Communications in Algebra

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Category of G-Groups and its Spectral

María José Arroyo Paniagua & Alberto Facchini

To cite this article: María José Arroyo Paniagua & Alberto Facchini (2017) Category of G-Groups and its Spectral Category, Communications in Algebra, 45:4, 1696-1710, DOI: 10.1080/00927872.2016.1222409

To link to this article: http://dx.doi.org/10.1080/00927872.2016.1222409

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Download by: [UNAM Ciudad Universitaria] Date: 29 November 2016, At: 17:29 COMMUNICATIONS IN ALGEBRA® 2017, VOL. 45, NO. 4, 1696–1710 http://dx.doi.org/10.1080/00927872.2016.1222409

Category of G-Groups and its Spectral Category María José Arroyo Paniaguaa and Alberto Facchinib aDepartamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico, D. F., México; bDipartimento di Matematica, Università di Padova, Padova, Italy

ABSTRACT ARTICLE HISTORY Let G be a . We analyse some aspects of the category G-Grp of G-groups. Received 15 April 2016 In particular, we show that a construction similar to the construction of the Revised 22 July 2016 spectral category, due to Gabriel and Oberst, and its dual, due to the second Communicated by T. Albu. author, is possible for the category G-Grp. KEYWORDS Group actions; Semi-abelian categories; Spectral categories 2000 MATHEMATICS SUBJECT CLASSIFICATION 20J15

1. Introduction The original motivation of this paper was to give an adequate categorical setting to the techniques employed in [4]. In that paper, Andrea Lucchini and the second author obtained results concerning the direct product decompositions of a group G as a direct product of indecomposable normal employing techniques essentially based on the notion of normal homomorphism [12, p. 81]. Here, if N and M are normal subgroups of G, a homomorphism f : N → G/M is normal if it commutes with all inner of G. In particular, a construction reminiscent of the construction of the spectral category [5] was applied to the set of all normal endomorphisms of the group G. In order to place these techniques into a proper setting, we study groups, where now is the group G, and G acts on its normal subgroups N and its quotient groups G/M via inner automorphisms. To this , we consider the category GGrp of all pairs (H, ϕ), where H is any group and ϕ : G → Aut(H) is a . In other words, GGrp is the category whose objects are the socalled Ggroups. The notion of Ggroup is a classical one, and sometimes G is called the operator group acting on H ([1, Denition 5.5] and [13, Denition 8.1]). In this context, normal subgroups N of G and quotient groups G/M become objects of GGrp, and normal homomorphisms become in GGrp. Thus we focus on the study of the category GGrp, which is a semi in the sense of [9]; cf. [1] and [6]. We determine the free Ggroups and show that the injective objects in the category GGrp are only the trivial ones, like in the case of the category Grp of groups. There are two more categories in strict relation with the category GGrp. One is the category of all Gsemidirect products (i.e., the category of pointed objects over the group G [1, Section 5]) and the other is the category of all le kGmodule Hopf algebras. We adapt the GabrielOberst construction of spectral category and its ′ dual to the category GGrp, getting two categories Spec(GGrp) and CG. This setting works particularly well on the full CG of GGrp consisting of all objects (H, ϕ) of GGrp for which the of the group homomorphism ϕ : G → Aut(H) contains the group Inn(H) of all inner automorphisms C C′ of H. Furthermore, there is a local canonical of G to the Spec(GGrp) × G (Theorem 6.7). Also, we obtain that the endomorphism of an object H of the category Spec(G Grp) is a regular monoid whenever H is a Ggroup in CG.

CONTACT Alberto Facchini [email protected] Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy. © 2017 Taylor & Francis COMMUNICATIONS IN ALGEBRA® 1697

The reader will notice the analogy between the category GGrp of Ggroups and the category RMod of all le modules over a xed R. The category RMod has as objects the pairs (M, λ), where M is an additive and λ: R → End(M) is a ring . In the category GGrp the morphisms f : (H, ϕ) → (H′, ϕ′) are the group homomorphisms f : H → H′ such that f (gh) = gf (h) for every g ∈ G, h ∈ H, exactly as in the category RMod. We write mappings on the le (we stress this, because in mappings are oen written on the right, sometimes as exponents. Moreover, writing mappings on the le has as a consequence that le actions correspond to homomorphisms, and right actions correspond to antihomomorphisms. We will come back to this point in Remark 2.3). Let G be a group. As we have already said above, a (le) Ggroup is a pair (H, ϕ), where H is a group and ϕ : G → Aut(H) is a group homomorphism. Since we write mappings on the le, a Ggroup can be equivalently dened as a group H endowed with a mapping : G × H → H, (g, h) → gh, called le scalar multiplication, such that (a) g(hh′) = (gh)(gh′) (b) (gg′)h = g(g′h) (c) 1Gh = h for every g, g′ ∈ G and every h, h′ ∈ H. As far as terminology is concerned, it is convenient to be very precise. The notion of Ggroup is a particular case of the notion of group as treated in [12, p. 28] (for any set , an group is a pair (H, ϕ), where H is a group and ϕ : → End(H) is a mapping). At the same time, this notion of group is a particular case of the more general notion of a group with multiple operators studied in [7]. Also, recall that a (le) Gset is a pair (X, σ), where X is a set and σ : G → SX is a group homomorphism into the SX on X. Equivalently, a Gset is a set X with a mapping : G × X → X, ′ ′ ′ (g, x) → gx, such that (gg )x = g(g x) and 1Gx = x for every g, g ∈ G and x ∈ X.

2. G-groups and their morphisms The rst part of this section is devoted to establishing the elementary terminology about Ggroups. A Ggroup morphism f : (H, ϕ) → (H′, ϕ′) is a group homomorphism f : H → H′ such that f (gh) = gf (h) ′ ′ for every g ∈ G, h ∈ H. We will denote by HomG(H, H ) the set of all Ggroup morphisms of H into H . Thus Ggroups form a category, which we will denote by GGrp. Notice that the category Grp of groups and the category 1Grp, where 1 is the (with one element), are isomorphic categories. This is the analog of the fact that the category Ab of abelian groups and the category Z−Mod of modules over the ring Z of are isomorphic categories, because 1 and Z are the initial objects in the and the , respectively. It is clearly possible to generalize our study of the category GGrp to a more general setting, though this will not be our primary aim in this paper. The generalization is the following. Let C be a category and M a monoid. We can obviously dene the category CM of all Mobjects in C as the category of all pairs (C, ϕ), where C is an object in C and ϕ : M → End(C) is a monoid homomorphism. If we view M as a category with one object, CM is the category of all M → C. When M is a group, every homomorphism M → End(C) factors uniquely though Aut(C), and so we can write ϕ : M → Aut(C) instead of ϕ : M → End(C). When C is the category Grp of groups and G is a group, CG turns out to be the category GGrp of Ggroups. Now for any category C and any monoid M, there is a U : CM → C, which associates to every object (C, ϕ) of CM (the category of Mobjects in C) the object C of C. Suppose now that C has Mindexed . This is the case, for instance, for M = G and C = Grp, where, for any family of groups Hg indexed by the elements g of G, the Mindexed is the of the groups Hg. The forgetful functor U : CM → C has a le adjoint F : C → CM. Here F is dened on objects by F(C) = (M C, ϕ), where M C is the Mindexed coproduct of |M| copies of C, and ϕ : M → End(M C) 1698 M. J. A. PANIAGUA AND A. FACCHINI is constructed as follows. For every m ∈ M, let ιm : C → MC be the coproduct injection. Thus, for every ′ ′ family of morphisms fm : C → C , where m ∈ M and C is an object of C, there exists a unique morphism ′ f : M C → C such that f ιm = fm for every m ∈ M. Now if m ∈ M is xed and we consider the family ′ of morphisms fm′ := ιmm′ : C → M C, m ∈ M, we get a unique morphism f := ϕ(m): M C → M C ′ such that ϕ(m)ιm′ = ιmm′ for every m ∈ M. Thus we get a mapping ϕ : M → End(M C) such that ′ ϕ(m)ιm′ = ιmm′ for every m, m ∈ M, and it is easily checked that ϕ is a monoid morphism. On the morphisms of C, F is dened as follows. Let C, C′ be two objects in C and α : C → C′ be a morphism. Consider F(C) = (M C, ϕ) and F(C′) = (M C′, ϕ′). For each m ∈ M, let ′ ′ ′ ιm : C → M C and ιm : C → M C ′ ′ be the coproduct injections. Given the family of morphisms αm = ιmα : C → M C , F(α) is the ′ ′ unique morphism F(α): M C → M C such that F(α)ιm = αm = ιmα for every m ∈ M, so that ϕ′(m)F(α) = F(α)ϕ(m) for every m ∈ M. The forgetful functor U : CM → C has both a right adjoint and a le adjoint, provided that C has Mindexed coproducts and products, which is the case for M a group and C the category of groups. For the reader’s convenience, we recall that, for a ring R, the forgetful functor U : R−Mod → Ab has as a right adjoint the functor HomZ(R, −): Ab → R−Mod and as a le adjoint the functor R ⊗Z −: Ab → R−Mod [10, Exercise 2, p. 89]. We now pass to study Ggroups from an algebraic point of view. We begin with an example, the −1 example of the regular Ggroup (G, α). Here α : G → Aut(G), g → αg, where αg(x) = gxg for every g, x ∈ G. ′ ′ ′ Recall that a G H of a Ggroup H, denoted H ≤G H, is a subgroup H of H such that gh′ ∈ H′ for every g ∈ G, h′ ∈ H′. For instance, the Gsubgroups of the regular Ggroup G are the ′ ′ ′ normal subgroups of G. If f : (H, ϕ) → (H , ϕ ) is a Ggroup morphism, A ≤G H implies f (A) ≤G H ′ ′ −1 ′ and A ≤G H implies f (A ) ≤G H.

An equivalence relation ∼ on H is compatible with the Ggroup structure on H if, for every g ∈ G and h1, h2, h3, h4 ∈ H, h1 ∼ h2 and h3 ∼ h4 imply h1h3 ∼ h2h4 and gh1 ∼ gh2. The following lemma is a special case of a wellknown property of groups with operators.

Lemma 2.1. Let H be a Ggroup. Then there is a canonical onetoone correspondence between the set EH of all equivalence relations ∼ on H compatible with the Ggroup structure on H and the set NG(H) of all ′ ′ normal subgroups H of H with H ≤G H. If ∼ ∈ EH, the corresponding in NG(H) is the ′ equivalence [1H]∼ of 1H modulo ∼. If H ∈ NG(H), the corresponding compatible equivalence on H −1 ′ is the equivalence relation ∼H′ dened, for every h1, h2 ∈ H, by h1 ∼H′ h2 if h1 h2 ∈ H .

′ ′ If H ∈ NG(H), the Ggroup structure on H induces a Ggroup structure on the H/H in such a way that the canonical projection H → H/H′ turns out to be a Ggroup morphism. The le action of G on H/H′ is dened by g(hH′) = (gh)H′ for every g ∈ G and h ∈ H. For every Ggroup ′ ′ morphism f : (H, ϕ) → (H , ϕ ), the ker f belongs to NG(H). Let us recall a few other standard observations adapted to Ggroups. Trivial Ggroups, that is, the Ggroups with only one element, are the null objects in the category GGrp. The in GGrp are precisely the Ggroup morphisms that are injective mappings (to see this, compose a non injective Ggroup morphism f : H → H′ with the embedding and the zero mapping of ker f into H). The kernels in the category GGrp are the Ggroup morphisms that are injective mappings and have a normal image. The are precisely the bijective Ggroup morphisms. In the next theorem, we show that the in GGrp are exactly the surjective mappings, as in Grp [10, Exercise 5, p. 21].

Theorem 2.2. Let G be a group and (H, ϕ), (H′, ϕ′) be Ggroups. A Ggroup morphism f : H′ → H is an in the category GGrp if and only if it is a surjective mapping. COMMUNICATIONS IN ALGEBRA® 1699

Proof. Clearly, any surjective Ggroup morphism is an epimorphism in GGrp, so that it suces to prove the converse. We go back to the general setting of the category CM described at the beginning of this section. We have see that the forgetful functor U : GGrp → Grp has a right adjoint. Thus U is a le adjoint functor, and le preserve colimits [10, Section V.5,p. 119] and, in particular, preserve coproducts and epimorphisms. Thus U : GGrp → Grp preserves epimorphisms. Thus if f is an epimorphism in GGrp, U(f ) is an epimorphism in Grp, hence a surjective mapping.

′ ′ ′ ′ If) f : (H, ϕ → (H , ϕ ) is a Ggroup morphism, the normal closure of f (H) in H belongs to NG(H ). Thus f has a categorytheoretic given by the quotient group of H′ modulo the normal closure of f (H) in H′. It is well known that the categorytheoretic product in the category Grp of groups is just the . Now if (H1, ϕ1), (H2, ϕ2) are Ggroups, then the group morphisms ϕi : G → Aut(Hi), i = 1, 2, dene a group morphism ϕ1 × ϕ2 : G → Aut(H1) × Aut(H2) ⊆ Aut(H1 × H2). The Ggroup (H1 × H2, ϕ1 × ϕ2) turns out to be the direct product of (H1, ϕ1) and (H2, ϕ2) in the category GGrp. The action of g on H1 × H2 is dened, equivalently, by g(h1, h2) = (gh1, gh2). Recall that as far as coproducts are concerned, the categorytheoretic coproduct of two groups H1, H2 in Grp is their free product H1 ∗ H2. Let εi : Hi → H1 ∗ H2, i = 1, 2, denotes the canonical embeddings. ′ Now let (H1, ϕ1) and (H2, ϕ2) be Ggroups. Their coproduct in the category GGrp is (H1 ∗ H2, ϕ ). ′ ′ Here, for every g ∈ G, ϕ (g) is the unique group endomorphism of H1 ∗ H2 such that ϕ (g)εi = εiϕi(g) for both i = 1 and i = 2. Equivalently, the action of G on H1 ∗ H2 can be described as follows. Each element [h] ∈ H1 ∗ H2 can be written uniquely in its normal form [h] = [h1][h2] [hr], (r ≥ 0), where

1 = hi ∈ Hλi , λi ∈ {1, 2} and λi = λi+1; if r = 0, [h] = 1 [12, p. 170]. Then g[h] = g([h1][h2] [hr]) = [gh1][gh2] [ghr]. It is easy to prove that the category GGrp has equalizers, hence it is nitely complete. It is also

a semiabelian category in the sense of [9]; cf. [1, Denition 1.3] and Gran [6]. Every semiabelian category C contains an abelian full subcategory Ab(C) whose objects are all the abelian objects of C. Here, an object C of a semiabelian category C is abelian when the identity of C commutes with itself, or, equivalently, when C can be provided with the structure of an abelian group, that is, there exists a morphism m: C × C → C in C such that m ◦ (idC, 0) = m ◦ (0, idC) = idC. In the particular case where C = GGrp and the object C is a Ggroup H, the mapping m: H × H → H such that m ◦ (idC, 0) = m ◦ (0, idC) = idC is necessarily the multiplication on H [1, Proof of Example 6.2]. This is a group morphism if and only if the group H is abelian. When H is a Ggroup which is an abelian group, then the group morphism multiplication m: H × H → H is a Ggroup morphism, because m(g(h1, h2)) = m(gh1, gh2) = (gh1)(gh2) = g(h1h2) = gm(h1, h2). It follows that the abelian objects in the category GGrp are the objects (H, ϕ) of GGrp with H an abelian group. We call them abelian Ggroups. The full subcategory Ab(GGrp) of GGrp whose objects are all abelian Ggroups is clearly equivalent to the category ZG−Mod of all le modules over the ZG, and ZG−Mod is an abelian category. It should be noted that the category of Gsets is a Boolean (which does not satisfy the Axiom of Choice), and the category of Ggroups is the category of groups of this topos. We are grateful to Professor Janelidze for this remark and several other enlightening comments on this paper. His suggestions greatly improved the quality of the exposition. He also noticed that some results of this paper can be generalized from the category of Ggroups to a wide class of semiabelian and more general categories.

Remark 2.3. Instead of considering le Ggroups, as we do in this paper, we could have dened and studied right Ggroups. Because of our original choice of writing mappings on the le, a right Ggroup must be dened as a pair (H, ϕ′), where H is a group and ϕ′ : G → Aut(H) is a group antihomomorphism. Equivalently, a right Ggroup is a group H endowed with a mapping : H×G → H, (h, g) → hg, called right scalar multiplication, such that (a) becomes (hh′)g = (hg)(h′g), etc. We can thus construct the category of GrpG of all right Ggroups. Let Gop be the opposite group of the group 1700 M. J. A. PANIAGUA AND A. FACCHINI

G. From a le Ggroup, we can also construct a right Gopgroup, and so we see that the category GGrp is isomorphic to the category GrpGop and GrpG is isomorphic to GopGrp. It is interesting to note that the categories GGrp and GrpG are always isomorphic. More precisely, let (H, ϕ) be a le Ggroup. Then ϕ′ : G → Aut(H), dened by ϕ′(g) = ϕ(g−1) for every g ∈ G, is a group antihomomorphism, so that (H, ϕ′) turns out to be a right Ggroup. This assignment gives a category GGrp → GrpG, in which the regular le Ggroup GG corresponds to the regular right Ggroup GG. Thus the situation is completely dierent from that of le and right modules over a ring R. Namely, for a ring R, denoting by Rop its opposite ring, the category R−Mod is isomorphic to the category Mod−Rop, the category Mod−R is isomorphic to the category Rop−Mod, and the category R−Mod is isomorphic to the category Mod−R when R is a commutative ring. In general, the categories R−Mod and Mod−R are not equivalent.

3. Free G-groups and injective objects From the point of view of Universal Algebra, Ggroups are algebras −1 H, , , 1H, (ϕg)g∈G, −1 where is a binary operation, is unary, 1H is nullary, and each ϕg is unary, such that the following −1 hold: H, , , 1H is a group, ′ ′ ′ ′ ϕg(hh ) = (ϕg(h))(ϕg(h )), ϕgg (h) = ϕg(ϕg (h)) and ϕ1G (h) = h for g, g′ ∈ G, h, h′ ∈ H.

Proposition 3.1. Let G be a group and X be a set. The free Ggroup over the set X is the FG×X over the cartesian product G × X on which the group G acts via g(g′, x) = (gg′, x) for every g, g′ ∈ G and

every x ∈ X.

Proof. We go back to the general setting of the category CM described at the beginning of Section 2, where now M is the group G and C is the category Set of sets, so that CM turns out to be the category G Set of all Gsets. As we have seen there, the forgetful functor U : GSet → Set has a le adjoint functor F : Set → GSet. The functor F is dened on objects by F(X) = (G×X, ϕ), where the cartesian product G × X is the Gindexed coproduct of |G| copies of the set X and ϕ : G → Aut(G × X) is dened by ϕ(g)(g′, x) = (gg′, x). The forgetful functor U′ : GGrp → GSet also has a le adjoint F′ : GSet → GGrp. Given any ′ ′ ′ object (X, ϕ) of GSet, F is dened on objects by F (X, ϕ) = (FX, ϕ ), where FX is the free group with free ′ ′ set X of generators and ϕ : G → Aut(FX) is the group morphism for which ϕ (g) is the ′ of FX that extends the ϕ(g): X → X on the set X of free generators of FX. Thus F F : Set → G ′ ′ Grp, X → (FG×X, ϕ ), is the le adjoint of the forgetful functor UU : GGrp → Set. This proves the proposition.

Remark 3.2. We will use the “standard additive notation” for mappings between multiplicative groups. That is, if H and H′ are multiplicative groups and α, β : H → H′ are any two mappings, we denote by α + β : H → H′ the mapping dened by (α + β)(h) = α(h)β(h) for every h ∈ H. If α and β are group morphisms, the mapping α + β turns out to be a group morphism if and only if the two morphisms α and β commute in the category of groups [1, Example 6.2], that is, if and only if α(h1) commutes with β(h2) for all h1, h2 ∈ H. The identity endomorphisms and the trivial homomorphisms will be denoted by 1 and 0, respectively. Now let G be a group and H a Ggroup. The set HH of all mappings H → H, endowed with the addition + dened above and the composition of mappings ◦ as multiplication, is a right near ring (that COMMUNICATIONS IN ALGEBRA® 1701 is, a group with respect to +, a monoid with respect to ◦, right distributivity holds, but le distributivity does not in general.) It is tempting to say that HH is a “right near Gring”, in the sense, that it is a right near ring, which is also a Ggroup (viewing HH as a product of copies of the Ggroup H, so that the action of G on HH is dened by (gf )(h) = g(f (h)) for every g ∈ G, f ∈ HH, h ∈ H), and the two structures are compatible, in the sense that g(f ◦ f ′) = (gf ) ◦ f ′ for g ∈ G and f , f ′ ∈ HH. H The subset EndG(H) := HomG(H, H) ⊆ H is a partial ring with identity, in the following sense. Set S = { (α, β) ∈ EndG(H) × EndG(H) | α(h1) commutes with β(h2) for all h1, h2 ∈ H }, so that the H addition + is a mapping +: S → EndG(H). Then EndG(H) is a multiplicative submonoid of H , and End(G) also has the partially dened operation +. Also, the equality 0+α = α+0 = α always holds, and the identities α +β = β +α, (α +β)+γ = α +(β +γ ), α(β +γ ) = αβ +αγ , (α +β)γ = αγ +βγ hold, for α, β, γ ∈ End(G), whenever both members of the equalities are dened. Notice that both distributivity laws hold in EndG(H), when the operations are dened. Finally, EndG(H) is a ring if and only if H is abelian.

It is interesting to notice that the automorphisms in GGrp of the regular object G are exactly the central automorphisms of G, that is, the automorphisms of G that belong to the kernel of the canonical mapping Aut(G) → Aut(G/ζ(G)) [4, Proposition 4.4(d)]. Here ζ(G) denotes the of G. Thus the automorphisms of G in GGrp are exactly the automorphisms of the group G of the form 1 + ϕ for some endomorphism ϕ of G with ϕ(G) ⊆ ζ(G) [4, Lemma 4.3]. It is easy to see that if ϕ is a group endomorphism of G with ϕ(G) ⊆ ζ(G), then 1+ϕ is an automorphism of G if and only if the restriction of 1 + ϕ to ζ(G) is an automorphism of the abelian group ζ(G). In Section 5, we will present a construction very similar to the construction of the spectral category of a Grothendieck category due to Gabriel and Oberst [5]. In the case of a Grothendieck category C, the spectral category Spec(C) reects the behaviour of injective objects in the category, because two objects A, B of C are isomorphic in Spec(C) if and only if their injective envelopes E(A), E(B) are isomorphic C in . The situation for the category GGrp is necessarily completely dierent, because in this category the unique injective objects are the trivial Ggroups with one element, as we will see in Theorem 3.3. This is exactly what happens in the category Grp, see e.g., Eilenberg and Moore [2, p. 21–22]. Recall that an object E of a category C is injective if for any f : A → B in C and any morphism ℓ: A → E, there exists a morphism m: B → E such that mf = ℓ.

Theorem 3.3. If H is a Ggroup that is an injective object in the category GGrp, then H = 1.

Proof. Recall that any functor, which has a le adjoint that preserves monomorphisms, preserves injective objects [8, Theorem 3.8.12∗]. Thus let us go back to the general setting of the category CM described at the beginning of Section 2 , where M is the group G and C is the category Grp, so that the forgetful functor U : GGrp → Grp has F : Grp → GGrp as a le adjoint. The functor F maps any group morphism α : C → C′ to the Ggroup morphism F(α): G C → G C′ such that ′ ′ F(α)ιg = ιgα : C → G C for every g ∈ G. In particular, if α : C → C′ is a monomorphism in Grp, that is, an injective morphism, then F(α): G C → G C′ is also clearly an injective mapping, hence a monomorphism in GGrp. Thus F preserves monomorphisms, so that U preserves injective objects. Therefore if H is an injective object in the category GGrp, then U(H) is an injective object in the category Grp, so U(H) = 1, hence H = 1.

4. Other categories related to G-Grp 4.1. The category G-Sdp of all G-semidirect products Recall that a group P is a of its normal subgroup H and its subgroup G, written P = H ⋊ G, if and only if there exists a homomorphism P → G with kernel H which is the identity 1702 M. J. A. PANIAGUA AND A. FACCHINI on G. Hence we can introduce a category GSdp of all Gsemidirect products dened as follows. Fix a group G. The objects of GSdp are the triples (P, α, β), where P is a group and α : G → P, β : P → G are morphisms such that the composite mapping βα is the identity automorphism idG of the group G. The morphisms from (P, α, β) to (P′, α′, β′) in GSdp are all group morphisms a: P → P′ which make the diagram

α β G −→ P −→ G ↓ a α′ β′ G −→ P′ −→ G commute. Here, the vertical arrows on the right and on the le are the identity morphism idG of G. Thus the diagram commutes if and only if aα = α′ and β′a = β. The category GSdp is also called the category of pointed objects over the Ggroup G [1, p. 28]. Let (H, ϕ) be a Ggroup. Let P be the semidirect product H ⋊ G, that is, the cartesian product H × G with the operation dened by (h1, g1)(h2, g2) = (h1ϕ(g1)(h2), g1g2). Let α : G → P and let β : P → G be dened by α(g) = (1, g) and β(h, g) = g. Then α and β are group morphisms and βα = idG, so that (P, α, β) is an object of GSdp. ′ ′ If (H, ϕ) and (H , ϕ ) are Ggroups, (P, α, β) and (P′, α′, β′) are the corresponding objects of GSdp e e and f : (H, ϕ) → (H′, ϕ′) is a Ggroup morphism, dene f : P = H ⋊ G → P′ = H′ ⋊ G by f (h, g) = e (f (h), g) for every (h, g) ∈ P = H ⋊ G. Then f is a morphism in the category GSdp. The assignments e (H, ϕ) → (P, α, β) and f → f dene a functor F : GGrp → GSdp. Conversely, let L: GSdp → GGrp be the functor that assigns to any object (P, α, β) of GSdp the pair (ker β, ϕ), where ϕ : G → Aut(ker β) is dened by ϕ(g)(k) = kα(g) = α(g)kα(g)−1 for every g ∈ G, k ∈ ker β. If a: (P, α, β) → (P′, α′, β′) is a morphism in GSdp, L assigns to a the morphism L(a): ker β → ker β′ dened by L(a)(k) = a(k) for every k ∈ ker β. Then L is a quasiinverse to F, so that the categories GGrp and GSdp are equivalent [1, Proposition 5.7]. In particular, the category

GSdp is semiabelian. We will now examine the correspondence, induced by the functor L: GSdp → GGrp, between of H in GGrp and subobjects of P := H ⋊ G in GSdp.

Proposition 4.1. Let H be a Ggroup. Then: (a) The Gsubgroups of H correspond to the subgroups of P containing α(G). (b) For any subgroup P′ of P containing α(G), we have that P′ = (P′ ∩ ker β) ⋊ G. (c) If a: (P, α, β) → (P′′, α′′, β′′) is a morphism in GSdp,P′ is the kernel of a in GSdp and N is the kernel of L(a): ker β → ker β′′, then P′ = N ⋊ G. (d) The Gsubgroups of H that are normal subgroups of H correspond to the subgroups P′ = N ⋊ G of P, where N ranges in the set of the normal subgroups of H.

4.2. The category of left A-module Hopf algebras In the theory of group representations, there is a standard technique of associating to every group representation ρ : G → GLn(k) (G a group, n ≥ 1 an , k a eld) a le module over the group algebra kG of dimension n as a vector space over k. This yields an equivalence between the category of all representations of a group G over a eld k and the full subcategory of kG−Mod whose objects are the le kGmodules nitedimensional over k. In the case of Ggroups (H, ϕ), we have group morphisms ϕ : G → Aut(H), and thus we are in a situation very similar to that of a group representation (i.e., group morphism) ρ : G → GLn(k). Hence it is tempting to try to nd an analog for Ggroups of the technique mentioned in the previous paragraph COMMUNICATIONS IN ALGEBRA® 1703 for group representations of G. The analog for Ggroups seems to be the following, in the setting of Hopf algebras over a xed eld k. Cf. [11, p. 40].

Denition 4.2. Let (A, mA, uA, A, εA, SA) be a Hopf algebra. A Hopf algebra (M, mM, uM, M, εM, SM) is a le Amodule Hopf algebra if (a) M is a le Amodule, i.e., a le module over the algebra (A, mA, uA), via A⊗M → M, h⊗a → ha. (b) mM, uM, M and εM are le Amodule morphisms.

We understand that the term we use, “le Amodule Hopf algebra", is somewhat misleading, because it could be interpreted as a Hopf algebra in the category of le Amodules (such a concept makes sense if A is a quasitriangular Hopf algebra). But we have not been able to nd a better name for our le A module Hopf algebras. Notice that a le Amodule Hopf algebra M is a Hopf algebra, a le Amodule such that M is a le Amodule algebra (i.e., mM and uM are Amodule morphisms) and a le Amodule coalgebra (i.e., M and εM are Amodule morphisms). Notice that from this it follows that SM is a le Amodule morphism. Indeed, it is easy to check that SMmM is a le inverse, and mM(I ⊗ SM) is a right inverse for mM in the algebra Hom(A ⊗ M, M) with the convolution product, so that mM(I ⊗ SM) = SMmM, i.e., SM is a a le Amodule morphism. We denote by A−ModH the category of all le Amodule Hopf algebras. The morphisms in this category are the mappings that preserve both the structure as a le Amodule and the structure as a Hopf algebra. Given any Ggroup H, the group algebras kG and kH are Hopf algebras and kH turns out to be a le kGmodule Hopf algebra simply extending by kbilinearity the le scalar multiplication G × H → H to a le scalar multiplication kG ⊗ kH → kH. In this way, we get a faithful functor k from the category GGrp to kG−ModH, which is dened on objects by H → kH, i.e., associates to each object H of GGrp the le kGmodule Hopf algebra kH. Thus the category GGrp turns out to be equivalent

to a subcategory of kG−ModH. Now, for every Hopf algebra A, the set G(A) := { g ∈ A | (g) = g ⊗ g and ε(g) = 1 } of group like elements of A is a group, and the elements of G(A) are kindependent [11, p. 4]. If A is a Hopf algebra and M is a le Amodule Hopf algebra, le multiplication λg : M → M by a grouplike element g ∈ G(A) is a Hopf algebra automorphism of M, hence it induces a group automorphism λg|G(M) of G(M). The assignment g → λg|G(M) is a group homomorphism G(A) → Aut(G(M)), so that G(M) is a G(A)group. In this way, we get a functor G : A−ModH → G(A)Grp. Cf. [11, Example 4.1.6]. For a xed group G and the Hopf algebra kG, we have that G(kG) = G. The group algebra functor k : GGrp → kG−ModH is a le adjoint to G : kG−ModH → GGrp, hence there are natural onetoone correspondences ∼ HomkG−ModH(kH, M) = HomGGrp(H, G(M)) for every Ggroup H and le kGmodule Hopf algebra M. In other words, every Ggroup morphism H → G(M) extends by klinearity to a le kGmodule Hopf algebra morphism kH → M. It is well known that groups can be thought of as Hopf algebras over the eld k with one element ∗. Groups can be axiomatized by the same diagrams as Hopf algebras, where G is taken to be a set instead of a vector space. It suces to take as mG the multiplication in the group, as uG : k → G the mapping ∗ → 1G, as comultiplication G the diagonal mapping, as counit εG the mapping 1G → ∗, and as antipode the inverse g → g−1. From this point of view, Ggroups turn out to be le kGmodule Hopf algebras.

5. Spectral category

Let H bea Ggroup and let SubG(H) be the set of all Gsubgroups of H. Partially order SubG(H) by set inclusion. It is easily seen that the intersection of any family of Gsubgroups of H is a Gsubgroup of H. Hence SubG(H) turns out to be a complete bounded . 1704 M. J. A. PANIAGUA AND A. FACCHINI

If H is a Ggroup and A is a Gsubgroup of H, we say that A is Gessential in H if, for every Gsubgroup B of H, A ∩ B = 1 implies B = 1. In this case, we will write A ≤Ge H.

′ ′ ′ ′ −1 ′ Lemma 5.1. (a) Iff : (H, ϕ) → (H , ϕ ) is a Ggroup morphism and A ≤Ge H , then f (A ) ≤Ge H. ′ ′ (b) If H is a Ggroup and A, A ≤Ge H, then A ∩ A ≤Ge H. ′ ′ (c) If A ≤Ge A and A ≤Ge H, then A ≤Ge H.

′ ′ ′ ′ Proof. Let) f : (H, ϕ → (H , ϕ ) be a Ggroup morphism and A ≤Ge H . Suppose B ≤G H and f −1(A′) ∩ B = 1. Then A′ ∩ f (B) = 1, because if b ∈ B and f (b) ∈ A′, then b ∈ f −1(A′), so that f −1(A′) ∩ B = 1 implies b = 1. From A′ ∩ f (B) = 1, it follows that f (B) = 1, because A′ is Gessential in H′. Then B ⊆ ker f ⊆ f −1(A′) and 1 = f −1(A′) ∩ B = B. This proves (a). The proofs of (b) and (c) are trivial.

Now we will present a construction similar to the construction of the spectral category of a Grothendieck category. Let G be a group and GGrp the category of all Ggroups. Dene a new category Spec(GGrp), called the spectral category of GGrp, as follows. The objects of Spec(GGrp) are the same ′ ′ as the objects of GGrp. Let H , H be two Ggroups. By Lemma 5.1(b), the set { Hi | i ∈ I} of all ′ ′ ′ ′ Gessential Gsubgroups of H is downward directed. If Hi, Hj are Gessential Gsubgroup of H and ′ ′ ′ ′ ′ ′ Hi ⊆ Hj, the embedding Hi ֒→ Hj induces a mapping HomG(Hj, H) → HomG(Hi, H). Thus we ′ ′ ′ obtain an upward directed family of sets HomG(Hi, H) and mappings HomG(Hj, H) → HomG(Hi, H). Set

Hom (H′, H) := lim Hom (H′, H), Spec(GGrp) −→ G i ′ ′ where Hi ranges in the set of all Gessential Gsubgroups of H . The composition in Spec(GGrp) is ′ → ℓ ′′ → ′ Grp dened as follows. If f : H H and : H H are morphisms in the category Spec(G ), then ′ ′ f is represented by a Ggroup morphism f : Hf → H and ℓ is represented by a Ggroup morphism ′ ′′ ′ ′ ′ ′′ ′′ ′−1 ′ ℓ : Hℓ → H for suitable Hf ≤Ge H and Hℓ ≤Ge H . By Lemma 5.1(a), ℓ (Hf ) is a Gessential ′′ ′−1 ′ ′′ Gsubgroup of Hℓ . By Lemma 5.1(c), ℓ (Hf ) is a Gessential Gsubgroup of H . The composite morphism f ℓ in Spec(GGrp) is the image in Hom (H′′, H) := lim Hom (H′′, H) of the Spec(GGrp) −→ G i composite homomorphism

′ ′ ′−1 ′ ℓ ′ f ℓ (Hf ) −→ Hf −→ H. There is a canonical functor P: GGrp → Spec (GGrp), which is the identity on objects and maps any Ggroup morphism f : H′ → H to its canonical image in Spec(GGrp).

Lemma 5.2. A Ggroup H is a null object in GGrp if and only if P(H) is a null object in Spec(GGrp).

Proof. As we have already remarked, the null objects in GGrp are exactly the trivial groups. It is also easily seen that if H is a Ggroup with one element, then P(H) is a null object in Spec(GGrp). Conversely, let H be a Ggroup with P(H) a null object. Then the identity mapping ι: H → H and the trivial morphism 0: H → H are Ggroup morphisms that coincide in the direct , hence there exists a Gessential Gsubgroup H′ of H such that the embedding H′ ֒→ H and the trivial morphism H′ → H coincide. This implies that H′ = 1. Thus 1 is a Gessential Gsubgroup of H. So 1 ∩ H = 1 implies H = 1, as we wanted to prove.

Since P(1) isthenullobjectinSpec(GGrp), the P(H) → P(H′) in Spec(GGrp) P(0) P(0) is the composite morphism P(H) −→ P(1) −→ P(H′), i.e., it is P(0): P(H) → P(H′), where 0 now denotes the zero morphism of H into H′. COMMUNICATIONS IN ALGEBRA® 1705

Hence we can consider kernels in Spec(GGrp), where the kernel of a morphism f : P(H) → P(H′) in Spec(GGrp) is the equalizer of f : P(H) → P(H′) and P(0): P(H) → P(H′). We leave to the reader to ′ ′ ′ check that if f : P(H) → P(H ) is a morphism in Spec(GGrp) represented by a morphism f : Hf → H ′ ′ ′ in GGrp, with H ≤Ge H, and ε : ker(f ) → H is the embedding, then the kernel of f : P(H) → P(H ) in Spec(GGrp) is P(ε). In the special case in which Hf = H, we get that:

Proposition 5.3. The functor P: GGrp → Spec(GGrp) preserves kernels, that is, is le exact.

The next lemma is easy.

Lemma 5.4. Let G be a group and H be a nontrivial Ggroup. The following two conditions are equivalent: (a) The intersection of any two nontrivial Gsubgroups of H is nontrivial. (b) Every nontrivial Gsubgroup of H is Gessential.

We say that a nontrivial Ggroup H is uniform if it satises the equivalent conditions of Lemma 5.4. Recall that a division monoid is a monoid M = 1 with zero such that U(M) = M \{0} [4]. For instance, if H is a Ggroup that is a nite , then the endomorphism monoid of H in the category GGrp and the endomorphism monoid of H in the category Grp are division .

Proposition 5.5. If H is a uniform Ggroup, then EndSpec(GGrp)(P(H)) is a division monoid.

Proof. It suces to show that every morphism

f ∈ EndSpec(GGrp)(P(H)) is either the zero morphism or an invertible element in EndSpec(GGrp)(P(H)). Now if f ∈ EndSpec(GGrp)

′ (P(H)), f is the image in the of a Ggroup morphism f : Hf → H for a suitable Hf ≤Ge H. The morphism f ′ is either injective or noninjective. If f ′ is noninjective, it has a nontrivial kernel ′ ′ ker(f ), which is a Gsubgroup of Hf , hence of H. But H is a uniform Ggroup, so that ker(f ) ≤Ge H. Thus the restriction of f ′ to the Gessential Gsubgroup ker(f ′) of H is zero, so that f = 0 in EndSpec(GGrp)(P(H)). ′ Hence, it remains to show that if f : Hf → H is injective, then f is invertible in EndSpec(GGrp)(P(H)). ′ ′ f (Hf ) Now if f : Hf → H is injective, then its corestriction f | : Hf → f (Hf ) is an isomorphism. Let ′ f (Hf ) −1 (f | ) be its inverse and ε : Hf → H be the embedding. Then it is easily seen that the inverse of f in EndSpec(GGrp)(P(H)) is represented by the composite homomorphism of the isomorphism ′ f (Hf ) −1 (f | ) : f (Hf ) → Hf and the embedding ε : Hf → H.

6. The dual construction It is possible to dualize the construction of the spectral category of a Grothendieck category [3]. Recall that in a category C, a monomorphism f : A → B is essential if for every object C and every morphism g : B → C, if gf is a monomorphism, then g is a monomorphism. Dually, an epimorphism f : A → B is superuous if for every object C and every morphism g : C → A, if fg is an epimorphism, then g is an epimorphism. Let G be a group. In the category of Ggroups, a monomorphism f : H → H′ is essential if and only if its image is a Gessential Gsubgroup of H′. Dually, an epimorphism f : H → H′ is superuous if and only if its kernel ker f is a Gsuperuous Gsubgroup of H, in the following sense. Let H be a Ggroup and A a Gsubgroup of H. We say that A is Gsuperuous in H if, for every G subgroup B of H, AB = H implies B = H. In this case, we will write A ≤Gs H. For any group H, the group of all inner automorphisms of H will be denoted by Inn(H). The group Inn(H) is a subgroup of Aut(H). 1706 M. J. A. PANIAGUA AND A. FACCHINI

′ ′ Lemma 6.1. (a) Iff : H → H is a Ggroup morphism and A ≤Gs H, then f (A) ≤Gs H . (b) If A ≤Gs B and B ≤Gs H, then A ≤Gs H. (c) If (H, ϕ) is a Ggroup, ϕ(G) ⊇ Inn(H),N ≤G M ≤G H are Gsubgroups of H, N ≤Gs H and M/N ≤Gs H/N, then M ≤Gs H.

Proof. For (a), let A be a Gsuperuous Gsubgroup of H. Let B′ be a Gsubgroup of H′ with f (A)B′ = H′. We must show that B′ = H′. We have that Af −1(B′) = H, because if h ∈ H, then f (h) = f (a)b′ for suitable a ∈ A and b′ ∈ B′, so that a−1h ∈ f −1(B′). Thus h ∈ Af −1(B′). From Af −1(B′) = H and the fact that A is Gsuperuous in H, it follows that f −1(B′) = H, hence f (f −1(B′)) = f (H). Thus B′ ⊇ f (H) ⊇ f (A). It follows that B′ = f (A)B′ = H′. This concludes the proof of (a). The proofs of (b) and (c) are easy and le to the reader. Notice that the hypothesis ϕ(G) ⊇ Inn(H) implies that every Gsubgroup of H is normal in H, so that it is possible to construct the quotient Ggroup H/N.

We now introduce two more categories. The rst category is the full subcategory CG of GGrp whose objects are all the Ggroups (H, ϕ) with ϕ(G) ⊇ Inn(H), where Inn(H) is the subgroup of Aut(H) of all ′ ′ inner automorphisms of H. So, if H is an object of CG and H is a Gsubgroup of H, then H is a normal subgroup of H and it is possible to construct the quotient Ggroup H/H′.

′ ′ ′ Lemma 6.2. If H is an object of CG and H is a Gsubgroup of H, then H and H/H are objects of CG.

C′ C The second category G we now introduce has the same objects as G, but its morphisms are dened as follows. The set of all Gsuperuous Gsubgroups of a Ggroup H is upward directed, because the product of any two Gsuperuous Gsubgroups of H is a Gsuperuous Gsubgroup of H. Let H′ and H be two Ggroups. For any two Gsuperuous Gsubgroups N1, N2 of H with N1 ⊆ N2, composition with ′ ′ the canonical projection H/N1 → H/N2 induces a mapping HomG(H , H/N1) → HomG(H , H/N2). ′ ′ Thus we get an upward directed family of sets HomG(H , H/N) and mappings HomG(H , H/N1) → ′ HomG(H , H/N2). Set ′ ′ HomC′ (H , H) := lim HomG(H , H/N), G −→ C′ where N ranges in the set of all Gsuperuous Gsubgroup of H. The composition in G is dened as ′ ′′ ′ C′ follows. If f : H → H and g : H → H are two morphisms in G, then f is represented by a G ′ ′ ′ ′′ ′ group morphism f : H → H/Nf and g is represented by a Ggroup morphism g : H → H /Ng ′ ′ for suitable Gsuperuous Gsubgroups Nf ≤Gs H and Ng ≤Gs H . Set f (Ng) := Mf ,g/Nf . Then ′ ′ Mf ,g is Gsuperuous in H (Lemma 6.1 ). The Ggroup morphism f : H → H/Nf induces a G ′′ ′ C′ group morphism f : H /Ng → H/Mf ,g, and the composite morphism fg in G is the image, in ′′ ′′ HomC′ (H , H) := lim HomG(H , H/N), of the composite mapping G −→ ′ ′′ ′′ g ′ f H −→ H /Ng −→ H/Mf ,g. C C′ There is a canonical functor Q: G → G which is the identity on objects and maps any Ggroup ′ ′ morphism f ∈ HomC (H , H) to its canonical image in HomC′ (H , H). G G

C C C′ Lemma 6.3. An object H of G is a null object in G if and only if Q(H) is a null object in G.

Proof. Clearly, the null objects of GGrp are the trivial groups. It is easily seen that if H is a trivial G C′ group, then Q(H) is a null object in G. Conversely, let H be a Ggroup such that Q(H) is a null object. Then the identity mapping ι: H → H and the trivial morphism 0: H → H are Ggroup morphisms that coincide in the direct limit, hence there exists a Gsuperuous Gsubgroup N of H such that the canonical projection H → H/N and the trivial morphism H → H/N coincide. Thus their kernels N and H are equal. Hence H is Gsuperuous in H, so H = 1. COMMUNICATIONS IN ALGEBRA® 1707

Lemma 6.4. The following conditions are equivalent for a Ggroup H = 1: (a) The product of any two proper Gsubgroups of H is a proper subgroup of H. (b) Every proper Gsubgroup of H is Gsuperuous in H.

A nontrivial Ggroup H is couniform if it satises the equivalent conditions of Lemma 6.4.

Lemma 6.5. The following conditions are equivalent for an object H of CG: (a) H is a couniform Ggroup. (b) For every N ≤G H,N = H, the quotient Ggroup H/N is indecomposable as a direct product in the category GGrp.

Proposition 6.6. bIf( HC ∈ O ) is a couniform Ggroup, then EndC′ (Q(H)) is a division monoid. G G

Proof. We must show that every morphism f ∈ EndC′ (Q(H)) is either the zero morphism or invertible. G ′ Now f ∈ EndC′ (Q(H)) is the image in the direct limit of a Ggroup morphism f : H → H/N for a G f ′ suitable Gsuperuous Gsubgroup Nf of H. The morphism f is either surjective or nonsurjective. ′ ′ If f is nonsurjective, then f (H) = M/Nf for some Nf ≤G M

If C and D are categories, we say that a functor F : C → D is local if, for every morphism f : C → C′ in C, if F(f ): F(C) → F(C′) is an isomorphism in D, then f is an isomorphism in C.

Theorem 6.7. For every group G, the product functor ′ P × Q: CG → Spec(GGrp) × CG is a local functor.

′ ′ Proof. Let f : H → H be a Ggroup morphism with H, H objects of CG, P(f ) an isomorphism in C′ Spec(GGrp) and Q(f ) an isomorphism in G. Since P(f ): P(H) → P(H′) is invertible in Spec(GGrp), there must exist a Ggroup morphism ′ ′ ′ ′ f : Hf → H, with Hf ≤Ge H , such that the image in Spec(GGrp) of the composite homomorphism ′ −1 ′ f ′ f f (Hf ) −→ Hf −→ H is the identity morphism of the object P(H) of Spec(GGrp). Thus there −1 ′ exists a Gessential Gsubgroup N of H, contained in f (Hf ), such that the composite homomorphism ′ f ′ f N −→ Hf −→ H is the embedding of N in H. Hence the restriction of f to N is injective, that is, N ∩ ker f = 1. Since N is Gessential in H, it follows that ker f = 1. This proves that f is injective. ′ C′ Dually, Q(f ): Q(H) → Q(H ) is an isomorphism in the category G, so that there is a Ggroup ′ ′ C′ morphism g : H → H/Ng, with Ng ≤Gs H, whose image in G is an inverse of Q(f ). Set f (Ng) := Mf ,g, ′ ′ ′ so that the Ggroup morphism f : H → H induces a Ggroup morphism f : H/Ng → H /Mf ,g ′ C′ and the composite morphism of Q(f ): Q(H) → Q(H ) and the image of g in G is the image in ′ ′ ′ ′ ′ HomC′ (H , H ) := lim HomG(H , H /N ) of the composite mapping G −→ ′ ′ ′ g f ′ H −→ H/Ng −→ H /Mf ,g. 1708 M. J. A. PANIAGUA AND A. FACCHINI

′ ′ Since this is the identity of Q(H ), there exists a Gsuperuous Gsubgroup M of H containing Mf ,g ′ ′ ′ g f ′ can ′ such that the composite mapping H −→ H/Ng −→ H /Mf ,g −→ H /M is the canonical projection ′ ′ ′′ ′ ′ H → H /M. It follows that the homomorphism f : H/Ng → H /M induced by f : H → H is an epimorphism. Thus f (H)M = H′. Since M is Gsuperuous in H′, we get that f (H) = H′, so that f is surjective. Thus the Ggroup morphism f is a , hence an isomorphism in CG.

7. Further properties of the objects in CG

In this section, we examine some further properties of the objects in CG, that is, the Ggroups (H, ϕ) with ϕ(G) ⊇ Inn(H).

Proposition 7.1. Let H be a Ggroup in CG. For every Gsubgroup N of H, there exists a Gsubgroup C of H with N ∩ C = 1 and NC ≤G−e H.

Notice that all Gsubgroups of H are normal in H, so that N ∩ C = 1 implies that NC = N × C is a direct product.

Proof. Let N bea Gsubgroup of H. Set

F := { X ≤G H | X ∩ N = 1 }. Then F is nonempty because it contains the trivial subgroup. Partially order F by inclusion. We can apply Zorn’s lemma, so there exists an element C maximal in F. It suces to show that NC = N × C is essential in H. Let X ≤G H, and assume that (N × C) ∩ X = 1. Then we have that the Gsubgroup

NCX = N × C × X of H is a direct product. Thus C × X ∈ F. By the maximality of C, it follows that X = 1.

A monoid M is regular provided that for every f ∈ M there exists g ∈ M such that fgf = f .

Proposition 7.2. For every Ggroup H in CG, the endomorphism monoid of P(H) in the category Spec(G Grp) is a regular monoid.

′ ′ Proof. Let f be an endomorphism of P(H). Then f is represented by a Ggroup morphism f : Hf → H ′ ′ ′ for some Gessential Gsubgroup Hf of H. Apply Proposition 7.1 to the Gsubgroup ker f of Hf . This is possible, because ker f ′ is a Gsubgroup of H, hence it is xed by the inner automorphisms induced by ′ ′ C ′ the elements of H, hence of Hf , so that ker f is an object of G. By Proposition 7.1, there exists a C ≤G Hf ′ ′ ′ ′ with C ∩ ker f = 1 and C ker f = C × ker f ≤G−e H. The restriction of f to C is an injective mapping of C into H, hence it induces an isomorphism f ′′ : C → f ′(C). By Proposition 7.1 again, there exists a G ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ subgroup C of H with f (C)∩C = 1 and f (C)C = f (C)×C ≤G−e H. Let g : f (C)×C → C×ker f be the Ggroup morphisms that is equal to (f ′′)−1 on f ′(C) and is zero on C′. Let g be the image of g′ in the endomorphism monoid of P(H). Then fgf = f .

We are ready to determine the monomorphisms in the categories Spec(GGrp) and CG.

Proposition 7.3. Let A, B be Ggroups, A′ a Gessential Gsubgroup of A and f ′ : A′ → B a Ggroup morphism. Then the image f : P(A) → P(B) of f ′ in Spec(GGrp) is a monomorphism in Spec(GGrp) if and only if f ′ is a monomorphism in GGrp (that is, if and only if f ′ is an injective mapping; cf. the paragraph between Lemma 2.1 and Theorem 2.2). COMMUNICATIONS IN ALGEBRA® 1709

Proof. ( ⇒) Assume that f is a monomorphism in Spec(GGrp). The kernel ker f ′ of f ′ is a Gsubgroup of A′. Let ε, 0: ker f ′ → A′ be the embedding and the zero Ggroup morphism, respectively. Then f ′ε = f ′0, so that fP(ε) = fP(0). It follows that P(ε) = P(0). Hence, there exists a Gessential G ′′ ′ ′′ ′′ ′ ′ subgroup A of ker f for which ε|A′′ = 0|A′′ . Thus A = 1. But A is Gessential in ker f , so ker f = 1, and f ′ is injective. (⇐) Let f ′ be a monomorphism in GGrp. Let ℓ, m: P(C) → P(A) be two morphisms in Spec(G ′ ′ ′ ′ Grp) with f ℓ = fm. Then ℓ and m are represented by Ggroup morphisms ℓ , m : C → A with C ≤G−e ′ ′ A′ ′ ′ A′ ′′ ′ C. From f ℓ = fm, it follows that f ◦(ℓ |C′′ ) = f ◦(m |C′′ ) for some C ≤G−e C. As f is a monomorphism, ′ ′ ′ A ′ A ′ ′′ ′ ′′ ℓ |C′′ = m |C′′ , so ℓ |C = m |C , hence ℓ = m. C′′ C Let G be the full subcategory of Spec(GGrp) whose objects are those of G.

′ ′ ′ Proposition 7.4. Let A and B be Ggroups in CG, let A be a Gessential Gsubgroup of A and f : A → B ′ C′′ C′′ be a Ggroup morphism. Then the image f : P(A) → P(B) of f in G is a monomorphism in G if and only if f ′ is a monomorphism in GGrp (that is, if and only if f ′ is an injective mapping).

Proof. ( ⇒) The proof is essentially the same as the proof of Proposition 7.3. It suces to notice that the ′ ′ kernel ker f of f is in CG by Lemma 6.2. (⇐) Every monomorphism in a category that is a morphism in a subcategory remains a monomor phism in the subcategory.

Clearly, the functor P: GGrp → Spec(GGrp) respects nite products. In the next theorem, we show another analogy of the category Spec(GGrp) with the spectral category of a Grothendieck category.

C′′ Theorem 7.5. Every of an object P(H) of G is a direct factor of P(H). That is, for every G ∼ subgroup N of a Ggroup H ∈ Ob(CG), there exists a Gsubgroup C of H such that P(H) = P(N) × P(C).

Proof. Let H bea Ggroup in CG. By Proposition 7.4, the subobjects of P(H) are of the form P(N) → P(H) for some Gsubgroup N of H. By Proposition 7.1, there is a Gsubgroup C of H with N ∩ C = 1 and NC ≤G−e H. Thus NC = N × C (Gsubgroups are normal subgroups) and P(H) = P(NC) = ∼ C′′ P(N × C) = P(N) × P(C) in G (equivalently, in Spec(GGrp)).

We conclude the paper with some remarks. As we have said in the Introduction, the original motivation of this work was that of giving a proper categorical setting to the researches in [4] and, more generally, to the investigations concerning directproduct decompositions of a group G as a direct product of nitely many indecomposable groups. The correct categorical setting seems to be the following. Fix a group G and construct the category GGrp of Ggroups, with its regular object G. Then all normal subgroups N of G and all quotient groups G/M of G become subobjects and quotient objects of the regular object G in the category GGrp. This is the case, in particular, for all direct factors of G, and hence GGrp is the proper category in which to study nite directproduct decompositions of G. Also, all these objects G, N and G/M belong to the category CG (Lemma 6.2), which is particularly satisfactory. The “normal homomorphisms” and the “normal endomorphisms” in [4], now become the morphisms and the endomorphisms of these objects in the categories GGrp and CG. Some results of [4], for example [4, Lemma 4.5], now become trivial, and the monoid (partial ring) NEnd(G) turns out to be the endomorphism monoid of the regular object G in the category GGrp. If H is a Ggroup in CG, the automorphisms of H in GGrp are central automorphisms of H (cf. the paragraph aer Remark 3.2). As a consequence, if H, L are isomorphic Ggroups in CG and α, β : H → L are two isomorphisms in GGrp, then there exists a Ggroup morphism ϕ : H → L such that ϕ(H) ⊆ ζ(L) and α = β + ϕ. 1710 M. J. A. PANIAGUA AND A. FACCHINI

By the Krull–Remak–Schmidt Theorem, given two decompositions G = G1 × × Gr = H1 × × Hs of a group G with the maximal and minimal conditions on normal subgroups as a product of indecomposable factors, then r = s and there exist a permutation σ of {1, 2, ... , r} and a central automorphism α of G such that α(Hi) = Gσ(i) for every i = 1, 2, ... , r [12, 3.3.8]. The existence of the central automorphism is a part of the theorem due to Robert Erich Remak (Berlin, 1888 Auschwitz, 1942). The reason of the presence of the central automorphism α in this statement corresponds exactly to the fact that the right categories for the study of group decompositions of the group G are the category GGrp and CG, where automorphisms are exactly the central automorphisms of G.

Acknowledgment

We would like to thank the referee for his useful suggestions.

Funding

The research of the second author was supported in part by Università di Padova (Progetto ex 60% “Anelli e categorie di moduli”) and Fondazione Cassa di Risparmio di Padova e Rovigo (Progetto di Eccellenza “Algebraic structures and their applications”).

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