On the structure of abelian profinite groups
aa r X i v : . [ m a t h . G R ] N ov ON THE STRUCTURE OF ABELIAN PROFINITE GROUPS
M. FERRER AND S. HERN ´ANDEZ
Abstract.
A subgroup G of a product Q i ∈ N G i is rectangular if there are subgroups H i of G i such that G = Q i ∈ N H i . We say that G is weakly rectangular if there are finitesubsets F i ⊆ N and subgroups H i of L j ∈ F i G j that satisfy G = Q i ∈ N H i . In this paperwe discuss when a closed subgroup of a product is weakly rectangular. Some possibleapplications to the theory of group codes are also highlighted. Introduction
For a family { G i : i ∈ N } of topological groups, let L i ∈ N G i denote the subgroupof elements ( g i ) in the product Q i ∈ N G i such that g i = e for all but finitely many indices i ∈ N . A subgroup G ≤ Q i ∈ N G i is called weakly controllable if G ∩ L i ∈ N G i is dense in G ,that is, if G is generated by its elements of finite support. The group G is called weaklyobservable if G ∩ L i ∈ N G i = G ∩ L i ∈ N G i , where G stands for the closure of G in Q i ∈ N G i for the product topology. Although the notion of (weak) controllability was coined byFagnani earlier in a broader context (cf. [3]), both notions were introduced in the areaof of coding theory by Forney and Trott (cf. [7]). They observed that if the groups G i are locally compact abelian, then controllability and observability are dual propertieswith respect to the Pontryagin duality: If G is a closed subgroup of Q i ∈ N G i , then it is Date : November/19/2018.The first-listed author acknowledges partial support by the Universitat Jaume I, grant P1171B2015-77. The second-listed author acknowledges partial financial support by the Spanish Ministry of Science,grant MTM2008-04599/MTM; and the Universitat Jaume I, grant P1171B2015-77. weakly controllable if and only if its annihilator G ⊥ = { χ ∈ \ Q i ∈ N G i : χ ( G ) = { }} is aweakly observable subgroup of Q i ∈ N c G i (cf. [7, 4.8]).In connection with the properties described above, the following definitions wasintroduced in [11]. Definition 1.1.
A subgroup G of a product Q i ∈ N G i is rectangular if there are subgroups H i of G i such that G = Q i ∈ N H i . We say that G is weakly rectangular if there are finitesubsets F i ⊆ N and subgroups H i of L j ∈ F i G j that satisfy G = Q i ∈ N H i . We say that G is a subdirect product of the family { G i } i ∈ I if G is weakly rectangular and G ∩ L i ∈ I G i = L i ∈ N H i .The observations below are easily verified.(1) Weakly rectangular subgroups and rectangular subgroups of Q i ∈ N G i are weaklycontrollable.(2) If each G i is a pro- p i -group for some prime p i , and all p i are distinct, then everyclosed subgroup of the product Q i ∈ N G i is rectangular, and thus is a subdirectproduct.(3) If each G i is a finite simple non-abelian group and all G i are distinct, thenevery closed normal subgroup of the product Q i ∈ N G i is rectangular, and thus asubdirect product.The main goal addressed in this paper is to study to what extent the converse ofthese observations hold. In particular, we are interested in the following question (cf.[11]): Problem 1.2.
Let { G i : i ∈ N } be a family of compact metrizable groups, and G aclosed subgroup of the product Q i ∈ N G i . If G is weakly controllable, that is, G ∩ L i ∈ N G i N THE STRUCTURE OF ABELIAN PROFINITE GROUPS 3 is dense in G , what can be said about the structure of G ? In particular, under whatadditional conditions is the group G a subdirect product of { G i : i ∈ N } , that is, weaklyrectangular and G ∩ L i ∈ I G i = L i ∈ N H i , where each H i is a subgroup of L j ∈ F i G j for some F i ⊆ N ? In connection with this question, the following result was established in [4].
Theorem 1.3.
Let I be a countable set, { G i : i ∈ I } be a family of finite abelian groupsand G = Q i ∈ I G i be its direct product. Then every closed weakly controllable subgroup H of G is topologically isomorphic to a direct product of finite cyclic groups. Unfortunately, this result does not answer Problem 1.2, which remains open tothe best of our knowledge. Finally, it is pertinent to mention here that the relevanceof these notions stem from coding theory where they appear in connection with thestudy of convolutional group codes [7, 12]. However, similar notions had been studiedin symbolic dynamics previously. Thus, the notions of weak controllability and weakobservability are related to the concepts of irreducible shift and shift of finite type ,respectively, that appear in symbolic dynamics. Here, we are concerned with abelianprofinite groups and our main interest is to clarify the overall topological and algebraicstructure of abelian profinite groups that satisfy any of the properties introduced above.In the last section, we shall also consider some possible applications to the study ofgroup codes. 2.
Basic facts
Pontryagin-van Kampen duality.
One of the main tools in this research isKaplan’s extension of Pontryagin van-Kampen duality to infinite products of locally
M. FERRER AND S. HERN ´ANDEZ compact abelian (LCA) groups. In like manner that Pontryagin-van Kampen dualityhas proven to be essential in understanding the structure of LCA, the extension ac-complished by Kaplan for cartesian products and direct sums [9, 10] (and some othersubsequent results) have established duality methods as a powerful tool outside theclass of LCA groups and they have been widely used in the study of group codes.We recall the basic properties of topological abelian groups and the celebratedPontryagin-van Kampen duality.Let G be a commutative locally-compact group. A character χ of G is a con-tinuous homomorphism χ : G −→ T where T is the multiplicative group of complexnumbers of modulus 1. The characters form a group b G , called dual group , which isgiven the topology of uniform convergence on compact subsets of G . It turns out that b G is locally compact and there is a canonical evaluation homomorphism E G : G −→ bb G. Theorem 2.1.
The evaluation homomorphism E G is an isomorphism of topologicalgroups. Some examples: b T ∼ = Z , b Z ∼ = T , b R ∼ = R , \ ( Z /n ) ∼ = Z /n. (some groups are self dual, such as finite abelian groups or the additive group of thereal numbers)Pontryagin-van Kampen duality establishes a duality between the subcategoriesof compact and discrete abelian groups. If G denotes a compact abelian group and Γdenotes its dual group, we have the following equivalences between topological proper-ties of G and algebraic properties of Γ: N THE STRUCTURE OF ABELIAN PROFINITE GROUPS 5 (i) weight ( G ) = | Γ | (metrizablity ⇔ | Γ | ≤ ω );(ii) G is connected ⇔ Γ is torsion free;(iii)
Dim ( G ) = 0 ⇔ Γ is torsion; and(iv) G is monothetic ⇔ Γ is isomorphic to a subgroup of T d .In general, it is said that a topological abelian group ( G, τ ) satisfies
Pontryaginduality (is
P-reflexive for short,) if the evaluation map E G is a topological isomorphismonto. We refer to the survey by Dikranjan and Stoyanov [2] and the monographs byDikranjan, Prodanov and Stoyanov [1] and Hofmann, Morris [8] in order to find thebasic results about Pontryagin-van Kampen duality.The following result, due to Kaplan [9, 10] is essential in the applications ofduality methods. Theorem 2.2 (Kaplan) . Let { G i : i ∈ I } be a family of P -reflexive groups. Then:(i) The direct product Q i ∈ I G i is P -reflexive.(ii) The direct sum L i ∈ I G i equipped with a suitable topology is a P -reflexivegroup.(iii) It holds: ( Y i ∈ I G i ) b ∼ = M i ∈ I c G i ( M i ∈ I G i ) b ∼ = Y i ∈ I c G i Kaplan also set the problem of characterizing the class of topological Abeliangroups for which Pontryagin duality holds.Let g ∈ G and χ ∈ b G , it is said that g and χ are orthogonal when h g, χ i = 1.Given S ⊆ G and S ⊆ b G we define the orthogonal (or annihilator) of S and S as S ⊥ = { χ ∈ b G : h g, χ i = 1 ∀ g ∈ S } M. FERRER AND S. HERN ´ANDEZ and S ⊥ = { g ∈ G : h g, χ i = 1 ∀ χ ∈ S } . Obviously G ⊥ = { e b G } and b G ⊥ = { e G } .The following result has also many applications in connection with duality theory. Theorem 2.3.
Let S and R be subgroups of a LCA group G such that S ≤ R ≤ G .Then we have d R/S ∼ = S ⊥ /R ⊥ . Corollary 2.4.
Let H be a closed subgroup of a LCA group G . Then [ G/H ∼ = H ⊥ and b H ∼ = b G/H ⊥ . Abelian profinite groups.
Our main results concern the structure of abelianprofinite groups that appear in coding theory. Firstly, we recall some basic definitionsand terminology. For every group G let us denote by ( G ) p the largest p -subgroup of G and P G = { p ∈ P : G contains a p − subgroup } where p ∈ P G and P is the set ofall prime numbers. An element g of a p -primary group G is said to have finite heightin G h if this is the largest natural number n such that the equation p n x = g has asolution x ∈ G . We say that g has infinite height if the solution exists for all n ∈ N .Here on, the symbol G [ p ] denotes the subgroup consisting of all elements of order p . Itis well known that G [ p ] is a vector space on the field Z ( p ).3. Order controllable groups
Definition 3.1.
Let { G i : i ∈ N } be a family of topological groups and C a subgroupof Q i ∈ N G i . We have the following notions: C is weakly controllable if C T L i ∈ N G i is dense in C . C is (uniformly) controllable if for every i ∈ N there is n i ∈ N such that if c ∈ C there exists c ∈ C such that c | [1 ,i ] = c | [1 ,i ] and c | ] n i , + ∞ [ = 0 (we assume that n i is N THE STRUCTURE OF ABELIAN PROFINITE GROUPS 7 the less natural number satisfying this property). This implies that there exists c ∈ C such that c = c + c , supp ( c ) ⊆ [1 , n i ] and supp ( c ) ⊆ [ i + 1 , + ∞ [. C is order-controllable if for every i ∈ N there is n i ∈ N such that if c ∈ C thereexist c and c in C such that c = c + c , with supp ( c ) ⊆ [1 , n i ], supp ( c ) ⊆ [ i + 1 , + ∞ [and order ( c ) ≤ order ( c | [1 ,n i ] ) (again, we assume that n i is the less natural numbersatisfying this property). As a consequence, we also have that order ( c ) ≤ order ( c ).Here, the order c is taken in the usual sense, considering c as an element of the group C . Every controllable group is weakly controllable and, if the groups G i are finite,then the notions of controllability and weakly controllability are equivalent (see [4]).The following result partially answers Problem 1.2 for p -groups. The proof can befounded in [5]. Theorem 3.2.
Let { G i : i ∈ N } be a family of finite, abelian, p -groups and let G = Q i ∈ N G i . If C is an infinite closed subgroup of G which is order-controllable, then C isweakly rectangular. In particular, there is a sequence { y m : m ∈ N } ⊆ C T L i ∈ N G i suchthat C is topologically isomorphic to Q m ∈ N < y m > . This result extends directly to general products of finite abelian groups and givesa partial answer to Question 1.2.
Theorem 3.3.
Let C be an order-controllable, closed, subgroup of a countable product G = Q i ∈ N G i of finite abelian groups G i . Then C is weakly rectangular. In particular,there is a sequence { y ( p ) m : m ∈ N , p ∈ P G } ⊆ G ∩ ( L i ∈ N G i ) such that { y ( p ) m : m ∈ N } ⊆ ( G ∩ ( L i ∈ N G i )) p and G is topologically isomorphic to Q m ∈ N p ∈ P G h y ( p ) m i . M. FERRER AND S. HERN ´ANDEZ
Proof.
Since each group G i is finite and abelian, it follows that it is a finite sum offinite p -groups, that is G i = L p ∈ P i ( G i ) p and P i = P G i is finite, i ∈ N . Note that P G = ∪ P i . Then Q i ∈ N G i ∼ = Q i ∈ N ( Q p ∈ P i ( G i ) p ) ∼ = Q p ∈ P G ( Q i ∈ N p ( G i ) p ) where N p = { i ∈ N : G i has a p − subgroup } . Consider the embedding j : G ֒ → Q p ∈ P G ( Q i ∈ N p ( G i ) p ) and thecanonical projection π p : Q p ∈ P G ( Q i ∈ N p ( G i ) p ) → Q i ∈ N p ( G i ) p .Set G ( p ) = ( π p ◦ j )( G ), that is a compact group. Since G ∩ ( L i ∈ N G i ) is dense in G , itfollows that ( π p ◦ j )( G ∩ ( L i ∈ N G i )) = G ( p ) ∩ ( L i ∈ N p G i ) p is dense in G ( p ) . Observe that if p ∈ P G then G ( p ) ∩ ( L i ∈ N p G i ) p = ( G ∩ ( L i ∈ N G i )) p (otherwise it is the neutral element). ApplyingTheorem 3.2, for each p ∈ P G there is a sequence { y ( p ) m : m ∈ N } ⊆ ( G ∩ ( L i ∈ N G i )) p such that G ( p ) ∼ = Q m ∈ N h y ( p ) m i . Then the sequence { y ( p ) m : m ∈ N , p ∈ P G } verifies theproof. (cid:3) The notion of rectangular and weakly rectangular subgroup of an infinite productextend canonically to subgroups of infinite direct sums. In this direction, we have:
Theorem 3.4.
Let C be an order-controllable subgroup of L k ∈ N G k such that every group G k is finite and abelian. Then C is weakly rectangular. In particular, there is a sequence ( y n ) ⊆ C such that C ≃ M n ∈ N h y n i Group codes
According to Forney and Trott [7], a group code is a set of sequences that hasa group property under a componentwise group operation. In this general setting, agroup code may also be seen as the behavior of a behavioral group system as given byWillens [13, 14]. It is known that many of the fundamental properties of linear codesand systems depend only on their group structure. In fact, Forney and Trott, loc.
N THE STRUCTURE OF ABELIAN PROFINITE GROUPS 9 cit., obtain purely algebraic proofs of many of their results. In this section, we followthis approach in order to apply the results in the preceding sections to obtain furtherinformation about the structure of group codes in very general conditions.Without loss of generality, assume, from here on, that a group code is a subgroupof a (sequence) group W , called Laurent group , that has the generical form W = W f × W c , where W f is a direct sum of abelian groups (locally compact in general)and W c a direct product. More precisely, let I ⊆ Z be a countable index set and let { G k : k ∈ I } be a set of symbol groups, a product sequence space is a direct product W c = Y k ∈ I G k equipped with the canonical product topology. A sum sequence space is a direct sum W f = M k ∈ I G k equipped with the canonical sum (box) topology. Sequence spaces are often defined tobe Laurent sequences W L = ( M k< G k ) × ( Y k ≥ G k ) . The character group of W L is W aL = ( Y k< c G k ) × ( M k ≥ c G k ) . Thus, a group code C is a subgroup of a group sequence space W and is equippedwith the natural subgroup topology. Next we recall some basic facts of this theory(cf. [3, 6, 7]). These notions are used in the study of convolutional codes that are wellknown and used currently in data transmission (cf. [6]).Let C be a group code in the product sequence space W = Q k ∈ I G k . Accordingto Fagnani, C is called weakly controllable if it is generated by its finite sequences. In other terms, if C = C ∩ W f . The group code C is called pointwise controllable if for all w , w ∈ C and k ∈ I ,there exist L ( k ) ∈ N and w ∈ C with w ( i ) = w ( i ) ∀ i < k , and w ( i ) = w ( i ) ∀ i ≥ k + L ( k ).With the notation introduced above, Let C be a group code in W . For any k ∈ I and L ∈ N , we set C k ( L ) := { c ∈ C : there exists w ∈ C with w ( i ) = 0 ∀ i < k and w ( i ) = c ( i ) ∀ i ≥ k + L } and C k := [ L ∈ N C k ( L ) . Obviously C k (1) ⊆ C k (2) ⊆ · · · C k ( L ) ⊆ · · · ⊆ C k .We have the following equivalence, whose verification is left to the reader. Proposition 4.1. C is controllable if and only if C = T k ∈ I S L ∈ N C k ( L ) . Given a group code C , the subgroup C c := \ k ∈ I [ L ∈ N C k ( L )is called the controllable subcode of C . A code C is called uniformly controllable whenfor every k ∈ I , there is L k such that C = T k ∈ I C k ( L k ). If there is some L ∈ N suchthat C k = C k ( L ) for all k ∈ I , it is said that C is L -controllable . Finally, C is stronglycontrollable if it is L -controllable for some L . If C is uniformly controllable and thesequence ( L k ) is bounded, then C is strongly controllable and the least such L is the controllability index (controller memory) of C . N THE STRUCTURE OF ABELIAN PROFINITE GROUPS 11
Using the same words as in [7], the core meaning of “controllable” is that anycode sequence can be reached from any other code sequence in a finite interval. Thefollowing property is clarifying in this regard. In the sequel C :[ k,k + L ) = { w ∈ C : w ( j ) = 0 ∀ j / ∈ [ k, k + L ) } . Proposition 4.2. C is controllable if and only if for any w ∈ C , there is a sequence ( L k ) contained in N such that w ∈ P k ∈ I C :[ k,k + L k ) .Proof. Let w ∈ C and let k i ∈ I be the first index such that w ( k ) = 0. Then thereis w ∈ C and L ⊂ I such that w ( k ) = w ( k ) and w ( i ) = 0 if i ≥ L + 1. Take k = L + 2 and let w ∈ C satisfying w ( i ) = ( w − w )( i ) for all i < k and w ( i ) = 0for all i ≥ k + L . In general we select w n ∈ C such that w n ( i ) = 0 if i < k n − , w n ( i ) = ( w − w − · · · w n − )( i ) for all i < k and w n ( i ) = 0 if i ≥ k n + L n . We havethat w = P n ∈ N w n is the product topology and furthermore the sum P n ∈ N w n ( i ) is finitefor all i ∈ I . (cid:3) Analogous notions are defined regarding the observability of a group code. Thegroup code C is called weakly observable if C ∩ W f = C ∩ W f . Let C a group code in W , we set( C f ) k [ L ] := { c ∈ W f : c | [ k,k + L ] ∈ C | [ k,k + L ] } . The group code C is called pointwise observable if C f = \ k ∈ I \ L ∈ N ( C f ) k [ L ] . If C is a group code, then the supergroup C ob := C ∪ ( \ k ∈ I \ L ∈ N ( C f ) k [ L ])is called the observable supercode of C . A code C is called uniformly observable when forevery k ∈ I , there is L k such that C f = T k ∈ I ( C f ) k [ L k ]. If there is some L ∈ N such that C f = T k ∈ I ( C f ) k [ L ] for all k ∈ I , it is said that C is L -observable . Finally, C is stronglyobservable if it is L -observable for some L . Obviously, if C is uniformly observable andthe sequence ( L k ) is bounded, then C is strongly observable and the least such L is the observability index (observer memory) of C .Recently, Pontryagin duality methods have been applied systematically in thestudy of convolutional abelian group codes. In this approach, a dual code C ⊥ is as-sociated to every group code C using Pontryagin-van Kampen duality in such a waythat the properties of C can be reflected ( dualized ) in C ⊥ . Along this line, the followingduality theorem provides strong justification for the use of duality in convolutionalgroup codes (see [7]). Theorem 4.3. [7]
Given dual group codes C and C ⊥ , then C is (resp. weakly, strongly)controllable if and only if C ⊥ is (resp. weakly, strongly) observable, and vice versa. Using duality, we obtain the following additional equivalences (cf. [7]).
Proposition 4.4.
For any group code C we have(1) ( C c ) ⊥ ∼ = ( C ⊥ ) o .(2) C is controllable if and only if C ⊥ is observable.(3) C is uniformly controllable if and only if C ⊥ is uniformly observable. Therefore, we can put our attention on the controllability of a group code wlog.In this direction, the following result was proved in [4].
N THE STRUCTURE OF ABELIAN PROFINITE GROUPS 13
Theorem 4.5 ([4]) . Let
C ≤ Q k ∈ N G k be a complete group code such that every group G k is finite (discrete). Then the following conditions are equivalent:(1) C is weakly controllable.(2) C is controllable.(3) C is uniformly controllable. In [3] Fagnani proves that, if C is a closed, time invariant, group code in G Z , with G being a compact group, then the properties of weak controllability, controllability andstrong controllability are equivalent. A different proof of this result follows easily usingthe ideas introduced above. Indeed, suppose that C is a weakly controllable, compactgroup code in W . By Theorem 4.6, we know that C is controllable and therefore C = T k ∈ I S L ∈ N C k ( L ). Suppose, in addition, that C is time invariant, then C k ( L ) = C ( L )for all k ∈ I . Furthermore, using Baire category theorem and the compactness of C , itfollows that there must be some L ∈ N such that C = C ( L ), which means that C isstrongly controllable.The results formulated above do not hold in general. In fact, an example of agroup H that is weakly controllable but not controllable is provided in [4]. Furthermore,using Theorem 4.3, we obtain that the group H ⊥ is is weakly controllable but notcontrollable.As a consequence of the preceding results we obtain the following relation betweenweakly controllable and controllable group codes (cf. [4]). Theorem 4.6. If C is a group code in W = W f × W c = ( M i< G i ) × ( Y i ≥ G i ) . Then the following assertions hold: (a) If every group G i is discrete, then C is controllable if and only if C is weaklycontrollable.(b) If every group G i is finite (discrete), then C is weakly controllable if and only if C is uniformly controllable.(c) If every group G i is a fixed compact group G , and C is a time-invariant, closedsubgroup of W c , then C is controllable if and only if C is strongly controllable. In case the groups in the family { G i : i ∈ I } are abelian, Theorem 4.3 yields asimilar result for observable group codes, using Pontryagin duality. Theorem 4.7. If C is a group code in W = W f × W c = ( M i< G i ) × ( Y i ≥ G i ) . Then the following assertions hold:(a) If every group G i is discrete abelian, then C is observable if and only if C isweakly observable.(b) If every group G i is finite (discrete) abelian, then C is weakly observable if andonly if C is uniformly observable.(c) If every group G i is a fixed discrete abelian group G , and C is a time-invariantsubgroup of W f , then C is observable if and only if C is strongly observable. Conclusion
To conclude, let us point out that, so far, the applications of Harmonic Analysisand duality methods to the study of group codes have basically reached the abeliancase (via Pontryagin duality and Fourier analysis). The non-commutative case has notyet been fully studied, but it can be expected that the application of duality techniques
N THE STRUCTURE OF ABELIAN PROFINITE GROUPS 15 in the study of non-abelian group codes could provide some results analogous to thosealready known in the Abelian case (see the work of Forney and Trott, op.cit). However,the nonabelian duality requires more complicated tools such as Kreˇin algebras, vonNeumann algebras, operator spaces, etc.). Therefore, it is first necessary to develop an appropriate nonabelian duality that can be applied in a similar way to how it is donein the Abelian case.
Acknowledgment:
The authors thank Dmitry Shakahmatov for several helpfulcomments.
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Universitat Jaume I, Instituto de Matem´aticas de Castell´on, Campus de Riu Sec,12071 Castell´on, Spain.
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