Generators and closed classes of groups

Publ. Mat. 65 (2021), 431–457 DOI: 10.5565/PUBLMAT6522102 GENERATORS AND CLOSED CLASSES OF GROUPS Ram´ on Flores and Jos´ e L. Rodr´ ıguez Abstract: We show that, in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits, and extensions is also singly-gener- ated under isomorphisms and direct limits, and in particular is co-reflective. In this way, we extend to the group-theoretic framework the topological analogue proved by Chach´olski, Parent, and Stanley in 2004. We also establish several new relations between singly-generated closed classes. 2010 Mathematics Subject Classification: Primary: 20E34; Secondary: 20E22, 55P60. Key words: co-reflections, cellular covers, direct limits, extensions, radicals. 1. Introduction In a category of R-modules over a certain ring R, the study of the closure of different classes under different operations has been a hot research topic for decades (see [26] for a thorough approach). The operations include extensions, subgroups, quotients, etc. In the case of R=Z, we are dealing, of course, with operations in the category of abelian groups. There is another different but related problem. Given a collection D of objects in a class C, the smallest class that contains Dand is closed under certain operations is called the class generated by Dunder these operations. When this class is equal to C, it is said that Cis generated by Dunder the operations, and the elements of Dare called generators of C. In this situation, if Dconsists of only one object, the class Cis said to be singly-generated under the operations. Now the question is clear: given a class Cdefined by a certain property, is it possible to construct a generator for it? A standard reference for this topic is [1], and related problems have been recently treated in [10]. In this paper we will deal with different versions of these two problems in the category of groups. Rather than from the vast literature in Module and Category Theory related to these topics, our motivation and strategy The first author was partially supported by FEDER-MEC grants MTM2010- 20692 and MTM-2016-76453-C2-1-P. Second-named author was supported by grant MTM2016-76453-C2-2-P and grant FQM-213 of the Junta de Andaluc´ıa. 432 R. Flores, J. L. Rodr´ ıguez come from the field of Homotopy Theory. Since the nineties, Bousfield, Chach´olski, and Farjoun ([6], [13], [18]), among others, developed an ambitious program whose main goal was to classify spaces and spectra in terms of classes defined by certain operations, particularly homotopy direct limits (cellular classes) and fibrations (periodicity classes). Taking account of this framework, the fundamental group functor gives a natural (but nontrivial) way to understand closed classes of groups from the same viewpoint. We are mainly interested in three classes of operations in the class of groups, namely taking direct limits, constructing extensions from a given kernel and cokernel, and taking quotients. The relation between the closed classes under these operations has been already studied in [14] from a different point of view. We focus on singly-generated classes, and in fact our first main result is a generation result. Given a group A, we denote by C(A) the class generated by Aunder direct limits and isomorphisms, and by C(A) the class generated by Aunder direct limits, extensions, and isomorphisms. The elements of C(A) are usually called A-cellular groups, and the elements of C(A) are called A-acyclic groups. Then: Theorem 3.12. For any group A, there exists a group Bsuch that C(A) = C(B). Moreover, we are able to describe Bin terms of groups of the class C(A). This statement allows us to generalize results of [34], where the case in which the Schur multiplier of Ais trivial was studied, and also of [3] for the case of groups with trivial ordinary homology. It can also be seen as the group-theoretic version of the main theorem of [16], and it is in the spirit of the generators of radicals described in [11]; see Section 5. We remark that the proof of [16] cannot be translated to the category of groups, because the simplicial structure of the category of spaces is crucial in it. Section 4 is devoted to establish relations between the closed classes under the operations mentioned above. The hierarchy is the following: C(A)C(A) Cq(A)Ct(A) where Cq(A) is the closure of C(A) under quotients and Ct(A) is the closure of C(A) under quotients and extensions. All these classes turn out to be co-reflective. Furthermore, we find an example where the four classes are distinct (see Example 4.14). Generators and Closed Classes of Groups 433 The co-reflections associated to the classes C(A), Cq(A), and Ct(A) are given respectively by the A-cellular cover CA, the A-socle SA, and the A-radical TA, which are well-known notions in Group/Module Theory (see definitions in Section 2). However, for the class C(A), the existence of the corresponding co-reflection DAhad been previously established ([34]) only for groups for which there exists a two-dimensional Moore space, and this goal was achieved only by topological methods. This gap in the theory is filled in this paper making use of Theorem 3.12, that guarantees the existence of DAfor any group A. Roughly speaking, the philosophy is the following: given a group G, a singly-generated class C, and the co-reflection Fassociated to a certain operation, the kernel and the cokernel of the homomorphism F(G)→Gmeasure how far Gis from belonging to the class generated from Cunder the operation. We prove how the co-reflections define the corresponding closed classes, describe when possible the kernels, and discuss, for different instances of the generator, the (in)equality between the corresponding classes. In particular, we prove the following: Proposition 4.17. Suppose that for a group Athe class C(A)is closed under extensions, i.e. C(A) = C(A). Then Cq(A) = Ct(A). The converse of this result (stated as Question 4.19) is not true in general. Section 5 is devoted to give a counterexample that involves the Burnside idempotent radical and the Burnside group B(2, q). It is also an interesting question to find groups for which this converse is true. In fact, we are not even aware of an example of an abelian group A such that Cq(A) = Ct(A), but C(A)6=C(A) (or equivalently SA=TA and CA6=DA); see Proposition 4.16 and the subsequent discussion. All the classes that appear in this paper will be supposed to be closed under isomorphisms. Moreover, the axioms of ZFC will be assumed throughout the paper. 2. Co-reflections 2.1. Background on co-reflections. In this section we introduce the main notions that will be used throughout the rest of our paper. We will start with the classical categorical definition of co-reflection, that will be crucial for us. Definition 2.1. Given an inclusion of categories C0⊆C, where C0is full in C,C0is co-reflective in Cif the inclusion functor possesses a right adjoint (F, µ), called co-reflection. 434 R. Flores, J. L. Rodr´ ıguez A good survey about the topic from the categorical point of view can be found in [1]. We recall here some basic facts and notations that will be used later. (1) Given a co-reflective subcategory C0⊆C, the associated co-reflec- tion F:C→C0is unique up to natural isomorphism. (2) Objects Xsatisfying FX ∼ =Xare called F-colocal, and the morphisms X→Yinducing isomorphisms FX →FY are called F-colocal equivalences. (3) An object Zis F-colocal if and only if C(Z, X)∼ =C(Z, Y ) for all F-colocal equivalences X→Y, and also, a morphism X→Yis an F-colocal equivalence if and only if C(Z, X)∼ =C(Z, Y ) of all F-colocal objects Z. In particular, η:FX →Xis an F-colocal equivalence, and FX is F-colocal. (4) An inclusion C1⊆C2of two co-reflective subcategories induces a natural transformation τ:F1→F2between the corresponding coreflections, and viceversa. Furthermore, τis unique up to natural equivalence. It is well-known that co-reflections preserve finite products in the categories of modules and spaces. Next we prove an analogous statement for arbitrary groups, as we are not aware of a proof of it in the literature (see [19, Lemma 2.3] for a proof in a concrete instance). Proposition 2.2. Let F:Groups → C be a co-reflection in the category of groups, with µG:FG →Gits counit. Then, F(G×H)∼ =FG ×FH. Proof: By (3) above, we have that µG×µH:FG ×FH →G×H is an F-colocal equivalence, as Hom(Z, F G ×FH)≃Hom(Z, FG)× Hom(Z, F H)≃Hom(Z, G)×Hom(Z, H)≃Hom(Z, G ×H), for all F-colocal groups Z, and the composition is induced by µG×µH. To prove that FG ×FH is F-colocal, set an ordering FH ={xi:i < α} for some ordinal α, and consider the linear sequence FG∗F H F G∗FH · · · FG∗F H F G∗FH · · · , f0f1fifi+1 where fi(g) = xigx−1 ifor g∈FG, and fi(h) = hfor h∈FH. The colimit is isomorphic to the product FG ×FH. Thus FG ×FH is a direct limit of F-colocal groups, and hence also F-colocal. Uniqueness, up to isomorphism, yields F(G×H)∼ =FG ×FH. This property will be useful in particular in Example 4.14. Generators and Closed Classes of Groups 435 2.2. Group-theoretic constructions. The following are well-known examples of group-theoretic constructions. As we shall see in Section 4, they give rise to different co-reflections over interesting (co-reflective) subcategories of the category of groups. Definition 2.3. Let Aand Hbe groups. (1) The A-socle SAHis the subgroup of Hgenerated by the images of all the homomorphisms A→H. (2) The A-radical TAHof His the normal subgroup of Hcharacterized by the following property: H/TAHis the largest quotient of Hsuch that Hom(A, H/TAH) = 0. Groups Gthat satisfy Hom(A, G)=0 are called A-reduced (or A-null). The projection ϕ:HH/TAH is called the A-reduction (or A-nullification), where H/TAHis A- reduced and ϕinduces a bijection Hom(H/TAH, G)≃Hom(H, G), for all A-reduced groups G. (3) The A-cellular cover of His the unique group CAH∈ C(A), up to isomorphism, that can be constructed from copies of A by iterated direct limits, and such that there exists a homomorphism CAH→Hwhich induces a bijection Hom(A, CAH)≃ Hom(A, H) (an A-equivalence). Observe that this is equivalent to Hom(G, CAH)≃Hom(G, H) being a bijection for every G∈ C(A). Remark 2.4.Note that similar definitions are possible if we change A by a collection Aof groups indexed over a certain set I. In this case, the generator would be the free product of isomorphism classes of elements of A. Moreover, it is possible to give versions of the socle and the radical relative to a class of epimorphisms of groups. See [1] for a general treatment of these topics. On the other hand, the A-reduction is a particular example of a localization functor Lfwith respect to the constant homomorphism A→ {1}(see for example [9] for background on localization of groups and spaces). For the convenience of the reader, we recall more constructive versions of the radical and the cellular cover. The A-radical TAHis the union TA(H) = SiTi(H) of a chain starting at T0(H) = SA(H) and defined inductively by Ti+1(H)/T i(H) = SA(H/T i(H)), and Tλ(H) = SαTα(H) if λis a limit ordinal. This process yields an A-reduced group H/TAH. Furthermore, if ϕ:H→Gis any homomorphism with G A-reduced, then it factors through H→H/TAH→G. To see this observe that, if SAH→H→Gis trivial, ϕfactors through H→H/SA(H)→G. Inductively, if i+ 1 is a successor ordinal, since SA(H/TiH)→H→G is trivial, then ϕfactors through H/T i+1H→G. If λis an ordinal limit, H/TαH→Gfor α < λ, and this yields H/TλH→G. 436 R. Flores, J. L. Rodr´ ıguez In turn, the A-cellular cover CAHis constructed in Theorem 1 of [15] in the following explicit way. Let Aand Hbe two groups, CH 0=?A→HH the free product indexed over all homomorphisms A→H, ev: CH 0→H the evaluation homomorphism, and KH 0the kernel of the evaluation. Then, CH A(H)∼ =(CH 0/[CH 0, KH 0])/TA(KH 0/[CH 0, KH 0]). An inductive description of CAHcan be found in Section 3 of [34]. Next we will check that previous constructions are functorial, and then they provide an important source of co-reflections (see Section 4): Proposition 2.5. Given a group A, the A-socle, the A-radical, and the A-cellular cover define functors in the category of groups. Proof: Consider a group homomorphism f:G→H. To check functoriality we should verify that frestricts to the corresponding A-socles, f:SAG→SAH. Let xbe an element of SAG. Then there exist a finite set of homomorphisms φi:A→G,i≤n, and elements xi∈Asuch that x=φ1(x1)· · · φn(xn). Hence, f(x)=(f◦φ1(x1)) · · · (f◦φ1(xn)), which implies f(x)∈SAH. By induction along the increasing chain TAH=SiTiH, one can prove functoriality of all the Tiand hence of TA. Let f:G→Hbe a group homomorphism. If i+ 1 is a successor ordinal, and assuming that frestricts to TiH→TiG, we obtain that finduces a homomorphism on the quotients H/TiH→G/TiG. Now the functoriality of SGyields the restriction SA(H/TiH)→SA(G/TiG), and therefore the restriction Ti+1H→Ti+1G. For a limit ordinal λ, if frestricts to TαH→TαGfor each α < λ, then it can be extended to TλH→TλG. Note that the previous argument also yields the functoriality of the quotients H→H/TiHand hence that of the A-reduction homomorphism H→H/TAH. To check functoriality of the A-cellular cover, consider a homomorphism f:H→H0. Following the notation of the previous construction of CAH,finduces a homomorphism ˜ f:CH 0→CH0 0and a commutative diagram KH 0CH 0H KH0 0CH0 0H0 ˜ f whose rows are group extensions. Now, as ˜ f(KH 0)⊆KH0 0, it is clear that ˜ f(and then f) induces a homomorphism CH 0/[CH 0, KH 0]→CH0 0/[C0H 0, KH0 0]. Now the result follows from the functoriality of the radical, taking account of the previous description of CA(H). Generators and Closed Classes of Groups 437 Remark 2.6.A thorough analysis of cellular covers and their functorial behaviour is undertaken in Section 2 of [5]. Concerning cellular covers, a different characterization was obtained by J. Scherer and the second author: Theorem 2.7 ([34, Theorem 3.7]).Given groups Aand H, there is a central extension (2.1) K CA(H)SA(H) such that Hom(Aab, K)=0and the natural map Hom(A, H)→H2(A;K) is trivial. Moreover, this extension is initial with respect to all the extensions with these two properties. 2.3. Covers of spaces. We will also need to review some constructions in the category of spaces. Analogously to groups, given a pointed space M, the cellular class C(M) generated by Mis the smallest class that contains Mand is closed under homotopy equivalences and pointed homotopy colimits. The corresponding augmentation CWMX→Xfor a pointed space X, called the M-cellular cover of X, was explicitly described by Chach´olski in his thesis [13] in the following way: consider the homotopy cofiber sequence (2.2) W[M,X]∗M X CofMX, ev where ev is the evaluation map. Denote by PMthe M-nullification functor, i.e. the homotopical localization with respect to the constant map M→ ∗ (see [18, Chapter 1]). Then, for any connected space X, CWMXfits into the following homotopy fiber sequence (2.3) CWMX X PΣM(CofMX). In [34, Theorem 3.3] the relation between cellular covers of groups and spaces was described: if Mis a 2-dimensional complex with π1M=A and Xis an Eilenberg–MacLane space K(H, 1), then π1CWMX=CAH for every group H. 3. Generating classes of groups Given a group A, the class C(A) is the smallest class that contains Aand is closed under direct limits and extensions. The elements of this class will be called A-acyclic groups. The main goal of this section will be to show that such a class is always singly-generated using only direct limits or, in other words, that this class possesses a cellular generator: 438 R. Flores, J. L. Rodr´ ıguez Definition 3.1. Given a class Cof groups closed under direct limits, a cellular generator for the class Cis a group Bsuch that C=C(B). In our context, Cwill usually be the class C(A) for some group A. These cellular generators are not unique and, in fact, several models for particular instances of Ahave been proposed in the literature ([3], [34]). We now present a general construction, which is inspired in the methodology of Chach´olski–Parent–Stanley [16], that solved successfully the same problem in the category of topological spaces. Concretely, we adopt to the group-theoretic context their idea of small space, and follow the general scheme of their proof; however, it should be remarked that the translation is highly nontrivial, as their methods are homotopical in nature and rely in the simplicial structure of space, and we deal with a completely different context. We start with a set-theoretic definition: Definition 3.2. For a given infinite regular uncountable cardinal λa group His called λ-small (or small with respect to λ) if #H < λ. In the sequel, unless stated otherwise, we will assume that λis regular infinite. Next we show the easiest example of λ-smallness. Example 3.3. For λ=ℵ1,λ-small means finite or countably infinite. Using the previous definition, we can define the class C[λ]. Definition 3.4. Given a class Cof groups closed under direct limits, we denote by C[λ] the subclass of Cdefined by the following property: Hbelongs to C[λ] if for each λ-small group K, every group homomorphism K→Hfactors as K→C→Hfor some λ-small group Cin C. Note that the definition of C[λ] only depends on Cand the cardinal λ. We remark that the group Cdepends on the homomorphism K→H (and in particular depends on Kand H). The following characterization says that we can restrict ourselves to subgroups of H. Proposition 3.5. Let λbe an infinite regular cardinal. Then, a group H is in C[λ]if and only if for each λ-small subgroup S < H, there exists a λ-small subgroup Cin Csuch that S < C < H. Proof: Let K→Hbe any group homomorphism with K λ-small. The image ϕ(K) is a λ-small subgroup of H. Assuming the necessary condition, there exists a λ-small subgroup C∈ C such that ϕ(K)< C < H. Hence, K→ϕ(K)→C→His the desired factorization. Generators and Closed Classes of Groups 439 For the converse, suppose that Sis any λ-small subgroup of H, and let S−→ C0 φ0 −→ Hbe a factorization of the inclusion S ,→H, where C0is λ-small and belongs to C. Then φ0(C0)< H is λ-small and S≤φ0(C0). Denote K0=φ0(C0). If S=K0, then Sis a retract of C0, hence it belongs to C. Therefore, we can take C=Sand we are done. If S < K0, we can proceed in the same way with the inclusion K0< H, obtaining a factorization K0−→ C1 φ1 −→ H. Iterating the process if necessary, we obtain a string of homomorphisms such that for every ithere is a strict inclusion KiKi+1: S C0K0C1K1C2K2· · · , where Ciis λ-small in Cfor every i. Passing to the countable limit we get that the composition colimi<ω Kicolimi<ω Ci+1 colimi<ω Ki+1 is the identity, so colim Kiis a retract of colim Ci, hence it belongs to C. Thus, C= colim Kiis the desired subgroup of H. The following proposition determines the scope of the class C[λ]. Proposition 3.6. Let λbe an infinite regular cardinal. A group His in C[λ]if and only if it is isomorphic to a direct limit of λ-small groups in C. Proof: By the previous proposition, for each λ-small subgroup Sof H there exists a λ-small subgroup in Csuch that S < C < H. Hence, His isomorphic to the direct system of all λ-small subgroups of Hthat belong to C. Conversely, let Hbe a direct limit of λ-small groups in Cand let us prove that Hbelongs to C[λ]. Following [12, 4.3], and taking into account that π1commutes with colimits, every direct limit of groups can be obtained as an iteration of limits over linear categories (chains) and push-outs, so we need to check the statement in these two cases. Observe that Hbelongs to C. Let Hbe the direct limit of the λ-small groups {Gα}over a linear category I, i.e. G1→G2→ · · · → H,κ-indexed for a certain ordinal κ. As His a regular cardinal, it makes sense to distinguish between the following two cases. If Iis λ-filtered, a standard compactness argument states that every homomorphism K→Hfrom a λ-small group factors through Gαfor a certain α. On the other hand, if Iis not λ-filtered, this means that the cardinality of κis smaller than λ, and then the diagram 446 R. Flores, J. L. Rodr´ ıguez Lemma 4.10. Let Abe a group and Ba cellular generator of C(A). Then SB=TA. Proof: Both SBand TAare co-reflections over the class Ct(A). The result follows from uniqueness of co-reflections. Corollary 4.11. For every group A, we have DA=DATA. Proof: It is well-known that CA=CASAfor every group A. Now, if Bis a cellular generator of C(A), CB=DAand SB=TA. So we are done. In particular, in the computation of DAHfor a group Hit is enough to take as input the A-radical of H. The following proposition provides concrete descriptions of DA. Proposition 4.12. Let Hand Gbe finite nontrivial groups, with H simple, |H||G|, and such that there is exactly one subgroup of Gthat is isomorphic to H. Then DHG=H, and G/H does not belong to C(H). Proof: As His simple and normal in G,SHG=THG=H. Thus, DHG=DHTHG=DHH=H, as His clearly H-acyclic. Moreover, if G/H belonged to C(H), then this would also be the case for G, and that is impossible because DHG6= G. Note that this result gives information for classes C(A) with H2A6= 0. For example, if H=An,G= Σn, and n≥5, the proposition implies that DAnΣn=An, and hence neither Σnnor Z/2 can be constructed out of Anby direct limits and extensions. The groups for which H2A6= 0 are beyond the scope of Theorem 4.5, as the existence of the Moore space implies that H2Ashould be trivial there. Remark 4.13.We note that the use of the word “acyclic” in our context comes from Homotopy Theory, and especially from Bousfield, who thoroughly studied the classes of spaces for which a homology theory vanishes. We would like to emphasize that the acyclic cover owes its name to the fact that, in the cases studied by Rodr´ıguez–Scherer in [34], it is the fundamental group of the homotopy fibre of a localization; in general this group may have nontrivial ordinary homology, so it is not acyclic in the usual sense of Group Homology. 4.2. Comparing the classes. From now on we will concentrate in measuring the difference between the classes above. Fix a group A, and let Fbe one of the four co-reflections. Then given a group G, FG measures in general how close the group Gis from belonging to Generators and Closed Classes of Groups 447 the corresponding class. Moreover, the homomorphisms SAG→G, TAG→G, and SAG→TAGare always injective, and it is a consequence of [34, Proposition 3.1] and Corollary 4.7 above that CAG→SAGand DAG→TAGare always surjective. Hence, we have a commutative diagram (4.3) CAG SAG DAG TAG which generalizes diagram (4.2) above to the case of a general A. Note that in general, CAG→DAGis neither surjective nor injective. Using an example, we start by showing that the four inclusions of classes in diagram (4.1) can be strict. Of course, the previous results imply that two such classes are identical if and only if their associated co-reflections are the same. Example 4.14. Consider Gthe direct product of the Mathieu group M12 with the cyclic group Z/9. We use Proposition 2.2 to compute the value of the co-reflections over this group. As the socle and the radical are always normal subgroups and M12 is simple and contains elements of order 3, we have SZ/3G=M12 ×Z/3 and TZ/3G=G. Moreover, taking into account that H2(M12) = Z/2 (see for example Section 11 in [5]) we have that CZ/3M12 =f M12, the universal cover of M12. Observe that the alternative CZ/3M12 =Z/2×M12 is not possible as Z/2×M12 is not Z/3-cellular. Hence, CZ/3G=f M12 ×Z/3. On the other hand, DZ/3G=f M12 ×Z/9, as Gis Z/3-constructible. Next we will discuss when the socular class Cq(A) is closed under extensions, and hence equal to the radical class Ct(A). For example, if A=Z/p for some prime p, it is known that there are many groups G such that G/SZ/pGhas again p-torsion, and the same happens if we change Z/p by ∗j≥1Z/pj. This contrasts with the case of abelian groups, in which for A=Lj≥1Z/pj,TAN=SANfor every abelian group. When Ais the additive group of the rational numbers, the situation is similar. If Nis abelian, SQN=TQN, because the image of every homomorphism Q→Nis divisible and then splits out of N. However, if Nis not abelian, the equality is not true in general. We have found no example of this situation in the literature, so we describe one in the sequel: 448 R. Flores, J. L. Rodr´ ıguez Example 4.15. Let Gbe the push-out of inclusions Q←-Z,→Z[1/p]. Following [29, Chapter 5], a presentation of the group (Q,+) is given by: {x1, x2, x3, . . . |xn n=xn−1for n≥2}. If we look at the rationals as a subgroup of the reals, the fraction 1 m! corresponds to the generator xmfor every m≥1. Analogously, a presentation of Z[1/p] is: hy1, y2, y3, . . . |yp n=yn−1,for n≥2i. Now identifying Z[1/p] with the irreducible fractions of Rwith denominator a power of p, the fraction 1 pmcorresponds to ym. Using Seifert–van Kampen’s theorem, we obtain that a presentation for the group Gis given by: hx1, y1, x2, y2, . . . |xn n=xn−1, yp n=yn−1, x1=y1,for n≥2i. Note that the amalgamation is reflected in the relation x1=y1. Consider now the normal subgroup Nof Ggenerated by the copy of Q inside Ggiven by the generators {xi}. This group is generated by conjugates of this copy of Q, and therefore N < SQG. On the other hand, as Gis torsion-free, every homomorphism φ:Q→Gis injective, and then Im φis isomorphic to Q. Now invoking a result of Moldavanskii ([31]; see also [30, p. 229]), and taking into account that Q≮Z[1/p], we obtain that Im φ<Nfor every φ:Q→G, and this already implies that N=SQG. Next we will describe the quotient G/N. A presentation of this group is obtained by trivializing the generators {xi}in the previous presentation of G(which in particular implies trivializing y1); so we get (after a renumbering of generators): hy1, y2, . . . |yp 1= 1, yp n=yn−1,for n≥2i. This is a classical presentation of the quasicyclic group Z/p∞, being the generator of every Z/pnidentified with yn. As there exists an epimorphism Q→Z/p∞, we conclude that G=TQG, and then SQGis strictly contained in TQG, as claimed. In fact, G=TQG, as the only nontrivial quotients of the quasicyclic group are isomorphic to it. Using amalgams Z[1/q]←-Z,→Z[1/p], with q6=p, and taking into account that there are always nontrivial homomorphisms Z[1/q]→ Z/p∞, it can be proved in an analogous way that SZ[1/q]6=TZ[1/q]for every prime q. Moreover, similar arguments can be used to prove that for a nontrivial set of primes J, the co-reflection SZ[J−1]is never equal to TZ[J−1]. In particular we obtain: Generators and Closed Classes of Groups 449 Proposition 4.16. If Ais a nontrivial additive subgroup of Q,SA=TA if and only if A=Z. Proof: By the work of Beaumont–Zuckerman [2], the additive subgroups of Qare the rational subgroups of rank one. As stated in Example 6.2 of [34] (see also [21]), the only nontrivial subgroups Aamong them for which SAcan be equal to TAare those of the shape A=Z[J−1] for a set of primes J. But if Jis non-empty and p∈J, the same argument of Example 4.15 above proves that the Z[J−1]-socle of the pushout Z[J−1]←-Z,→Z[1/p] is strictly contained in its Z[J−1]-radical, which in turn equals the whole group. On the other hand, if J=∅, every group coincide with its Z-socle, and then SZ=TZ. So we are done. The moral here is that it is not easy to find examples for which SA= TA. However, there are at least two ways of constructing such examples. The first family arises in the context of varieties of groups. Consider a set of equations and the verbal subgroup Wdefined by these equations in the free group F∞. It is proved in [11] that for every Wthere exists a locally free group Asuch that the homomorphisms of Winto any group G identify the W-perfect radical and, in particular, this fact guarantees that SA=TA. The inspiring (and first) “generator” of this kind was the group constructed by Berrick–Casacuberta in [3], and a closely related group will be the crucial ingredient of Proposition 5.3 below. The second family of examples can be constructed as a byproduct of Theorem 3.12, taking account of the following two results: Proposition 4.17. Suppose that for a group Athe class C(A)is closed under extensions, i.e. C(A) = C(A). Then Cq(A) = Ct(A). Proof: In this case, CAG=DAGfor every group G, and then there is a diagram DAG SAG DAG TAG Id j As jshould be an epimorphism, and since it is always a monomorphism, it is an isomorphism. Corollary 4.18. Let Abe a group. Then for any cellular generator B of the class C(A)we have SB=TB. Recall that by Theorem 3.12, it is always possible to construct such B out of any given A, and this provides many examples of (big) groups for which SA=TA. It is natural and interesting to ask if there is a converse of the previous proposition, so we consider the following problem: 450 R. Flores, J. L. Rodr´ ıguez Question 4.19. If TA=SAfor a certain group A, is it true that C(A) = C(A)? We will see in the next section that the general answer is negative. Before, we will discuss in which cases such an answer can be positive. The following result is easy to prove and gives a sufficient condition. Proposition 4.20. Let Abe a group, and assume that C(A) = Cq(A) and SA=TA. Then C(A) = C(A). Proof: As the members of the classes are defined precisely as the groups for which the corresponding co-reflection is the identity, SA=TAimplies Cq(A) = Ct(A), and then by hypothesis C(A) = Ct(A). But the inclusions C(A)⊆ C(A)⊆ Ct(A) always hold, so the result follows. It can be seen that the condition of the proposition holds for the universal group of Berrick–Casacuberta. However, the following example shows that the conditions of the previous proposition are not necessary in general for cellular generators of closures under extensions: Example 4.21. Given a prime p, consider a cellular generator Bof the class C(Z/p). If C(B) = Cq(B), this would imply that CB=SB, and equivalently DZ/p =TZ/p. But this is not true in general, as can be deduced for instance from Example 4.14. Observe that we obtain in particular that C(B) = C(B)6=Cq(B) = Ct(B) in this case. In general, the relation between the group co-reflections CAand CB (= DA) has been investigated when H2A= 0, and it is quite well understood when Ais a cyclic group or a subring of Q(see for example [34]). As seen above, the latter is not useful in our context, as SA6=TAfor these groups. However, in the next section, we will take profit of some features of the case H2A= 0 to describe a counterexample for Question 4.19. We emphasize that in this proof we will not need the results of Section 3. 5. The Burnside radical and a free Burnside group Recall from [11] that the Burnside idempotent radical for a prime p of a group Gis the largest subgroup H < G which is generated by p-powers of its elements. As is established in [11, Theorem 3.3], given any group Gand a prime p, its corresponding Burnside radical can be always generated by the images of homomorphisms of a group F, called the generator of the radical. This group will be the key object in the proof of Proposition 5.3 below, so we will compile below some properties of it that will be useful in the sequel, in particular in the homology computations of Lemma 5.2. Generators and Closed Classes of Groups 451 Proposition 5.1. The following statements hold for the group F: (i) Fgenerates the Burnside radical. (ii) Fis an infinite free product of groups Fn, where nruns over all the non-decreasing sequences of integers. (iii) The groups Fnare countable. (iv) Every Fnis a directed system of finitely generated free groups {Fn,r}, and in particular is locally free and torsion-free. (v) For every nand r < s,rank Fn,r <rank Fn,s. (vi) The image of every bonding homomorphism φr:Fn,r →Fn,r+1 is always contained in a product of p-powers of generators of Fn,r+1. (vii) For every nand r, there exist distinguished systems {xn,r}of generators of Fn,r, such that for every i6=j,φr(xi,r)and φr(xj,r)are spanned by disjoint subsets of the system {xn,r+1}. (viii) Fis locally free and torsion-free. Proof: Statement (i) is exactly Theorem 3.3 in [11], and the remaining ones are established in the course of the proof of the mentioned result. We adopt the notation from there. The definition of Fas a free product of (ii) appears in the second paragraph of that proof. Just after it, the construction of Fnas a directed system of free groups Fn,r is undertaken, with a precise description of the generator of each group and the bonding homomorphisms φ. The fact (established in the proof of Theorem 3.3 in [11]) that the groups Fn,r are finitely generated implies (iii) and the first part of (iv), and the construction as a directed system of free groups gives locally freeness and torsion-freeness of Fn (rest of statement (iv)). The labeling of the generators in the second paragraph of the proof of the mentioned result implies the inequalities of (v), and the fact that in this case the only nontrivial word that defines the variety is w=xpgives (vi), by the definition of the product in that paragraph. As all the symbols that appear in the definition of xrappear again in the generators that span φr(xr) we have (vii). Finally, (viii) is a straight consequence of (iv), as being locally free is closed under free products. Now we are ready to compute the ordinary homology of F. Lemma 5.2. The first homology group H1(F)is a free module over Z[1/p] of infinite rank, and HiF= 0 if i > 1. Proof: We first compute the homology of every Fn. As these groups are filtered colimits of free groups, HiFn= 0 if i > 1. Moreover, the 452 R. Flores, J. L. Rodr´ ıguez construction of Fnin [11] as a telescope of homomorphisms that sends generators to products of p-powers (see statement (vi) of the previous proposition) implies that the first homology group is the direct limit of a system ZM I1 ZM I2 Z· · · , where: •Every direct sum is taken over a finite set of indexes (statement (iv) above). •If j < k, then Ij< Ik(statement (v)). •For every k, the homomorphism LIkZ→LIk+1 Ztakes the |Ik|components of the leftmost group to the first |Ik|of the rightmost group in their order, and the homomorphism is, up to a change of basis, multiplication by p(statements (vi) and (vii)). Hence, the direct system that defines H1Fnis a countable direct sum of copies of the system Zp −→ Zp −→ Zp −→ · · · and then this homology group is a free module over Z[1/p]. Finally, writing Fas the colimit (under the inclusion) of all the subgroups of the shape Fi1∗ · · · ∗ Finand taking the corresponding limit in homology, the result follows. Note that the rank of H1(F) as a free module over Z[1/p] is uncountable. Now given a prime q > 1075,q6=p, consider the free Burnside group B(2, p) in two generators. It is a group of exponent q, and by Corollary 31.2 in [32] the bound q > 1075 implies that its second homology group is free abelian in a countable number of generators. In the sequel we will denote B(2, q) by G. Proposition 5.3. For the group Fabove defined, the equality SF=TF holds, but CF6=DF. Proof: As Fgenerates the idempotent radical associated to a variety of groups, SF=TF. We will check now that CF6=DF. As Ghas exponent q and qis coprime to p, multiplication by pis an isomorphism. Then Gis equal to its idempotent Burnside radical, and then G=SFG=TFG. Hence, to prove that CFG6=DFGit is enough to check that the kernels of DFG→Gand CFG→Gare not isomorphic. We first compute the kernel of the epimorphism DFG→G. Since Fis locally free, there exists a 2-dimensional Moore space M(F, 1) and therefore, we can use Theorem 4.5 directly to compute such kernel. Consider the Chach´olski cofibration (5.1) _ [M,K(G,1)]∗ M K(G, 1) Y, Generators and Closed Classes of Groups 453 where Ydenotes the homotopy cofiber. We proved just after Theorem 4.5 that the kernel of DFG→Gis isomorphic to π2(PMY), which in turn is the Ext-p-completion of the group π2Yin the primes for which Fab is uniquely p-divisible ([7, Theorem 7.5]). By Lemma 5.2, Fab =H1Fis a Z[1/p]-free module, and hence we only need to complete in the prime p. Applying Mayer–Vietoris to (5.1) and the fact that π(Y) = H2(Y), as Yis simply-connected, we obtain an extension (5.2) A π2Y R. f Here A=H2(G) is free abelian in a countable number of generators and Ris the kernel of the homomorphism H1(W[M,K(G,1)]∗M)→H1K(G, 1). In this case R=LJZ[1/p] is a free Z[1/p]-module for some set Jof indexes, because H1(W[M,K(G,1)]∗M) is and Z[1/p] is a principal ideal domain. Taking now account of the fact that Hom(Z/p∞,Z) = 0 and [8, VI.2.5], we obtain another extension: Ext(Z/p∞, A) Ext(Z/p∞, π2Y) Ext(Z/p∞, R). As the Ext-p-completion of a free Z[1/p]-module is trivial, then so is the term on the right-hand side, and the homomorphism on the lefthand side is an isomorphism. Now observe that again by [8, VI.2.5], the extension Z,→AAgiven by the inclusion of a factor induces another extension Ext(Z/p∞,Z) Ext(Z/p∞, A) Ext(Z/p∞, A). As Ext(Z/p∞,Z) = Z∧ pwe obtain that the kernel of DFG→Gcontains at least a copy of the p-adic integers. Next we will prove that CFG6=DFG. In order to do so, we check that the kernel of CFG→Gcannot contain a subgroup isomorphic to Z∧ p. Again from the argument after Theorem 4.5 we have that this kernel is isomorphic to π2(PΣMY), and this is the quotient of π2Yby its F-radical T, which is in turn equal to the Z[1/p]-radical as π2Yis abelian ([11, Section 3]). Let us describe Tand (π2Y)/T more explicitly. For short, denote by S=SZ[1/p]π2Y, the Z[1/p]-socle of π2Y. Consider all the Z[1/p]-submodules of Rof rank one such that the inclusion Z[1/p],→Rlifts to π2Y. After a change of basis, we may assume that there is a set I < J (that can be empty) such that (a) the module R decomposes as a direct sum R=RI⊕RJ\I, being RI=LIZ[1/p] and RJ\I=LJ\IZ[1/p], and (b) every homomorphism Z[1/p]→RIlifts to π2Y. 454 R. Flores, J. L. Rodr´ ıguez Next we will prove S∼ =RI, via the restriction of fin (5.2). First, observe that the elements of Sare precisely the elements of π2Ythat are uniquely p-divisible, and every such element is in the image of some homomorphism Z[1/p]→π2Y. Now for every homomorphism g:Z[1/p]→ π2Ythe composition f◦gis again nontrivial; see 3F.11 in [27], for example, taking into account that Adoes not contain p-divisible elements. In particular, gis a lift of a homomorphism Z[1/p]→RI. Hence, the restriction of fto the socle takes values in RI. Moreover, if x∈RI, then xis in the image of some homomorphism Z[1/p]→RI, so by lifting this homomorphism we obtain a preimage of xin the socle, and f|Sis then surjective. To check injectivity observe that, fixing a homomorphism Z[1/p]→RI, the lifting to the socle is unique, because again by 3F.11 in [27] the induced homomorphism Hom(Z[1/p], π2Y)→ Hom(Z[1/p], RI) is injective. Then, the only homomorphism Z[1/p]→S that lifts the trivial one is trivial, and then f(x) = 0 implies x= 0 for x∈S. So we have proved that fmaps isomorphically the socle S onto RI. Now we have a commutative diagram, where rows and columns are exact: {1}S RI A π2Y R A(π2Y)/S RJ\I ≃ ≃ f Observe that (π2Y)/S has no uniquely p-divisible elements; otherwise, some homomorphism Z[1/p]→RJ\Icould be lifted to that group and this is not possible by the definition of I. This implies that S=T. Now it is immediate that (π2Y)/T cannot contain a copy of Z∧ p, taking into account for example that the p-adic integers admit nontrivial homomorphisms from Z[1/q] (qa prime different from p) and (π2Y)/T does not. Hence DFG6=CFGand we are done. Remark 5.4.In the previous proof we do not know if (π2Y)/T is a free abelian group, because we are unable to show if the aforementioned extension that defines π2Ysplits or not. Recall that Ext(Z[1/p],Z) = Z∧ p/Z is uncountable, so there is an uncountable number of abelian extensions of Z[1/p] by Z, but only the trivial extension of Zby Z[1/p] splits. We conclude by pointing out that the arguments of Proposition 5.3 are also valid when approaching CFand DFfor the perfect groups defined by Generators and Closed Classes of Groups 455 Berrick–Miller in [4] for which the Schur multiplier is free abelian. Moreover, mutatis mutandis, they may also be used to describe the Z/p-acyclic cover of these Burnside groups of high exponent. This is related with recent results about cellular covers of Burnside groups ([25], [28] [22], [23]), that in particular have been useful to solve some conjectures by Farjoun [17]. On the other hand, note that we have approached the description of the acyclic and cellular covers of a non-nilpotent group with respect to an uncountable non-nilpotent group. This kind of problems has been widely studied in the recent literature, but usually under restrictive hypotheses of finiteness or nilpotency of the groups involved. See for example [5], [19], [20], [24]. Acknowledgments. We warmly thank J´erˆome Scherer for several discussions about the paper and for his thorough revision of the manuscript, which has led to a substantial improvement of the presentation. We also thank Pedro Guil and Fernando Muro for useful discussions. The first author expresses his gratitude to the Department of Mathematics of the University of Almer´ıa for their kind hospitality. We thank the referee for her/his detailed and sharp report. References [1] J. Ad´ amek and J. Rosick´ y,“Locally Presentable and Accessible Categories”, London Mathematical Society Lecture Note Series 189, Cambridge University Press, Cambridge, 1994. DOI: 10.1017/CBO9780511600579. [2] R. A. Beaumont and H. S. Zuckerman, A characterization of the subgroups of the additive rationals, Pacific J. Math. 1(2) (1951), 169–177. DOI: 10.2140/ pjm.1951.1.169. [3] A. J. Berrick and C. Casacuberta, A universal space for plus-constructions, Topology 38(3)(1999), 467–477. DOI: 10.1016/S0040-9383(97)00073-6. [4] A. J. 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