Accessible Published by De Gruyter August 12, 2020

The main decomposition of finite-dimensional protori

Wayne Lewis, Peter Loth and Adolf Mader
From the journal Journal of Group Theory

Abstract

A protorus is a compact connected abelian group of finite dimension. We use a result on finite-rank torsion-free abelian groups and Pontryagin duality to considerably generalize a well-known factorization of a finite-dimensional protorus into a product of a torus and a torus free complementary factor. We also classify direct products of protori of dimension 1 by means of canonical “type” subgroups. In addition, we produce the duals of some fundamental theorems of discrete abelian groups.

1 Introduction

All groups in this article are abelian, and “group” means “abelian group”. Pontryagin duality is a duality on the category LCA of locally compact groups and restricts to a duality between the category of discrete abelian groups and the category of compact abelian groups; see [5, Theorem 7.63, page 358], [5, Proposition 7.5 (i), page 303], [4, Theorem (24.3), page 377], [4, Theorem (23.17), page 362]. Being dual categories, one might expect parallel theories developing, but this is not the case at all. Probably motivated by the basis theorem for finite abelian groups, the main interest in abelian group theory has been on direct decompositions, indecomposable groups and classification. In topological group theory, the topology adds many more notions to the algebraic concepts, and direct products, the concept dual to direct sums, are more or less a side issue.

We are particularly concerned with the duality between the category TFFR of discrete torsion-free groups of finite rank and the category of protori, i.e., compact connected groups of finite dimension induced by the Pontryagin duality. The prime examples of indecomposable torsion-free groups are the groups of rank 1. These are classified up to isomorphism by their types (see below). The duals of rank-1 groups are the protori of dimension 1, called solenoids. These are not even discussed in [5], while [4, (25.3), page 403] contains a fairly involved representation. A simple description of solenoids and their properties is contained in [10]; see Theorem 3.2. In the discrete category, the next step was to consider completely decomposable groups, i.e., groups isomorphic to direct sums of rank-1 groups. These have been classified up to isomorphism by cardinal invariants involving “type subgroups” ([3, Theorem 3.5, pages 415, 416]). The dual concept is that of fully factorable groups, i.e., direct products of solenoids. These have been classified up to category isomorphism by Loth [11] using the dual concept of types, called patterns, which were introduced earlier by Leopold [8, 9] when studying the duals of Butler groups. In Theorem 3.11, we establish the classification of fully factorable groups up to topological isomorphism in terms of critical type sets. Instead of using patterns as in [11, Corollary 2.4] and [6, Corollary 7.5.4, page 296], we refer directly to types, obtaining a result that is closer to Baer’s original approach in [2]. Beyond that, the news is bad about direct decompositions of torsion-free groups. There exist an abundance of indecomposable torsion-free groups, even some of arbitrarily large cardinality [3, Theorem 4.9, page 435], and the decompositions of groups in TFFR can be quite “pathological”. The most striking and attractive result in that direction is [3, Theorem 5.2, page 439]. However, there also is a positive result whose basic idea is to peel off well-understood summands from a group in TFFR and then concentrate on the complementary summand. The “main decomposition” of torsion-free groups of finite rank says that every such group is the direct sum of a completely decomposable summand and a complementary summand that has no completely decomposable direct summands, and this decomposition has strong uniqueness properties [13, Theorem 2.5]. Section 4 contains the main factorization of protori, Theorem 4.5, that, for protori, vastly generalizes [5, Corollary 8.47, page 415]. We also dualize a number of substantial results from discrete group theory made possible by using the new “type subgroups” of compact groups.

2 Notation and background

As usual, denotes the set of positive integers, is the set (or group) of all integers, is the set (or group) of rational numbers and denotes the set (or group or field) of real numbers. By , we denote the set of all primes. The group of p-adic integers is denoted by ^p, and ^=p^p is the group of profinite integers.

In the category of topological groups, one has to distinguish between purely algebraic homomorphisms and both algebraic and continuous morphisms. We denote by Hom(G,H) the group of algebraic homomorphisms, and by cHom(G,H), we denote the subgroup of continuous homomorphisms. In the following, AB means that A,B are algebraically isomorphic, while GtH means that the topological groups G,H are topologically isomorphic, i.e., isomorphic in the category of topological groups. The Pontryagin dual of a discrete group A is

A=Hom(A,𝕋)

equipped with the compact-open topology, and the Pontryagin dual of a compact group G is the discrete group G=cHom(G,𝕋), where 𝕋:=/.

For any topological group G, the construct 𝔏(G):=cHom(,G) is its Lie algebra. It is well known that 𝔏(G) is a real topological vector space via the stipulation (rf)(x):=f(rx), where f𝔏(G) and r,x, and with the topology of uniform convergence on compact sets [5, Definition 5.7, page 117, Proposition 7.36, page 340]. We define the dimension dim(G) of G by

dim(G)=dim𝔏(G).

For a compact group G of positive dimension, dim(G)=rk(G); see [5, Theorem 8.22, page 390]. The identity component of G is denoted by G0. All topological groups are assumed to be Hausdorff.

Remark 2.1.

Let G be a compact group. Then the following are equivalent.

  1. (1)

    dim(G)=0.

  2. (1)

    G is totally disconnected.

  3. (1)

    G is a torsion group.

  4. (1)

    rk(G)=0.

Proof.

(1) (2): [5, Proposition 9.46, page 490].

(2) (3): [5, Corollary 8.5, page 377].

(3) (4): trivial. ∎

2.1 Short exact sequences

Let G,H,K be topological groups. The sequence E:H𝑓G𝑔K is exact if

  • E is exact as a sequence in the category of discrete groups, i.e., f and g are homomorphisms, f is injective, g is surjective and Im(f)=Ker(g),

  • f and g are continuous.

Lemma 2.2.

The following statements hold.

  1. (1)

    Let E:A𝑓B𝑔C be an exact sequence of discrete groups. Then

    E:CgBfA

    is exact as a sequence of topological groups, where C,B,A are compact groups. Furthermore, the maps g and f are open maps onto their images.

  2. (1)

    Let E:H𝑓G𝑔K be an exact sequence of topological groups, and assume further that H,G,K are compact groups. Then

    E:KgGfH

    is exact in the category of discrete groups.

Proof.

(1) is a consequence of Pontryagin duality, that A=Hom(A,𝕋) with the compact-open topology and that 𝕋 is a divisible (injective) group. The claim that g,f are open maps onto their images follows from the open mapping theorem [5, EA1.21, page 704].

(2) Here we use [15, Theorem 2.1]. This theorem requires “proper” maps, i.e., continuous maps that are open maps onto their images. The maps f and g are proper because H and G are compact so that the open mapping theorem applies. ∎

2.2 Direct sums and products

We will also have to dualize direct sums of discrete groups and products of compact groups.

Lemma 2.3.

The following statements hold.

  1. (1)

    Let A=iIAi be a direct sum of discrete groups. Then AtiIAi, the topology being the product topology.

  2. (1)

    Let G=iIGi be the topological product of the compact groups Gi. Then GtiIGi in the category of discrete groups.

Proof.

(1) [5, Proposition 1.17, page 17].

(2) By (1), we have

(iIGi)tiIGitiIGi,

hence dualizing again iIGit(iIGi). ∎

Pontryagin duality is a duality of the category LCA of locally compact groups with itself. Discrete groups and compact groups are locally compact. Concepts and facts valid for locally compact groups are true for both discrete and compact groups, and formulating them in LCA avoids the necessity for separate statements.

Recall that a closed subgroup K of an LCA group G is said to split from G if G=KL (algebraic direct sum) for some closed subgroup L of G such that the map ϕ:K×LG:(k,l)h+k is a topological isomorphism. Given G=KL, the open mapping theorem shows that ϕ is automatically a topological isomorphism whenever G is compact. Therefore, we have the following corollary.

Corollary 2.4.

Let K be a closed subgroup of a compact group G. Then K splits from G if and only if G=KL for some closed subgroup L of G.

For an LCA group G and a subgroup K, let (G,K) or K denote the annihilator of K in G. We will use a simple observation that can be found in [1] or [6].

Lemma 2.5.

Let K be a closed subgroup of an LCA group G. Then K splits from G if and only if (G,K) splits from G.

3 Completely decomposable groups and fully factorable compact groups

Recall that a discrete group is completely decomposable if it is a direct sum of rank-1 groups, and dually, a compact group is fully factorable if it is the direct product of solenoids.

3.1 Types, rank-1 groups and solenoids

Torsion-free groups of rank 1 are well understood and can be classified by means of “types”. A height sequence is a sequence (h2,h3,,hp,) indexed by with entries non-negative integers or . Given a height sequence (hp), then

1php|p

is an additive subgroup of containing , a rational group, that is a rank-1 group. Here a generator 1p stands for the set {1pnn}. Every rank-1 group A is isomorphic to a rational group, and a rational group is easily seen to be generated by fractions 1php, where 0hp, and this produces a height sequence (hp). This indicates how every rank-1 group is associated with a height sequence and conversely. Two height sequences (hp) and (kp) are equivalent if hp= if and only if kp= and the integer entries of (hp) and (kp) differ at most at finitely many places.

A type is an equivalence class of height sequences. The type tp(A) of a rank-1 group A is the type containing the height sequence associated with A. For types σ and τ, we write στ if there are height sequences (hp)σ and (kp)τ such that (hp)(kp). An old result says that two rank-1 groups are isomorphic if and only if their types are equal [3, Theorem 1.1, page 411]. For a solenoid Σ=A, we define its type by tp(Σ)=tp(A).

Proposition 3.1 ([10, Proposition 3.2]).

Let A,B be rank-1 torsion-free groups with tp(A)=σ and tp(B)=τ, and let Σ,Υ be solenoids with tp(Σ)=σ and tp(Υ)=τ.

  1. (1)

    If fHom(A,B) and f0, then f is injective.

  2. (1)

    If gcHom(Υ,Σ) and g0, then g is surjective.

  3. (1)

    Hom(A,B)0στcHom(Υ,Σ)0.

  4. (1)

    ABσ=τΥtΣ.

We now review the explicit description of solenoids given in [10, Theorem 1.1].

Theorem 3.2 ([10, Theorem 1.1]).

Let τ=[(,hp,)] be a type, and let Στ be a solenoid of type τ. Let Z^(p)=Z^p, let Z^(pn) denote the cyclic group of order pn if nN, and let Z^(p0)={0}. Then

Στt(p,hp0^(php))×(𝐮,-1)

where

𝐮=(,1p,)hp0^(php).

In particular,

  1. (a)

    for ω=tp()=[(,,)],

    Σω=t(p^p)×(𝐮,-1)t^×(𝐮,-1);
  2. (b)

    for τ=tp(1p)=[(,0,,0,)],

    Στ=(1p)t^p×(1p,-1);
  3. (c)

    for σ=tp(1p|p)=[(,1,)],

    Σσt(p(p))×(𝐮,-1);
  4. (d)

    for ζ=tp()=[(1,0,)],

    Σζt^(2)×(12,-1)t𝕋.

We will choose a rank-1 group of type τ, say a rational group, and call it Sτ. Furthermore, we set Στ=Sτ. In particular, we choose Sζ= and Sω=. This will be convenient in Section 4.

3.2 Type subgroups and fully factorable groups

Completely decomposable groups are classified up to isomorphism by cardinal invariants [3, Chapter 12.1, Theorem 1.1, Theorem 3.5]. The classification uses the so-called type subgroups that are defined as follows. Let A be any torsion-free group and aA. Then a is contained in a maximal rank-1 subgroup a*A of A, namely the purification of the subgroup a of A generated by a. In fact, the identity a*A/a=tor(A/a) defines a*A; in particular, A/a*A is torsion-free. Then the type of a in A, tpA(a), is defined to be the type of a*A.

One then defines

  • A(τ):=aAtpA(a)τ, a pure subgroup of A,

  • A*(τ):=aAtpA(a)>τ that is not pure in general,

  • A(τ):=(A*(τ))*A, the purification of A*(τ) in A.

These “type subgroups” Ax(τ) (where x is either absent, or x=* or x=) are functorial subgroups, i.e., for all fHom(A1,A2), we have f(A1x(τ))A2x(τ); in particular, the type subgroups are fully invariant subgroups.

Let A be completely decomposable. Then A=ρTcr(A)Aρ, where Aρ is the direct sum of rank-1 summands of type ρ. It is easy to see that A(τ)=ρτAρ, A*(τ)=A(τ)=ρ>τAρ and AτA(τ)/A*(τ). This also shows that Tcr(A) is an invariant of the group A, called the “critical typeset” of A. The main result, due to Reinhold Baer, says that two completely decomposable groups A1 and A2 are isomorphic if and only if, for all τ, we have A1(τ)A1*(τ)A2(τ)A2*(τ).

For a discrete torsion-free group A, let 𝒮(A) be the set of all pure rank-1 subgroups of A. By ins, we denote the insertion map. Its dual (adjoint) in LCA is the restriction map denoted restr.

Lemma 3.3.

Let A be a discrete torsion-free group and G=A its dual group.

  1. (1)

    𝒮(A)=𝒮(A)=A. If S,S𝒮(A) and SS0, then S=S.

  2. (1)

    Let S𝒮(A). Then SinsAϕSA/S, ϕS the natural epimorphism, is exact with A/S torsion-free. The dual sequence

    (A/S)ϕSGrestrS

    is exact in the category of topological groups, φS is an embedding morphism, restr is a quotient morphism. Furthermore, ϕS((A/S)) is a closed connected subgroup of G, S is a solenoid and tp(S)=tp(S).

Proof.

(1) Every element aA is contained in S=a*A. If 0aSS, then S=a*A=S.

(2) Routine. ∎

Lemma 3.3 motivates the next definition. For a compact connected group G, let 𝒞(G) be the set of all connected closed subgroups C of G such that G/C is a solenoid. Let G=A for some torsion-free discrete group A. In establishing the connection between 𝒮(A) and 𝒞(G) using the annihilator mechanism [5, Theorem 7.64, page 359], we make use of Lemma 3.4.

Lemma 3.4.

The following statements hold.

  1. (1)

    Let KinsLϕKL/K be an exact sequence in LCA, where ins is the insertion morphism and ϕK is the natural quotient morphism. Then the dual sequence

    (L/K)ϕKLrestrK

    is an exact sequence in LCA, ϕK is an embedding morphism and restr is the restriction morphism. Furthermore,

    ϕK((L/K))=(L/K)ϕK=(L,K)=K.
  2. (1)

    Suppose that L is discrete, K is a rank- 1 subgroup of L and L/K is torsion-free. Then K is a solenoid and ϕK((L/K)) is a compact and connected subgroup of L.

  3. (1)

    Suppose that L is compact and K is a compact connected subgroup of L such that L/K is a solenoid. Then ϕK(L/K) is a rank- 1 subgroup of the discrete group L and K is torsion-free and discrete.

Proof.

(1) It is clear that

(L/K)ϕKLrestrK

is as claimed. It is easy to see that restr(f)=0 if and only if fS. Hence fS if and only if fKer(restr)=ϕK((L/K))=(L/K)ϕS.

(2) The subgroup ϕK((L/K))t(L/K) is compact and connected since (L/K) is the dual of a torsion-free discrete group and L/ϕS((A/S))K is a solenoid.

(3) The group ϕK((L/K))t(L/K) is a rank-1 subgroup of L since (L/K) is the dual of a solenoid. The quotient group L/ϕK((L/K))tK is discrete and torsion-free as K is the dual of the compact connected group K. ∎

Proposition 3.5.

Let A be a discrete torsion-free group and G=A its dual. Then

σ:𝒮(A)𝒞(G):σ(S)=S=(G,S)

is a bijective map. Thus C(G)={(G,S)|SS(A)}.

Proof.

By [5, Theorem 7.64 (v), page 360], σ(H)=H is an antiisomorphism of the lattice of closed subgroups of A and the lattice of closed subgroups of G. Let S𝒮(A). By Lemma 3.4 (2), σ(S)=S𝒞(G). Hence the restriction of σ to 𝒮(A) maps into 𝒞(G), and we have an injective map σ:𝒮(A)𝒞(G). This map is also surjective because, given C𝒞(G), by Lemma 3.4 (3), we get C𝒮(A) and σ(C)=C=C (see [5, Theorem 7.64 (4), page 361]). ∎

In analogy to the type subgroups of A, we define “type subgroups” of the compact group G as follows:

  • G((τ)):={CC𝒞(G),tp(G/C)τ},

  • G*((τ)):={CC𝒞(G),tp(G/C)>τ},

  • G((τ)):=(G*((τ)))0, the identity component of G*((τ)).

Remark 3.6.

Let G be a compact group, τ a type, and let

𝒞(G)(τ)={C𝒞(G)tp(G/C)τ}.

There is a well-defined morphism of topological groups

σ:G{G/CC𝒞(G)(τ)}:σ(g)=(,g+C,)C𝒞(G)(τ)

and Ker(σ)=G((τ)). Hence G/G((τ)) is embedded in {G/CC𝒞(G)(τ)}.

The following useful fact can be found in [9].

Lemma 3.7.

If H is a subgroup of a torsion-free group A, then

(A,H*A)=(A,H)0.

Theorem 3.8.

Let A be torsion-free, G=A and τ any type. Then the following statements are true.

  1. (1)

    G((τ))=A(τ)=(G,A(τ))=(G,G(τ)) and G((τ)) is a compact connected subgroup of G.

  2. (1)

    G*((τ))=A*(τ)=(G,A*(τ))=(G,(G)*(τ)). The group G*((τ)) is a compact subgroup of G, but need not be connected.

  3. (1)

    G((τ))=A(τ)=(G,A(τ))=(G,(G)(τ)) and G((τ)) is a compact connected subgroup of G.

  4. (1)

    A(τ)tGG((τ)), A*(τ)tGG*((τ)) and A(τ)tGG((τ)).

There is no loss of generality in assuming that G=A for some torsion-free group A, but it is true in general that

G((τ))=(G,G(τ)),G*((τ))=(G,(G)*(τ)),G((τ))=(G,(G)(τ)).

Proof.

We first consider the exact sequences

Ax(τ)insAϕτA/Ax(τ),

where x is either absent, or x=* or x=, and ϕτ is the quotient morphism. Then we have the exact sequence of compact groups

(A/Ax(τ))ϕτGrestr(Ax(τ)),

where, by Lemma 3.4 (1) with L=A and K=Ax(τ), we have

(Ax(τ))=(G,Ax(τ))t(A/Ax(τ)).

We have the topological isomorphism ηA:AA=G:ηA(a)(χ)=χ(a). So, in particular, ηA(Ax(τ))=(G)x(τ). Hence

(G,(G)x(τ))=(G,ηA(Ax(τ)))={gGηA(Ax(τ))(g)=0}={gGfor allaAx(τ):ηA(a)(g)=g(a)=0}=(G,Ax(τ)).

(1) We apply [5, Theorem 7.64 (7), page 360] and Proposition 3.5 to get

A(τ)=(G,A(τ))=(G,{S𝒮(A)tp(S)τ})={SS𝒮(A),tp(S)τ}={CC𝒞(G),tp(G/C)τ}=G((τ)).

We have that A/A(τ) is torsion-free, so A(τ)t(A/A(τ)) is both compact and connected.

(2) We argue as in (1), but A*(τ) need not be pure, i.e., A/A*(τ) need not be torsion-free, and hence G*((τ)) need not be connected.

(3) Lemma 3.7 shows that (G,A(τ))=(G,(A*(τ))*A)=(G,A*(τ))0. By (2), this group coincides with (G*((τ)))0 which equals G((τ)).

(4) These isomorphisms are immediate consequences of our exact sequences using that (A/Ax(τ))t(G,Ax(τ))=Gx((τ)). ∎

A functorial subgroup is a functor F assigning to each group X a subgroup F(X) in such a way that, for every morphism f:XY, it is true that

f(F(X))F(Y).

This notion is due to B. Charles [3, page 35]. We will show that the type subgroups are functorial subgroups. This follows from a general lemma.

Lemma 3.9.

Suppose that F is a functorial subgroup on the category of discrete groups. Let GLCA. Then G(G,F(G)) is a functorial subgroup on LCA.

Proof.

Let f:GH be a morphism of LCA groups. Then f:HG is a homomorphism of discrete groups, and by hypothesis, f(F(H)F(G). It follows that

f(G,F(G))={f(g)for allχF(G):χ(g)=0}{f(g)for allξf(F(H)):ξ(g)=0}={f(g)for allηF(H):(ηf)(g)=0}{hHfor allηF(H):η(h)=0}=(H,F(H).

Corollary 3.10.

The type subgroups G((τ)), G*((τ)) and G((τ)) are functorial.

We can now state and prove the classification of fully factorable compact groups. A fully factorable compact group G is by definition a product of solenoids with no intrinsic order of factors. We can therefore collect the factors of type τ to get a factor Gτ that is the product of all solenoid factors of type τ of the given factorization of G. We obtain the “homogeneous factorization” G=τTcr(G)Gτ, where Tcr(G)={τGτ0} is the “critical typeset” of G.

Theorem 3.11.

Let G be a fully factorable compact group and G=τTcr(G)Gτ, where Gτ is a product of solenoids all of type τ. Then GτtG((τ))/G*((τ)), i.e., the isomorphism class of Gτ is independent of the particular factorization of G.

(*) Two fully factorable compact groups G1 and G2 are isomorphic as topological groups if and only if, for all types τ, it is true that

G1((τ))/G1*((τ))tG2((τ))/G2*((τ)).

Proof.

Without loss of generality, G=A for some completely decomposable group A=τTcr(A)Aτ so that G=τTcr(G)Gτ with Gτ=Aτ. We have commutative diagrams with natural maps and exact rows and columns as follows:

A(τ)insAϕAA(τ)insλA*(τ)insAψAA*(τ)insA(τ)A*(τ)  (AA(τ))ϕGrestr(A(τ))λrestr(AA*(τ))ψGrestr(A*(τ))restr(A(τ)A*(τ)).

It follows that Gτ=Aτt(A(τ)A*(τ))tG*((τ))G((τ)). The last expression is an invariant of G, and hence, in any homogeneous factorization of G, the factors Gτ are unique up to topological isomorphism. The remaining claim (*) is an immediate consequence. ∎

Remark 3.12.

Let G=τTcr(G)Gτ be a fully factorable topological group. Then the following statements are true.

  1. (1)

    G*((τ))=G((τ)).

  2. (1)

    GτtG((τ))/G*((τ)) is determined up to topological isomorphism by its type and the cardinality of its solenoid factors.

On the basis of these results, we consider fully factorable compact groups to be well understood.

4 The main factorization G=K×T

The comprehensive treatise [5] contains the following result.

Theorem 4.1.

Let G be a compact group such that G is countable. Then G is metric and GtT×K, where K is torus-free and T is a characteristic maximal torus. In particular, cHom(T,K)=0.

Proof.

[5, Theorem 8.45, page 414] and [5, Corollary 8.47, page 415]. ∎

Theorem 4.1 says that G factors with a well-understood factor T and an unknown factor K.

It is well known that the injectives in the category of compact abelian groups are the tori [5, Theorem 8.78, page 436], and as such, sub-tori will be direct factors. The interesting part is the existence of a maximal subtorus, which implies that K is “torus-free”. This in turn means that cHom(𝕋,K)=0 because, 𝕋 being a solenoid, any non-zero morphism in cHom(𝕋,K) would be an embedding and contradict the assumption that T is the maximal torus of G. Trivially, cHom(𝕋,K)=0 means that K is torus-free.

In the category of discrete abelian groups, the injectives are the divisible groups, and the projectives are the free groups. Every group A contains a maximal divisible group and decomposes as A=AdivB, where Adiv is the maximal divisible subgroup and B is a “reduced” complement, containing no non-zero divisible subgroup. A group B is reduced if and only if Hom(,B)=0. Moreover, Adiv is a functorial subgroup, in particular fully invariant. There are usually many reduced complements. In fact, there is a lemma providing a good description of possible complements in a decomposition.

Lemma 4.2 ([7, Lemma 0.4, page 3]).

Let M=AB be a direct decomposition of R-modules. Denote by C the set of all direct complements of A in M. Then

Hom(B,A)𝒞:φBφ:={φ(b)+bbB}

defines a bijective mapping and

φ+1:BBφ:bφ(b)+b

is an isomorphism.

Although formulated for modules, Lemma 4.2 applies to topological abelian groups. As Adiv is injective, the group Hom(B,Adiv) is large as a rule, and there are many different complements B. It can happen that Hom(B,Adiv)=0 and the decomposition A=AdivB is unique, e.g., if Adiv is torsion-free and B is a torsion group.

The structure of divisible groups is well known: they are of isomorphism type

(𝔞)p(p)(𝔞p).

Dualizing, we obtain that the projective compact groups are of isomorphism type (Σω)𝔞×p(^p)𝔞p in agreement with [5, Theorem 8.78, page 436], and every compact abelian group has a decomposition G=Gproj×H, where Gproj is unique and, as a rule, there are many complements H.

The projective groups in the category of discrete abelian groups are the free groups isomorphic to (𝔞). Hence the injectives in the category of compact groups are the tori 𝕋𝔞 in agreement with [5, Theorem 8.78, page 436]. If A/B is free, then B is a direct summand of A. If A/B1 and A/B2 are free quotients, it is in general not clear how the summands Bi are related. When A is countable, there is a theorem due to K. Stein [3, Corollary 8.3, page 114] that evidently extends to any mixed A provided that A/tor(A) is countable.

Theorem 4.3.

A group A such that A/tor(A) is countable can be decomposed as A=FN, where F is free and N={Ker(f)fHom(A,Z)} is the unique largest subgroup of A with Hom(N,Z)=0. Furthermore, N is a functorial subgroup of A.

Proof.

Let KA with Hom(K,)=0, and let fHom(A,). Then we have fKHom(K,)=0, so KKer(f) and KN.

Let A,B be groups with A/tor(A),B/tor(B) countable and ϕHom(A,B). We have the decompositions A=FANA and B=FBNB. Let π:BFB be the projection belonging to the decomposition B=FBNB. Then

(πϕ)(NA)FB.

As FB is free, (πϕ)(NA)=0; hence ϕ(NA)Ker(π)=NB. ∎

By duality, we obtain a slight generalization of [5, Corollary 8.47 (i), page 415].

Theorem 4.4.

Let G be a compact group such that G/tor(G) is countable (equivalently, the identity component G0 of G is metrizable). Then G=T×N, where TtTa is the unique maximal torus of G and N is a subgroup with cHom(T,N)=0.

If G is a protorus, then evidently there is a factorization G=H×K, where H is fully factorable and K has no solenoid factor; just factor out solenoids until there are no more solenoid factors. This process terminates because G has finite dimension. The “main factorization” theorem says that such a factorization has strong uniqueness properties.

A compact group is clipped if it has no factor that is a solenoid. A discrete torsion-free group is clipped if it has no rank-1 summand. Clearly, a compact connected group G is clipped if and only if G is clipped.

We call two compact groups G,Hnearly isomorphic if, for all p, there exist φcHom(G,H), ψcHom(H,G) such that φψ=n1H, ψφ=n1G and gcd(p,n)=1. The definition applies to discrete torsion-free groups of finite rank. Near-isomorphism is weaker than isomorphism but preserves important properties such as direct decompositions [3, Chapter 12, § 10, page 465].

Theorem 4.5 (Main factorization).

Let G be a compact connected group of finite dimension. Then G has a decomposition G=K×T such that T is fully factorable and K is clipped. In particular, cHom(T,K)=0 and cHom(K,Q)=0.

Suppose that G=T×K=T×K, where T,T are fully factorable and K,K are clipped. Then TtT and K and K are nearly isomorphic.

Proof.

Dual of [13, Theorem 2.5]. ∎

The result in (Theorem 4.1) is more general in as much as it is not assumed that G is connected and finite dimensional [5, Corollary 8.47 (i), page 415]. To illustrate the difference between the two results, we give an example.

Example 4.6.

Let G=𝕋m×K, where K is a finite product of solenoids but torus-free. Then G=𝕋m×K is the claimed decomposition of Hofmann and Morris where nothing is known about K except that it is torus-free. According to Theorem 4.5, we have the decomposition G=G×0 in which the obscure factor is trivial.

A special case is worth stating.

Corollary 4.7.

Every protorus G has a factorization G=T×P×K, where T is a torus and P a projective group such that

cHom(𝕋,K)=0𝑎𝑛𝑑cHom(K,Σω)=0.

If also G=T×P×K, where T is a torus and P a projective group such that cHom(T,K)=0 and cHom(K,Σω)=0, then TtT and PtP.

Proof.

Factor K and K further into a product of a completely factorable factor and a clipped factor, and apply Theorem 4.5. ∎

The injective solenoid Σζt𝕋 and the projective solenoid Σω play special roles; a compact group G has a direct factor topologically isomorphic to Σζ if and only if cHom(Σζ,G)0, and a compact connected G has a direct factor topologically isomorphic to Σω if and only if cHom(G,Σω)0. The “dual Baer lemma” provides a criterion for the existence of a direct factor Στ for any type τ.

Lemma 4.8 (Dual Baer lemma).

Let G be a compact connected abelian group containing a subgroup Kt(Στ)a with KG((τ))=0. Then K splits from G.

In the proof, we will use the following observation.

Lemma 4.9.

Let GLCA and K a subgroup of G. Recall the canonical isomorphism ηG:GG. Then ηG((G,K))=(G,K).

Proof.

Let g(G,K). Then, for all χK, we have χ(g)=0. Hence

ηG(g)(χ)=χ(g)=0,

so ηG((G,K))(G,K).

Let f(G,K). Then f=ηG(g) for some gG. Hence, for all χK, we have

0=f(χ)=ηG(g)(χ)=χ(g),

so g(G,K) and ηG((G,K))(G,K). ∎

Proof of the dual Baer lemma.

We use the following version of the discrete Baer lemma [12, Lemma 2.4.12, page 39].

Let B be a pure subgroup of the torsion-free group A such that A/BSτ(a) for some cardinal a, and A=B+A(τ). Then A=BC for some subgroup C of A.

Without loss of generality, let G=A, and let B:=K=(A,K). Using Theorem 3.8 (1), we have

A=(A,{0})=(A,KG((τ)))=(A,K)+(A,G((τ)))=K+G((τ))=B+A(τ)=B+A(τ).

We claim that B is pure in A.

The short exact sequence KinsG𝜓G/K, where K is compact and connected, implies the short exact sequence

(G/K)ψGrestrK

of discrete groups, where ψ((G/K))=(A,K)=ηA(A,K)=ηA(B). Hence

ABηA(A)ηA(B)Aψ((G/K))K,

and K is torsion-free, so B is pure in A. By the discrete Baer lemma, A=BC for some C. Then Lemma 2.5 shows that K=K=B=(G,B) splits from G. ∎

We conclude with some useful applications. A discrete torsion-free group A is τ-homogeneous if, for all S𝒮(A), we have tp(S)=τ. Dually, a compact group G is τ-homogeneous if, for all C𝒞(G), we have tp(G/C)=τ. A fully factorable group G is τ-homogeneous if and only if Gt(Στ)𝔞 for some cardinal 𝔞.

Theorem 4.10.

Let Gt(Στ)n be a τ-homogeneous group of finite dimension n, and let K be a connected closed subgroup of G. Then K splits from G.

Proof.

The exact sequence KinsG𝜓G/K of compact groups implies the short exact sequence of discrete groups

(G/K)ψGrestrK,

where K is torsion-free as K is connected. By [3, Corollary 3.7, page 427], it follows that G=ψ((G/K))C=KC for some subgroup C of G. It follows by Lemma 2.5 that G=(G,K)(G,C). The canonical isomorphism ηG:GG maps (G,K) isomorphically onto (G,K) and maps (G,C) isomorphically onto (G,C), and it follows that

G=(G,K)(G,C)=K(G,C)=K(G,C).

The following theorem says in particular (when τ=tp()) that the quotient of an injective compact group modulo a connected subgroup is again injective. Its discrete dual [3, Theorem 3.9, page 426] implies that subgroups of free groups are free.

Theorem 4.11.

Suppose that G is a τ-homogeneous fully factorable group and G/K is a τ-homogeneous quotient of G (e.g. if K is connected). Then G/K is fully factorable.

Proof.

By hypothesis, Gt(Στ)𝔞 for some cardinal 𝔞.

The short exact sequence KinsG𝜓G/K in LCA implies the exact sequence

(G/K)ψGrestrK.

By hypothesis, (G/K) is τ-homogeneous. (This is true if K is connected. If so, K is torsion-free and ψ((G/K)) is pure in G and hence also τ-homogeneous as is G.) By [3, Theorem 3.9, page 426], we have that ψ((G/K)) is τ-homogeneous and completely decomposable, hence so is (G/K), (G/K)t(Sτ)(𝔟), and it follows that G/Kt(G/K)t(Στ)𝔟. ∎

Theorem 4.12.

Let G be a fully factorable compact group and G=H×K. Then the factor H is fully factorable.

Proof.

We have G=(G,H)(G,K). By [3, Theorem 3.10, page 427], the summand (G,K) is completely decomposable. The short exact sequence KinsG𝜓G/K implies the short exact sequence

(G/K)ψGrestrK.

Thus ψ((G/K))=(G,K) is completely decomposable, and further, (G/K) is completely decomposable. We conclude that HtHt(G/K) is fully factorable. ∎

We note that Theorem 4.12 was also obtained in [6, Corollary 7.5.5, page 296] in a more general context.


Communicated by Michael Giudici


References

[1] D. L. Armacost, The Structure of Locally Compact Abelian Groups, Monogr. Textb. Pure Appl. Math. 68, Marcel Dekker, New York, 1981. Search in Google Scholar

[2] R. Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), no. 1, 68–122. Search in Google Scholar

[3] L. Fuchs, Abelian Groups, Springer Monogr. Math., Springer, Cham, 2015. Search in Google Scholar

[4] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I: Structure of Topological Groups. Integration Theory, Group Representations, Grundlehren Math. Wiss. 115, Academic Press, New York, 1963. Search in Google Scholar

[5] K. H. Hofmann and S. A. Morris, The Structure of Compact Groups. A Primer for the Student—a Handbook for the Expert, De Gruyter Stud. Math. 25, De Gruyter, Berlin, 2013. Search in Google Scholar

[6] C. Jacoby and P. Loth, Abelian Groups, Structures and Classifications, De Gruyter Stud. Math. 73, De Gruyter, Berlin, 2019. Search in Google Scholar

[7] F. Kasch and A. Mader, Rings, Modules, and the Total, Front. Math., Birkhäuser, Basel, 2004. Search in Google Scholar

[8] C. Leopold, Butlergruppen und die Pontrjagin-Dualität, Diplomarbeit, University of Erlangen-Nürnberg, 1995. Search in Google Scholar

[9] C. Leopold, Butler groups and Lord groups, Acta Sci. Math. (Szeged) 63 (1997), no. 3–4, 397–404. Search in Google Scholar

[10] W. Lewis and A. Mader, Compact connected abelian groups of dimension 1, Rend. Semin. Mat. Univ. Padova, to appear. Search in Google Scholar

[11] P. Loth, The duals of almost completely decomposable groups, Arch. Math. (Basel) 68 (1997), no. 5, 353–358. Search in Google Scholar

[12] A. Mader, Almost Completely Decomposable Groups, Algebra Logic Appl. 13, Gordon and Breach Science, Amsterdam, 2000. Search in Google Scholar

[13] A. Mader and P. Schultz, Completely decomposable direct summands of torsion-free abelian groups of finite rank, Proc. Amer. Math. Soc. 146 (2018), no. 1, 93–96. Search in Google Scholar

[14] S. A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Soc. Lecture Note Ser. 29, Cambridge University, Cambridge, 1977. Search in Google Scholar

[15] M. Moskowitz, Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc. 127 (1967), 361–404. Search in Google Scholar

Received: 2020-05-27
Revised: 2020-06-20
Published Online: 2020-08-12
Published in Print: 2021-01-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston