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.
All groups in this article are abelian, and “group” means “abelian group”. Pontryagin duality is a duality on the category 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 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 , while [4, (25.3), page 403] contains a fairly involved representation. A simple description of solenoids and their properties is contained in ; 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  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 . 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 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 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 , and 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 the group of algebraic homomorphisms, and by , we denote the subgroup of continuous homomorphisms. In the following, means that are algebraically isomorphic, while means that the topological groups are topologically isomorphic, i.e., isomorphic in the category of topological groups. The Pontryagin dual of a discrete group A is
equipped with the compact-open topology, and the Pontryagin dual of a compact group G is the discrete group , where .
For any topological group G, the construct is its Lie algebra. It is well known that is a real topological vector space via the stipulation , where and , 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 of G by
For a compact group G of positive dimension, ; see [5, Theorem 8.22, page 390]. The identity component of G is denoted by . All topological groups are assumed to be Hausdorff.
Let G be a compact group. Then the following are equivalent.
G is totally disconnected.
is a torsion group.
(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 be topological groups. The sequence 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 ,
f and g are continuous.
The following statements hold.
Let be an exact sequence of discrete groups. Then
is exact as a sequence of topological groups, where are compact groups. Furthermore, the maps and are open maps onto their images.
Let be an exact sequence of topological groups, and assume further that are compact groups. Then
is exact in the category of discrete groups.
(1) is a consequence of Pontryagin duality, that with the compact-open topology and that is a divisible (injective) group. The claim that 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.
The following statements hold.
Let be a direct sum of discrete groups. Then , the topology being the product topology.
Let be the topological product of the compact groups . Then in the category of discrete groups.
(1) [5, Proposition 1.17, page 17].
(2) By (1), we have
hence dualizing again . ∎
Pontryagin duality is a duality of the category 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 avoids the necessity for separate statements.
Recall that a closed subgroup K of an group G is said to split from G if (algebraic direct sum) for some closed subgroup L of G such that the map is a topological isomorphism. Given , the open mapping theorem shows that ϕ is automatically a topological isomorphism whenever G is compact. Therefore, we have the following corollary.
Let K be a closed subgroup of a compact group G. Then K splits from G if and only if for some closed subgroup L of G.
Let K be a closed subgroup of an group G. Then K splits from G if and only if splits from .
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 indexed by with entries non-negative integers or . Given a height sequence , then
is an additive subgroup of containing , a rational group, that is a rank-1 group. Here a generator stands for the set . Every rank-1 group A is isomorphic to a rational group, and a rational group is easily seen to be generated by fractions , where , and this produces a height sequence . This indicates how every rank-1 group is associated with a height sequence and conversely. Two height sequences and are equivalent if if and only if and the integer entries of and differ at most at finitely many places.
A type is an equivalence class of height sequences. The type of a rank-1 group A is the type containing the height sequence associated with A. For types σ and τ, we write if there are height sequences and such that . 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 , we define its type by .
Proposition 3.1 ([10, Proposition 3.2]).
Let be rank-1 torsion-free groups with and , and let be solenoids with and .
If and , then f is injective.
If and , then g is surjective.
We now review the explicit description of solenoids given in [10, Theorem 1.1].
Theorem 3.2 ([10, Theorem 1.1]).
Let be a type, and let be a solenoid of type τ. Let , let denote the cyclic group of order if , and let . Then
We will choose a rank-1 group of type τ, say a rational group, and call it . Furthermore, we set . In particular, we choose and . 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 . Then a is contained in a maximal rank-1 subgroup of A, namely the purification of the subgroup of A generated by a. In fact, the identity defines ; in particular, is torsion-free. Then the type of a in A, , is defined to be the type of .
One then defines
, a pure subgroup of A,
that is not pure in general,
, the purification of in A.
These “type subgroups” (where x is either absent, or or ) are functorial subgroups, i.e., for all , we have ; in particular, the type subgroups are fully invariant subgroups.
Let A be completely decomposable. Then , where is the direct sum of rank-1 summands of type ρ. It is easy to see that , and . This also shows that 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 and are isomorphic if and only if, for all τ, we have .
For a discrete torsion-free group A, let be the set of all pure rank-1 subgroups of A. By , we denote the insertion map. Its dual (adjoint) in is the restriction map denoted .
Let A be a discrete torsion-free group and its dual group.
. If and , then .
Let . Then , the natural epimorphism, is exact with torsion-free. The dual sequence
is exact in the category of topological groups, is an embedding morphism, is a quotient morphism. Furthermore, is a closed connected subgroup of G, is a solenoid and .
(1) Every element is contained in . If , then .
(2) Routine. ∎
Lemma 3.3 motivates the next definition. For a compact connected group G, let be the set of all connected closed subgroups C of G such that is a solenoid. Let for some torsion-free discrete group A. In establishing the connection between and using the annihilator mechanism [5, Theorem 7.64, page 359], we make use of Lemma 3.4.
The following statements hold.
Let be an exact sequence in , where is the insertion morphism and is the natural quotient morphism. Then the dual sequence
is an exact sequence in , is an embedding morphism and is the restriction morphism. Furthermore,
Suppose that L is discrete, K is a rank- 1 subgroup of L and is torsion-free. Then is a solenoid and is a compact and connected subgroup of .
Suppose that L is compact and K is a compact connected subgroup of L such that is a solenoid. Then is a rank- 1 subgroup of the discrete group and is torsion-free and discrete.
(1) It is clear that
is as claimed. It is easy to see that if and only if . Hence if and only if .
(2) The subgroup is compact and connected since is the dual of a torsion-free discrete group and is a solenoid.
(3) The group is a rank-1 subgroup of since is the dual of a solenoid. The quotient group is discrete and torsion-free as is the dual of the compact connected group K. ∎
Let A be a discrete torsion-free group and its dual. Then
is a bijective map. Thus .
By [5, Theorem 7.64 (v), page 360], is an antiisomorphism of the lattice of closed subgroups of A and the lattice of closed subgroups of G. Let . By Lemma 3.4 (2), . Hence the restriction of σ to maps into , and we have an injective map . This map is also surjective because, given , by Lemma 3.4 (3), we get and (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:
, the identity component of .
Let G be a compact group, τ a type, and let
There is a well-defined morphism of topological groups
and . Hence is embedded in .
The following useful fact can be found in .
If H is a subgroup of a torsion-free group A, then
Let A be torsion-free, and τ any type. Then the following statements are true.
and is a compact connected subgroup of G.
. The group is a compact subgroup of G, but need not be connected.
and is a compact connected subgroup of G.
, and .
There is no loss of generality in assuming that for some torsion-free group A, but it is true in general that
We first consider the exact sequences
where x is either absent, or or , and is the quotient morphism. Then we have the exact sequence of compact groups
where, by Lemma 3.4 (1) with and , we have
We have the topological isomorphism . So, in particular, . Hence
We have that is torsion-free, so is both compact and connected.
(2) We argue as in (1), but need not be pure, i.e., need not be torsion-free, and hence need not be connected.
(3) Lemma 3.7 shows that . By (2), this group coincides with which equals .
(4) These isomorphisms are immediate consequences of our exact sequences using that . ∎
A functorial subgroup is a functor F assigning to each group X a subgroup in such a way that, for every morphism , it is true that
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.
Suppose that F is a functorial subgroup on the category of discrete groups. Let . Then is a functorial subgroup on .
Let be a morphism of groups. Then is a homomorphism of discrete groups, and by hypothesis, . It follows that
The type subgroups , and 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 that is the product of all solenoid factors of type τ of the given factorization of G. We obtain the “homogeneous factorization” , where is the “critical typeset” of G.
Let G be a fully factorable compact group and , where is a product of solenoids all of type τ. Then , i.e., the isomorphism class of is independent of the particular factorization of G.
(*) Two fully factorable compact groups and are isomorphic as topological groups if and only if, for all types τ, it is true that
Without loss of generality, for some completely decomposable group so that with . We have commutative diagrams with natural maps and exact rows and columns as follows:
It follows that . The last expression is an invariant of G, and hence, in any homogeneous factorization of G, the factors are unique up to topological isomorphism. The remaining claim (*) is an immediate consequence. ∎
Let be a fully factorable topological group. Then the following statements are true.
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
The comprehensive treatise  contains the following result.
Let G be a compact group such that is countable. Then G is metric and , where K is torus-free and T is a characteristic maximal torus. In particular, .
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 because, being a solenoid, any non-zero morphism in would be an embedding and contradict the assumption that T is the maximal torus of G. Trivially, 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 , where 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 . Moreover, 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 be a direct decomposition of R-modules. Denote by the set of all direct complements of A in M. Then
defines a bijective mapping and
is an isomorphism.
Although formulated for modules, Lemma 4.2 applies to topological abelian groups. As is injective, the group is large as a rule, and there are many different complements B. It can happen that and the decomposition is unique, e.g., if is torsion-free and B is a torsion group.
The structure of divisible groups is well known: they are of isomorphism type
Dualizing, we obtain that the projective compact groups are of isomorphism type in agreement with [5, Theorem 8.78, page 436], and every compact abelian group has a decomposition , where 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 . Hence the injectives in the category of compact groups are the tori in agreement with [5, Theorem 8.78, page 436]. If is free, then B is a direct summand of A. If and are free quotients, it is in general not clear how the summands 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 is countable.
A group A such that is countable can be decomposed as , where F is free and is the unique largest subgroup of A with . Furthermore, N is a functorial subgroup of A.
Let with , and let . Then we have , so and .
Let be groups with countable and . We have the decompositions and . Let be the projection belonging to the decomposition . Then
As is free, ; hence . ∎
By duality, we obtain a slight generalization of [5, Corollary 8.47 (i), page 415].
Let G be a compact group such that is countable (equivalently, the identity component of G is metrizable). Then , where is the unique maximal torus of G and N is a subgroup with .
If G is a protorus, then evidently there is a factorization , 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 is clipped.
We call two compact groups nearly isomorphic if, for all , there exist , such that , and . 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 such that T is fully factorable and K is clipped. In particular, and .
Suppose that , where are fully factorable and are clipped. Then and K and are nearly isomorphic.
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.
Let , where K is a finite product of solenoids but torus-free. Then 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 in which the obscure factor is trivial.
A special case is worth stating.
Every protorus G has a factorization , where T is a torus and P a projective group such that
If also , where is a torus and a projective group such that and , then and .
Factor K and further into a product of a completely factorable factor and a clipped factor, and apply Theorem 4.5. ∎
The injective solenoid and the projective solenoid play special roles; a compact group G has a direct factor topologically isomorphic to if and only if , and a compact connected G has a direct factor topologically isomorphic to if and only if . The “dual Baer lemma” provides a criterion for the existence of a direct factor for any type τ.
Lemma 4.8 (Dual Baer lemma).
Let G be a compact connected abelian group containing a subgroup with . Then K splits from G.
In the proof, we will use the following observation.
Let and K a subgroup of . Recall the canonical isomorphism . Then .
Let . Then, for all , we have . Hence
Let . Then for some . Hence, for all , we have
so and . ∎
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 for some cardinal , and . Then for some subgroup C of A.
Without loss of generality, let , and let . Using Theorem 3.8 (1), we have
We claim that B is pure in A.
The short exact sequence , where K is compact and connected, implies the short exact sequence
of discrete groups, where . Hence
and is torsion-free, so B is pure in A. By the discrete Baer lemma, for some C. Then Lemma 2.5 shows that splits from G. ∎
We conclude with some useful applications. A discrete torsion-free group A is τ-homogeneous if, for all , we have . Dually, a compact group G is τ-homogeneous if, for all , we have . A fully factorable group G is τ-homogeneous if and only if for some cardinal .
Let be a τ-homogeneous group of finite dimension n, and let K be a connected closed subgroup of G. Then K splits from G.
The exact sequence of compact groups implies the short exact sequence of discrete groups
where is torsion-free as K is connected. By [3, Corollary 3.7, page 427], it follows that for some subgroup C of . It follows by Lemma 2.5 that . The canonical isomorphism maps isomorphically onto and maps isomorphically onto , and it follows that
The following theorem says in particular (when ) 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.
Suppose that G is a τ-homogeneous fully factorable group and is a τ-homogeneous quotient of G (e.g. if K is connected). Then is fully factorable.
By hypothesis, for some cardinal .
The short exact sequence in implies the exact sequence
By hypothesis, is τ-homogeneous. (This is true if K is connected. If so, is torsion-free and is pure in and hence also τ-homogeneous as is .) By [3, Theorem 3.9, page 426], we have that is τ-homogeneous and completely decomposable, hence so is , , and it follows that . ∎
Let G be a fully factorable compact group and . Then the factor H is fully factorable.
We have . By [3, Theorem 3.10, page 427], the summand is completely decomposable. The short exact sequence implies the short exact sequence
Thus is completely decomposable, and further, is completely decomposable. We conclude that is fully factorable. ∎
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