The canonical action of Ω on F_{i} factors through Ω_{F_{i−1}} and for each subgroup H ⊂ G, every ℤ-torsion element of Ω_{F_{i−1}}(H) is nilpotent. (3) There is an exact sequence F′′_{i} ⟶ F_{i} ⟶^{γ} F_{i−1} of natural transformations of Mackey functors (in particular, γ is not necessarily a natural transformation of Green ring functors) such that F′′_{i} is I(G)-hypercomputable and for each subgroup H ⊂ G, every ℤ-torsion element of Ω_{F_{i−1}}(H) is nilpotent. (4) There is an exact sequence F′′_{i} ⟶ F_{i} ⟶^{γ} F_{i−1} of Mackey functors such that F′′_{i} is I(G)-hypercomputable and γ is surjective. Then F_{0} is I(G)-hypercomputable ⟺ all the F_{i}^{,}s are I(G)-hypercomputable.

Proof. The assertion ⇐) is trivial. The assertion ⇒) is proved by induction on m. The case m = 0 is trivial. Assume that m > 0 and that the result is true for m − 1. Thus, F_{m−1} is I(G)-hypercomputable. Let F = F_{m} and F′ = F_{m−1}.

Suppose there is a natural transformation F′ ⟶ F as in (1). Then F is a Green module over F′. Since F′ is I(G)-hypercomputable, it follows from [D, (1.2)] that F is I(G)-hypercomputable.

Suppose (2) holds. Then F is a Green module over Ω_{F′} and by (1.2), Ω_{F′} is I(G)-hypercomputable. It follows now from [D, (1.2)] that F is I(G)- hypercomputable.

Let F′′ ⟶ F ⟶^{γ} F′ be an exact sequence as in (3). Let S and S′ be as in the definition of I(G)-hypercomputable. Consider the commutative diagram of short exact sequences.

∐_{H∈I(G)^{S}}S′^{−1}F′′(H) ⟶ ∐_{H∈I(G)^{S}}S′^{−1}F (H) ↠ ∐_{H∈I(G)^{S}}S′^{−1} image γ(H)

Ind_{I(G)^{S}}^{G}(S′^{−1}F′′) ↓  Ind_{I(G)^{S}}^{G}(S′^{−1}F) ↓  ↓ Ind_{I(G)^{S}}^{G}(S′^{−1} image γ)
S′^{−1}F′′(G)  ⟶  S′^{−1}F (G)  ↠ S′^{−1} image γ(G).

By [D, (1.2)], it suffices to show that the induction map Ind_{I(G)^{S}}^{G}(S′^{−1}F) is surjective. By assumption, the induction map Ind_{I(G)^{S}}^{G}(S′^{−1}F′′) is surjective and by (1.3), the induction map Ind_{I(G)^{S}}^{G}(S′^{−1}imageγ) is surjective. Thus, by the 5-Lemma, the induction map Ind_{I(G)^{S}}^{G}(S′^{−1}F) is surjective.

Let F′′ ⟶ F ⟶^{γ} F′ be an exact sequence as in (4). Then one gets a commutative diagram as above and deduces as above that F is I(G)- hypercomputable. Q.E.D.

The rest of this article is devoted to the proof of Theorem (1.1).

2. Proof of Theorem (1.1)

Throughout this section, G denotes a finite group and F a Green ring functor on G.

Definition 2.1. Let I(G) and R(G) be nonempty families of subgroups of G, which are closed under conjugation and taking subgroups of their members. I(G) is called relatively F -prime to R(G) if the 2-sided ideal image (Ind_{I(G)}^{G}F) is relatively prime to the 2-sided ideal Ker(Res_{R(G)}^{G}F), i.e., image (Ind_{I(G)}^{G}F) + Ker(Res_{R(G)}^{G}F) = F (G).

Lemma 2.2. Let F′ be a Green ring functor on G which is a Green module over F. If I(G) is relatively F -prime to R(G) then I(G) is relatively F′- prime to R(G).

Proof. Let x ∈ image (Ind_{I(G)}^{G}F) and y ∈ Ker (Res_{R(G)}^{G}F) such that x+y = 1_{F (G)}. Clearly, 1_{F′(G)} = 1_{F (G)}1_{F′(G)} = x1_{F′(G)} + y1_{F′(G)}. By Frobenius reciprocity, x1_{F′(G)} ∈ image (Ind_{I(G)}^{G}F′). Furthermore, y1_{F′(G)} ∈ Ker(Res_{R(G)}^{G}F′) because for H ∈ R(G), Res_{H}^{G}F′(y1_{F′(G)}) = (Res_{H}^{G}F (y))(Res_{H}^{G}F′ (1_{F′(G)})) = 0. Q.E.D.

Proposition 2.3. Let F′ be a Green ring functor on G. Let I(G) and R(G) be nonempty families of subgroups of G, which are closed under conjugation and taking subgroups of their members, such that I(G) is relatively F -prime to R(G). If F′ is a Green module over F and Ker(Res_{R(G)}^{G}F′) = 0 then every Green module on G over F′ is I(G)-computable.

Proof. By Dress’ theorem [D, (1.2)] (cf. [B, (12.13) (a)]), it suffices to show that the induction map Ind_{I(G)}^{G}F′ is surjective. But, this is a trivial consequence of Lemma (2.2) and the assumption that Ker(Res_{R(G)}^{G}F′) = 0. Q.E.D.

Let 𝒮(G) denote the set of all subgroups of G. If ℐ ⊂ 𝒮(G) is closed under conjugation, let Conj(ℐ) denote the set of conjugation classes (H) of elements H ∈ ℐ. Give Conj(𝒮(G)) the partial ordering defined by the rule (H) ≦ (K) ⇔ H is conjugate to a subgroup of K.

Definition 2.4. A linear ordering (H_{1}), (H_{2}), …, (H_{r}) of the elements of Conj(𝒮(G)) is called good if (H_{i}) ≦ (H_{j}) ⟹ i ≦ j. A nonempty, conjugate closed subset I(G) ⊂ 𝒮(G) is called initial if there is a good ordering (H_{1}), (H_{2}), …, (H_{r}) of Conj(𝒮(G)) such that for some k, Conj(I(G)) = {(H_{1}), (H_{2}), …, (H_{k})}.

The next lemma characterizes initial families of subgroups of G.

Lemma 2.5. Let I(G) be a nonempty, conjugate closed family of subgroups of G. Then I(G) is an initial family ⟺ I(G) is closed under taking subgroups of its members.

Proof The proof is an elementary exercise.

Key Lemma 2.6. Let Ω : Sub(G) ⟶ ((commutative associative rings with identity ))^{w} denote the Burnside ring (bi)functor. Let I(G) be a nonempty family of subgroups of G, which is closed under conjugation and taking subgroups of its members. Then I(G) is relatively ℚ ⊗_{ℤ} Ω-prime to I(G).

Proof. By Lemma (2.5), there is a good ordering (H_{1}), (H_{2}), …, (H_{r}) of Conj(𝒮(G)) such that {(H_{1}), (H_{2}), …, (H_{k})} = Conj(I(G)), for some k. For (H) ∈ Conj(𝒮(G)) and a finite G-set X, define ϕ_{(H)}(X) = |X^{H}| := number of elements in X which are left fixed by every element in H. It is easy to check, cf. [t.D, §1], that ϕ_{(H)} defines a ring homomorphism ϕ_{(H)} : Ω(G) ⟶ ℤ.

Let e_{i}(1 ≤ i ≤ r) denote the idempotent of ∏^{r}ℤ whose i′th coordinate is 1

and other coordinates are 0. Define ϕ : Ω(G) ⟶ ∏^{r}ℤ, X ⟼ ∏_{i=1}^{r} ϕ_{(H_{i})}(X)e_{i}.

Give the Burnside ring Ω(G) the ordered ℤ-basis [G/H_{1}], [G/H_{2}], …, [G/H_{r}] where [G/H_{i}] denotes the isomorphism class of the G-set G/H_{i} of left cosets

of H_{i} in G. Give ∏^{r}ℤ the ordered ℤ-basis e_{1}, e_{2}, …, e_{r}. One deduces easily from the paragraph preceeding [t. D, (1.2.3)] that ϕ is represented by an upper triangular matrix 𝕄(ϕ) whose i′th diagonal element is |N_{G}(H_{i})/H_{i}| where N_{G}(H_{i}) is the normalizer of H_{i} in G. In particular, the diagonal entries of 𝕄(ϕ) are nonzero natural numbers. Thus, ϕ is injective and ℚ ⊗ ϕ :

ℚ ⊗_{ℤ} Ω(G) ⟶ ∏^{r}ℚ is a ring isomorphism.

It is well known that the Burnside ring functor is a Green ring functor on G. Thus, the bifunctor ℚ ⊗_{ℤ} Ω : Sub(G) ⟶ ((commutative associative rings with identity))^{w}, H ⟼ ℚ ⊗_{ℤ} Ω(H), is also a Green ring functor. Let x ∈ ℚ ⊗_{ℤ} Ω(G). It follows from the definition of restriction for the Burnside ring functor that Res_{H}^{G}(ℚ ⊗_{ℤ} Ω)(x) = 1_{ℚ⊗_{ℤ}Ω(H)} ⟺ (ℚ ⊗ ϕ_{(K)})(x) = 1 for all subgroups K of H. Thus, I(G) is relatively ℚ ⊗_{ℤ} Ω-prime to I(G) ⟺(by definition) there are elements x ∈ image (Ind_{I(G)}^{G}(ℚ ⊗_{ℤ} Ω)) and y ∈ Ker(Res_{I(G)}^{G}(ℚ ⊗_{ℤ}Ω)) such that x + y = 1_{ℚ⊗_{ℤ}Ω(G)} ⟺ there is an element x ∈ image (Ind_{I(G)}^{G}(ℚ ⊗_{ℤ}Ω)) such that (Res_{H}^{G}(ℚ ⊗_{ℤ}Ω))(x) = 1_{ℚ⊗_{ℤ}Ω(H)} for all H ∈ I(G) ⟺ there is an element x ∈ image (Ind_{I(G)}^{G}(ℚ ⊗_{ℤ} Ω)) such that (ℚ ⊗ ϕ_{(H)})(x) = 1 for all H ∈ I(G). Since ℚ ⊗ ϕ is bijective, there is a unique element x ∈ ℚ ⊗_{ℤ} Ω(G) such that (ℚ ⊗ ϕ_{(H)})(x) = 1 for all H ∈ I(G) and (ℚ ⊗ ϕ_{(H)})(x) = 0 for all H ∉ I(G). Since 𝕄(ℚ ⊗ ϕ) = 𝕄(ϕ) is a triangular matrix with nonzero diagonal entries and since Conj(I(G)) =

{(H_{1}), (H_{2}), …, (H_{k})}, the element x above has the form ∑_{i=1}^{k} a_{i}[G/H_{i}] for

suitable rational numbers a_{i} ∈ ℚ. Thus, x ∈ image (Ind_{I(G)}^{G}(ℚ⊗_{ℤ}Ω)). Q.E.D.

Proof Of Theorem 1.1. By Dress induction [D, (1.2) and (1.7)] (a formulation in the language of the present article is given in [B, (12.13)]), it suffices to show that the induction map Ind_{I(G)}^{G}(ℚ ⊗_{ℤ} F) is surjective. But, this follows immediately from (2.6), (2.3), and the fact that ℚ ⊗_{ℤ} F is a Green module over ℚ ⊗_{ℤ} Ω, because any Mackey functor, for example F, is a Green module over Ω. Q.E.D.

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