locally compact near abelian groupsherfort/_khh_fgr/before_7-nov-2013/next13-10-19.pdflocally...

LOCALLY COMPACT NEAR ABELIAN GROUPS

WOLFGANG HERFORT, KARL H. HOFMANN,AND FRANCESCO G. RUSSO

Abstract. A locally compact group G is near abelian if it con-tains a closed abelian normal subgroup A such that every closedtopologically finitely generated subgroup of G/A is inductivelymonothetic and every closed subgroup of A is normal in G. Re-cent studies prove that projective limits of finite p-groups (p prime)are exactly those compact p-groups in which two closed subgroupscommute (that is, are “quasihamiltonian” or “M -groups”). Suchresults are extended to locally compact near abelian groups andtheir structure is studied. This requires a review of the structureof locally compact periodic groups and thos of their automorphismswhich leave all closed subgroups invariant. The new definition oflocally compact near abelian groups necessitates the definition ofinductively monothetic locally compact groups. Their structure iscompletely classified. The global structure of locally compact nearabelian groups is discussed.

Depending on the completion of a final version this abstract is in needof completion.

1. Introduction

The recent sources [7, 11] present studies of metabelian compactgroups G which are rich in commuting pairs of closed subgroups. Onthe one hand, a compact group C is called topologically quasihamilto-nian (or briefly, quasihamiltonian), if HK = KH for all closed sub-groups H and K of C. On the other hand, we quote the followingdefinition from [11]:

A compact group G is called near abelian, if it contains a closednormal abelian subgroup A such that G/A is monothetic and that allclosed subgroups of A are normal in G.

Date: October 25, 2013.2010 Mathematics Subject Classification. 22A05, 22A26, 20E34, 20K35.Key words and phrases. Quasihamiltonian compact groups, hamiltonian com-

pact groups, modular compact groups, monothetic groups, p-adic integers, projec-tive cover.

1

2 W. HERFORT, K. H. HOFMANN, AND F. G. RUSSO

For a prime number p, a compact abelian group G is called a compact

p−group if and only if its character group G is a (discrete) p−groupif and only if G is a pro-p−group if and only if G is a topologicalZp−module for the ring Zp of p−adic integers. The structure of nearabelian compact p−groups is explicitly described in [11, Theorems 4.21,5.11, 6.7, 7.1, 7.2]. Complications arise in the case p = 2. One of theprincipal results for p > 2 is this:

A compact p−group is quasihamiltonian if and only if it near abelian.

One of the tools of independent interest is the surprisingly nontrivialfact that

the class of compact near abelian p−groups is closed under the forma-tion of projective limits.

The theory for near abelian compact p−groups developed in [11] sum-marizes results from [4, 5, 24, 25, 26], and it exhibits certain formalsimilarties to the structure theory of connected real Lie groups thathave been called diagonally metabelian Lie groups (see [9]).

In the present article we generalize these results on compact nearabelian p−groups in two ways: Staying all the while within the cate-gory of locally compact topological groups, we free ourselves from therestriction to compactness, and from the restriction to p−groups. For apreliminary indication of our main results, we need to agree on certainelements of notation:

We say a topological group G is finitely generated provided thatthere is a finite subset X of G and G = 〈X〉. When P is a propertyof a topological group G (such as being monothetic, that is, having adense cyclic subgroup), then we call G inductively P , if every finitelygenerated subgroup of G is P .

We shall call a locally compact group G near abelian if it containsa closed normal abelian subgroup A such that every closed subgroupof A is normal in G and G/A is inductively monothetic, and we callsuch a subgroup like A a base subgroup of G. Such a situation clearlydefines a representation π : G/A → Aut(A) implemented by the innerautomorphisms of G acting on A.

Here, Aut(A) denotes the group of continuous and continuously in-vertible automorphisms of a locally compact group A. In most situa-tions, we consider it endowed with the compact open topology. How-ever, if Aut(A) fails to be a topological group, then we consider it tobe given a standard refinement, of the compact-open topology, oftencalled g-topology (cf. [16], see also [1], §3, no 5).

LOCALLY COMPACT NEAR ABELIAN GROUPS 3

Visualizing a locally compact abelian group A as additively writtenwe call the automorphisms leaving all closed subgroups of A invariantthe scalar automorphisms and denote the subgroup of scalar automor-phisms SAut(A). We note that SAut(A) contains the two elementsubgroup {idA,− idA} whose elements we call the trivial scalar auto-morphisms. A locally compact near abelian group G itself with be

called trivial if π(G/A) ⊆ {id,− id} for some base group A.

For a topological abelian group let comp(G) denote the subgroup

of all g such that 〈g〉 is compact (see [10], Definition 7.44). A locallycompact abelian group G is periodic if, firstly, G = compG and, sec-ondly, G is zero dimensional. The significance of this concept for ourdiscourse derives from Theorem 2.8 and Theorem 4.12 saying that theonly case that A can fail to be period occurs when G is a trivial nearabelian group. Theorem 4.12 provides insight into the theory of locallycompact nonabelian trivial near abelian groups G.

For understanding the base group A of a locally compact abeliangroup G, we therefore need a good understanding of periodic locallycompact groups and their scalar automorphism groups. We surveythese in Section 1, beginning with a quick review of Braconnier’s con-cept of local products of families of locally compact groups with copactopen subgroups applied to the primary decomposition of a locally com-pact abelian periodic group A (see 2.1), proceeding to the structuretheory of its scalar automorphism group SAut(A) (see 2.3, 2.4, 2.5,and, notably, 2.6 and Theorems 2.8 and 2.9 (Mukhin)).

The second part of a locally compact near abelian group G is thefactor group G/A modulo a suitable base group A. By definition, G/Ais an inductively monothetic locally compact group, or IMG. This re-quires a detailed discussion of IMG-s which we provide in Section 2, allthe way to a complete classification of locally compact IMG-s in Theo-rem 3.6 and its corollaries. For the structure theory of locally compactnear abelian groups it is of course of significance in which way an IMGcan act by scalar automorphisms on a given periodic group. This isdiscussed in what we call the “Gamma Theorem” 3.10 in desirable de-tail. Since these theorems tell us that considerable contingent of IMG-sconsists of discrete subgroups of Q or Q/Z, or indeed of the groups Zpof p−adic integers as in the predecessor study [11], it is of interest tous to understand how the concept of divisibility is compatible with thestructure of Zp, and we survey that question at the end of Section 2 inProposition 3.11.


The information collected up to that point is then applied in Sec-tion 3 to present first global results on the structure of locally compactnear abelian groups. After a presentation of some typical examples wediscuss the general structure of locally compact near abelian groupsbe addressing both the “nontrivial,” and the “trivial” case by present-ing, firstly, the “First Structure Theorem of Nontrivial Near AbelianGroups” 4.10 and, secondly, the “Structure Theorem of Trivial NearAbelian Groups” 4.12 which is the only place where nontrivial con-nected components can appear in the entire theory of locally compactnear abelian groups.

khh: I hope to see some expansion here on the generalstructure theorem for both locally compact AND compactnear abelian groups along the topic of WoHer’s recent lec-tures at conferences. It would be wonderful if these weremore structural than computational or algorithmic–but ifnecessary, we accept things as they are.

2. Preliminaries

2.1. Periodic abelian groups. In order to understand where periodiclocally compact abelian groups are positioned in the universe of alllocally compact abelian groups we recommend that the reader mightconsult Theorems 7.56 and 7.67 in [10], p. 354, respectively, p.365.Indeed every locally compact abelian group G has a periodic componentcomp(G)comp0(G)

where comp(G) is the union of all compact subgroups and

comp0(G) is the unique largest compact connected subgroup; both ofthese are fully characteristic subgroups of G.

Turning now to periodic locally compact abelian groups G, we let Gp

denote the union of all compact p-subgroups. Then Gp is a closed andfully characteristic subgroup of periodic group called the p-primarycomponent.

If C and K are compact open subgroups of a locally compact abeliangroup, then so are C + K and C ∩K, and (G + K)/K ∼= C/(C ∩K)is finite as are all other factor groups in the following diagram. In thissense, all compact open subgroups of a periodic group are close to eachother; we keep this in mind:


C +K

wwwwwwwww

GGGG

GGGG

G

C

GGGGGGGGG K

vvvvvvvvv

C ∩K

Let us rephrase a well-known result due to J. Braconnier in [2]:

Theorem 2.1. (J. Braconnier) Let G be a locally compact periodicabelian group and C any compact open subgroup of G. Then G isisomorphic to the local product∏loc

p(Gp, Cp).

Proof. Since G is topologically isomorphic to the dual group of G and

since G is periodic [8, (24.18) Corollary] implies that G is zero di-

mensional. Therefore G and G are both zero dimensional. Then [2,Theoreme 1, page 71] immediately yields the desired result. �

Remark 2.2. Theorem 2.1 implies that there is an exact sequence oflocally compact abelian groups

1→ C → A→ D → 1

where C =∏

pCp is profinite and D =⊕

pDp is a discrete torsion

group with Dp = Gp/Cp. The p-primary subgroup of A gives rise to anexact sequence

1→ Cp → Ap → Dp → 1

If G is a locally compact periodic abelian group, then every automor-phism (and indeed every endomorphism) α leaves Gp invariant. Wewrite αp = α|Gp : Gp → Gp. If C is a compact open subgroup, letAut(G,C) denote the subgroup of Aut(G) of all automorphisms leav-ing C fixed as a whole. In view of Theorem 2.1 we may identify G withits canonical local product decomposition of the pair (G,C).

Corollary 2.3. For a periodic locally compact abelian group G, thefunction

α 7→ (αp)p : Aut(G,C)→∏p

Aut(Gp, Cp)

is an isomorphism of groups, and α((gp)p) = (αp(gp))p.


Proof. It is straightforward to verify that the function is an injectivemorphism of groups. If (φp)p is any element of

∏p Aut(Gp), then the

morphism φ :∏

pGp →∏

pGp defined by φ((gp)p) = (φp(gp))p leaves

C =∏

pCp fixed as a whole and does the same with∏loc

p (Gp, Cp). Thus

the function α 7→ (αp)p is surjective as well. �

2.2. On scalar automorphisms. An automorphism α of a locallycompact abelian group is called scalar if it leaves all closed subgroupsinvariant. The subgroup of scalar automorphisms of G is denoted

SAut(G). By duality, the isomorphism α 7→ α : Aut(G) → Aut(G)

maps SAut(G) bijectively onto SAut(G). The subgroup {id,− id} ofSAut(G) is said to consist of trivial scalar automorphisms. All otherscalar automorphisms are called nontrivial.The set SAut(G) is a closed subgroup of Aut(G) where the latteris equipped with the coarsest topology containing the compact opentopology and turning Aut(G) into a topological group (the “g-topology”in [16]). If C is any compact open subgroup of G, then SAut(G) ⊆Aut(G,C).

Corollary 2.4. For a periodic locally compact abelian group G and anycompact open subgroup C, with the identification G =

∏locp (Gp, Cp), the

function

α 7→ (αp)p : SAut(G)→∏p

SAut(Gp)

is an isomorphism of groups.

Proof. Since every closed subgroup H of G satisfies Hp = H ∩Gp and

H =∏loc

p (Hp, Hp ∩ C), the assertion is a consequence of Corollary2.3. �

Thus we know SAut(G) if we know SAut(Gp) for each prime p. Alocally compact abelian p-group G is a Zp-module. For any invertiblescalar r ∈ Z×p the function µr : G→ G, given by µr(g) = r·g is a scalarautomorphism.

Lemma 2.5. For a locally compact p-group G the morphism

r 7→ µr : Z×p → SAut(G)

is surjective for all prime numbers p including p = 2.

Proof. Let α ∈ SAut(G). We must find an r ∈ Z×p such that α = µr. IfG is compact, this follows from Theorem 2.27 of [11]. Assume now thatG is not compact. Then it is the union of its compact subgroups. If C isa nonsingleton compact subgroup then by Theorem 2.27 of [11] there is


a unique scalar rC ∈ Z×p such that α|C = µrC |C. If K ⊇ C is a compactsubgroup containing C, then µrK |C = µrC |C. Since Z×p is compact, thenet (rC)C , as C ranges through the directed set of compact subgroupsof G, has a cluster point r ∈ Z×p ; that is, there is a cofinal subset C(j),j ∈ J for some directed set J such that r = limj rC(j). Since for everyg ∈ G there is a C with g ∈ C and then a j0 ∈ J such that j ≥ j0implies C ⊆ C(j) we conclude that α(g) = rC ·g = rC(j)·g for these j,and so α(g) = r·g. Thus α = µr and the proof is complete. �

We remark that the group Z×p is isomorphic to Zp × Z(p − 1) if p isodd and to Z2 × Z(2) if p = 2 (see e.g. [11], Lemma 2.1, respectively,Lemma 2.3), and thus is not monothetic iff p = 2. The factor SAut(Gp)of SAut(G) is not monothetic iff p = 2. The finite factor of Z×p is thegroup Cn for n = p − 1 if p > 2, respectively, n = 2 for p = 2, whereCn is the group of of n-th roots of unity. Thus Z×p contains a unique

element r of order 2; in view of 0 = 1− r2 = (1 + r)(1− r) this elementis always −1.

Now every periodic p-group G is a Z-module in a natural fashion. In-

deed, Z =∏

p Zp and G =∏loc

p (Gp, Cp) in a natural fashion so that for

z = (zp)p ∈ Z and g = (gp)p we have z·g = (zp·gp)p. Thus there is amorphism ζ of topological groups

z 7→ (µ(zp))p : Z× →∏p

SAut(Gp) = SAut(G).

Proposition 2.6. Let G be a periodic locally compact abelian group.Then we have the following conclusions:

(i) The natural map ζ : Z× → SAut(G) is surjective. In particular,

SAut(G) is a profinite group and homomorphic image of Z×.(ii) We have SAut(G) = {id,− id} only if the exponent of G is 2, 3, or4. For p = 2 the group SAut(G2) is not monothetic if the exponent ofG is at least 8.

Proof. (i) follows from the preceding Corollary 2.4 and Lemma 2.5.(ii) If the exponent of G divides 4 or is 3 then, any α ∈ SAut(G)satisfies (∀x ∈ G) α(x) = x or (∀x ∈ G) α(x) = −x. So such α is“trivial” in our sense according to the section named “Notation.”On the other hand, if the exponent is not in {2, 3, 4}, then either (a)p > 3, or (b) G3 has exponent at least 9, or (c) G2 has exponent atleast 8. In Case (a), Cp−1 acts non-trivially on Gp. In Case (b), themap x 7→ 4x is a nontrivial scalar automorphism of Gp, and in Case(c) the function x 7→ 5x is a non-trivial scalar automorphisms of Gp.


When p = 2 and the exponent is at least 8 then SAut(G2) (and soalso SAut(G)) is not monothetic as follows from the arguments in [11],Theorem 2.7. �

Remark 2.7. We use the opportunity to correct a typographical errorin the the proof of [11], Theorem 2.27. The formula for SAut(A) (calledAutscal(A) there) should read Autscal(A)=

Z×p ∼= Pp × Cp−1 if σA is faithful, p > 2,

Z×2 ∼= P4 × C2 if σA is faithful, p = 2,

Z(pn)× ∼= (Pp/P[n]p )× Cp−1 if σA is not faithful, p>2, n>1,

Z(2n)× ∼= (P4/P[n]4 )× Cp−1 if σA is not faithful, p=2, n>2.

This correction does not affect the validity of the argument.

A proof of Part (i) of the preceding proposition can also be obtainedas a corollary from a more general result by R. Winkler [27]:

Every (not necessarily continuous) endomorphism of a locally compactabelian group which leaves invariant every closed subgroup must act byscalars.

In the same line of structural result we point out a note by Moskalenko[17], which has an echo in the following sharper result.

Theorem 2.8. For a locally compact abelian group G, we consider thefollowing statements:

(1) G has nontrivial scalar automorphisms.(2) G is periodic.

Then (1) implies (2), and if G does not have exponent 2, 3, or 4, thenboth statements are equivalent.

Proof. We just remarked in Proposition 2.6 that (2) implies (1) underthe hypotheses stated there.Now assume (1). If G has a discrete subgroup ∼= Z, then every scalarautomorphism must be trivial. Hence Weil’s Lemma (see e.g. [10],Proposition 7.34) implies that G = comp(G) and we have to argue

that G0 = {0}. But (G0)⊥ in G is comp(G) (see [10], Theorem 7.67).

Since G has no nontrivial scalar automorphisms by (1), we know G =

comp(G) = (G0)⊥. This means G0 = {0}, as we had to show. �

Theorem 2.9 below was proved by Yu. N. Mukhin in [20], but may nowalso be obtained quickly by combining our results above. We need aspecific piece of notation: The universal zero dimensional compactifi-

cation of the integers is denoted Z; we know that Z ∼=∏

p Zp, where p


ranges through the set of all prime numbers. When, here and in thefollowing, in formulae contexts like

∏p Zp,

⊕p Z(p∞), (xp)p ∈

∏p Zp, it

will be understood that p ranges through the set of all primes wheneverthe context is clear.

Theorem 2.9 (Yu. N. Mukhin, Theorem 2 in [20]). Let G be a locallycompact abelian group. If G is not periodic then SAut(G) = {id,− id}.If, on the other hand, G is periodic, then SAut(G) =

∏p SAut(Gp)

and SAut(Gp) identifies with the group of units of either the ring Zpof p-adic integers or of the finite ring Zp/pmZp ∼= Z(pm) in case theexponent of Gp is pm. In particular, SAut(G) is a homomorphic image

of Z×.

As a consequence, SAut(G) is a profinite central subgroup of Aut(G).

Remark 2.10. In order to avoid confusion let us note that the “usualscalar multiplication” in R does not yield a “scalar automorphism” ofR in the present sense: if 0 6= r is any real number, then multiplicationby r is a scalar automorphism in our present sense if and only if r = ±1.

3. Inductively monothetic groups

The concept of an inductively monothetic topological group competeswith that of a monothetic topological group. While it will be sub-stantially used in our definition and treatment of locally compact nearabelian groups, we shall point out at an early stage that the former isneither more general nor mor special than the latter.We recall that a topological group G is said to have a property Pinductively if and only if for every finite subset F ⊆ G the subgroup〈F 〉 has property P .

Definition 3.1. We say that G is an inductively monothetic group,short an IMG, if every finitely generated subgroup is monothetic.

Obviously, a closed subgroup of an IMG is an IMG.For purely technical reasons we say that a topological group is

of class (I) if it is discrete,of class (II) if it is infinite compact, andof class (III) if it is locally compact but is neither

of class (I) nor of class (II).

Lemma 3.2. If G is an IMG of class (I), then it is isomorphic to asubgroup of Q or else of Q/Z, that is, if and only if it is either torsion-free of rank 1 or a torsion group of p-rank 1 for all primes.


Proof. Exercise in view of the fact that a finitely generated abeliangroup is a direct sum of cyclic ones which then must be cyclic. �

If G is class (I) and is the directed union of subgroups which are IMG-s,then G is an IMG.

Lemma 3.3. If G is an IMG of class (II), then it is monothetic andis is either connected of dimension 1, or it is infinite profinite.

Proof. By Proposition 2.42 of [10], a compact abelian group G is a Liegroup iff it is of the form Tn × F with a finite group F . For n ≥ 1,the monothetic torus Tn contains the subgroup Z(2)n which is an IMGonly if n = 1. If C ⊆ F is a nondegenerate cyclic subgroup of F andn = 1 then T contains an isomorphic copy C∗ of C and so G containsC∗ × C which fails to be an IMG. Thus G is a class (II) group and aLie group iff it is isomorphic to T or is a finite class (I) group in whichcase its p-rank is 1 for all primes p dividing the order of F . Thus G is a

Lie group of class (II) and an IMG iff G is a finitely generated discreteIMG.Now let G be an IMG of class (II). Then it is a strict projective limit

of its Lie group quotients which are IMG-s of class (II). Hence G is thedirected union of its finitely generated subgroups which are IMG-s of

class (I), and thus G is an IMG. By Lemma 3.2, G is then a subgroupof Q or of Q/Z. Now, by duality, G is of the kind asserted (cf. [10],822.ff.). �

The proof has shown that an infinite compact abelian group G is an

IMG of class (II) iff G is an infinite IMG of class (I).

Lemma 3.4. An IMG of class (III) is periodic.

Proof. (a) Suppose R were an IMG. But R is generated by 1 and√

2;thus R would have to be monothetic which it is not. Thus in an IMG ofclass (III) does not contain any vector subgroup and so the subgroup Ein Theorem 7.57 of [10] is trivial and G has a compact open subgroup,

(b) Since G is not discrete, there is an open infinite compact sub-group U which is an IMG. Suppose it were not 0-dimensional. Thenby Lemma 3.3 it would be isomorphic to a 1-dimensional compact con-nected group. Being divisible and open, it would be an algebraic andtopological direct summand; G being an IMG would imply G = Uwhich is impossible since G is of class (III). Thus U and hence G is0-dimensional.

(c) Suppose G is a class (III) IMG for which G 6= comp(G). Thenby [10], Corollary 4.5, there would be an infinite cyclic discrete group


D ∼= Z and for every compact open subgroup U the open sum U + Dis direct. Now G being an IMG implies that U has to be trivial. ThenG would be discrete, but it is of class (III). Therefore G = comp(G)and thus G is periodic. �

Now Theorem 2.1 reduces the classification of class (III) IMG-s to thecase that G is a p-group.

Lemma 3.5. Let G be a p-group which is a class (III) IMG. ThenG ∼= Qp.

Proof. Let U be a compact open subgroup of G. By Lemma 3.3, Uis a profinite monothetic group which cannot be finite since G is notdiscrete. Hence U ∼= Zp. Similarly, the discrete IMG and p-group G/U

is isomorphic to Z(p∞) =1

p∞ ZZ . Thus there is a surjective morphism

f : G→ Z(p∞) =

1p∞

ZZ

with kernel U . If we set

Um = f−1

(1pm

ZZ

)then G =

⋃∞m=0 Um and each Um is generated by the set consisting of

a generator of U and any element mapping by f onto a generator of1

pmZ

Z and thus is monothetic and hence isomorphic to Zp. Moreover,Um = p·Um+1. Thus the sequence

U = U0 ⊆ U1 ⊆ U2 ⊆ · · ·

is isomorphic to the sequence

Zp ⊆1

p·Zp ⊆

1

p2·Zp ⊆ · · · ,

and thus G = Qp follows. �

We summarize our findings:

Theorem 3.6. (Classification Theorem of Inductively MonotheticGroups) Let G be an inductively monothetic locally compact group.Then G is isomorphic to exactly one of the following groups:

(1) a subgroup of the discrete group Q.(2) a subgroup of the discrete group Q/Z,(3) a 1-dimensional compact connected abelian group,(4) an infinite monothetic group,


(5) a local product M =∏loc

p prime(Mp, Cp) with the p-primary com-

ponents Mp such that Mp/Cp ∼= Z(pnp) or ∼= Z(p∞) (in whichcase Mp = Qp) and that both

∏pCp and

⊕p(Mp/Cp) are infi-

nite.

The groups of type (3) and (4) of Theorem 3.6 are monothetic; othertypes may or may not be monothetic. An inspection of each or the fiveconditions yields at once the following

Corollary 3.7. Let G be a locally compact inductively monotheticgroup. Then the following conditions are equivalent:

(1) G = comp(G).(2) G is not isomorphic to a subgroup of the discrete group Q.

Also, the following two conditions are equivalent:

(3) G is periodic.(4) G is not connected and not isomorphic to a subgroup of the

discrete group Q.

As a consequence of these remarks and of Weil’s Lemma [10, Proposi-tion 7.43] we record

Corollary 3.8. Let G be inductively monothetic. Suppose that it con-tains a discrete subgroup D ∼= Z. Then G is discrete and torsion freeof rank one.

Examples of groups from the Classification Theorem 3.6 above are givenby each of the groups Qp or by local products of the type described inthe following

Example 3.9. Define G =∏loc

p (Zp, pZp). Then C =∏

p pZp is an open

subgroup of G isomorphic to Z =∏

p Zp, while its factor group G/C is

isomorphic to the infinite discrete group⊕

p Z(p). The inclusion map

induces an injective continuous morphism ε : G → Z =∏

p Zp withdense image which is not open onto its image.

Examples of this kind illustrate the considerable variety of groups oftype (5) in Theorem 3.6.

The following theorem will be crucial for the later developments.

Theorem 3.10. (The Gamma Theorem) Let Γ be a nonsingleton in-ductively monothetic locally compact group acting faithfully on a locallycompact abelian group G as a group of scalar automorphisms and as-sume that |Γ| 6= 2. Then G is periodic and Γ is either isomorphicto


(i) a nontrivial discrete proper discrete subgroup of Q,(ii) an infinite subgroup of Q/Z each of whose p-primary component

is finite,(iii) an infinite profinite monothetic group,

(iv) a noncompact local product∏loc

p (Gp, Cp) where no factor Gp isisomorphic to Qp and, consequently, all factors Gp are compact.

Proof. Let η denote the canonical morphism from Γ to SAut(G) that isguaranteed by the hypotheses. Since the action is faithful, η is injective.By Theorem 2.8, if G is not periodic, then |Γ| ≤ 2 contrary to thehypothesis that |Γ| 6= 2. Hence G is periodic. So by Theorem 2.9SAut(G) is a profinite group. Since a profinite group cannot containany nontrivial divisible subgroups, and since η is injective, Γ does notcontain any divisible subgroup. In particular, Γ0 = 1. Likewise, Γcannot contain subgroups isomorphic to either Q, Z(p∞), or Qp forany prime p. In view of the Classification Theorem 3.6, this completesthe proof. �

Case (ii) of Theorem 3.10 includes the possibility that Γ is a finite cyclicgroup.

Note that G is the union of its compact (and therefore Γ-invariant)subgroups. The case that G is a compact p-group and Γ is a monotheticp-group was extensively discussed in [11].

The following proposition is added to show which examples are allowedtypically by Theorem 3.10. Recall Zp ⊆ Qp and Q ⊆ Qp. For everynonzero rational number q there are unique sets of primes p1 < · · · < pkand nonzero integers n1, . . . , nk such that q = ±pn1

1 · · · pnkk , If pj 6= p

for j = 1, . . . , k then we say that q is reatively prime to p.We define Fp = 1

p∞·Z to be the subring of rationals of the form pnz,

n ∈ Z and z ∈ Z, and pF the subring of all rationals of the form pnzwith n ≥ 0 and z a rational relatively prime to p.

Proposition 3.11. (i) The additive group Zp does not contain anynontrivial divisible subgroups.(ii) Q ∩ Zp = pF.(iii) Q + Zp = Qp.(iv) Zp/pF ∼= R (as abstract groups).

(v) Zp/Z ∼= R × pFZ , where pF

Z∼=⊕

p 6=q Z(q∞). In particular, Zp/Z isdivisible.

Proof. (i) Zp is not the direct sum of two proper subgroups (see [6],p. 122ff.). A divisible subgroup of an abelian group is a direct summand(see e.g. [10] Corollary A1.36(i)). So (i) follows.


(ii) A number q ∈ Qp is of the form q = pnz with a unique smallestn ∈ Z and a unique z ∈ Zp. We have q ∈ Zp iff n ≥ 0. A rationalnumber is of the form ±pnz with a unique smallest n ∈ Z and a uniquerational number z relatively prime to p. Thus q ∈ Q∩Zp iff n ≥ 0 andz ∈ Q relatively prime to p. iff q ∈ pF. It also follows from this thatFp ∩ Zp = Fp ∩ pF = Z.

(iii) Fp +Zp is an additive subgroup of Qp containing Zp and such thatZ(p∞) ∼= Qp/Zp ⊇ (Fp + Zp)/Zp ∼= Fp/(Fp ∩ Zp) = Fp/Z = Z(p∞).Since Z(p∞) has no proper infinite subgroup, Qp/Zp = (Fp + Zp)/Zpfollows, and this implies Fp + Zp = Qp. A fortiori, Q + Zp = Qp.

(iv) Using (ii) above we compute Zp/pF = Zp

Zp∩Q∼= Q+Zp

Q = Qp/Q in

view of (iii) above. Now Qp is a rational vector spaces of dimensionc (the cardinality of the continuum) and Q is a 1-dimensional vectorsubspace. Thus Qp/Q is a rational vector space of dimension c andthus is isomorphic to R as a rational vector space.

(v) Firstly, we note that Zp/ZpF/Z

∼= Zp

pF∼= R. Secondly we observe that

pF/Z =⊕

p 6=q Z(q∞). Since this group is divisible and each divisible

subgroup is a direct summand (see [2], A1.36(i)) conclude Zp

Z∼= R× pF

Zas asserted. �

If we write the circle group additively as T = R/Z ∼= R × QZ , then we

notice that Zp/Z ∼= TZ(p∞)

and T ∼= Zp

Z × Z(p∞).

As was observed in [11], the additive group G of the compact ring Zpcontains a copy of Zp in its multipliative group of units whence SAut(G)contains a copy of Zp and therefore Zp, any discrete subgroup of Zp such

as pF, or the noncompact locally compact group group∏loc

p (Zp, pZp)of Example 3.9 as Γ for G = Zp in Theorem 3.10.

4. Locally compact near abelian groups

We extend to arbitrary locally compact groups the definition of nearabelian groups [11], introduced in [11] for profinite groups.

4.1. Definition and immediate consequences. We begin with thebasic definition.

Definition 4.1. A locally compact group G is near abelian providedit contains a closed normal abelian subgroup A such that

(1) G/A is inductively monothetic, and(2) every closed subgroup of A is normal in G.


We shall call A a base of the near abelian group G. If there is a closedsubgroup H such that G = AH, then we call H a scaling (sub)group.

Some remarks are in order at once. By Theorem 3.6, G/A is abelian,and so G′ ⊆ A. Every inner automorphism of G induces on A a scalarautomorphism by the very definition of a scalar automorphism in Para-graph 2.2. Thus we have a canonical morphism

ψ : G→ SAut(A), ψ(g)(a) = gag−1

whose kernel is kerψ is the centralizer CG(A) of A in G, containing Asince A is abelian by Definition 4.1. Thus we formulate the followingobservation for later use: the group

Γ := G/CG(A)

is a homomorphic image of G/A and therefore being inductively mono-thetic, acts faithfully via scalar automorphisms on A.Theorem 3.10 immediately implies

Lemma 4.2. If Γ acts nontrivially on A, then A is periodic.

Definition 4.3. If Γ acts by trivial scalar automorphisms, that is, bymultiplication by ±1, we say that G is a trivial near abelian groupotherwise we call G a nontrivial near abelian group

Note that an abelian near abelian group is trivial in this sense.

The following diagram may help to sort out this quantity of informa-tion:

inductively monothetic

G∣∣∣ }∼= Γ

injected−−−→ SAut(A)

CG(A)∣∣∣ }inductively monothetic

A∣∣∣G′∣∣∣{1}

periodic if G nontrivial

In the light of this diagram, the following observation is of interest:


Proposition 4.4. In a near abelian group G with any base group A,the centralizer CG(A) is an abelian normal subgroup of G containing Aand is maximal with respect to this property.

Proof. Let x, y ∈ CG(A). Since G/A and thus its subgroup CG(A)/Aare inductively monothetic, there is a z ∈ CG(A) such that x, y ∈〈z〉A. Since A and 〈z〉 are both abelian and 〈z〉 commutes elementwisewith A, the elements x and y commute. Thus CG(A) is commutativeand trivially contains A since A is abelian. If B is any commutativesubgroup of G containing A, then B ⊆ CG(A), trivially. Hence CG(A)is the largest such subgroup B. �

In particular, a base group A of a near abelian group G need not bea maximal abelian normal subgroup in general. This can be seen bytaking G = C2⊗C2 to be (near) abelian and either A = G and B = {0}or A = 〈(1, 0)〉 and B := 〈(0, 1)〉.

4.2. Examples of near abelian groups. Some examples may serveas a first illustration of these concepts. The first example shows thatany locally compact abelian group A can appear as the base group ofa (locally compact) near abelian group.

Example 4.5. Let A be an arbitrary locally compact abelian group.Then the multiplicative group C2 = {1,−1} ⊆ Z acts by trivial scalarautomorphisms on A and D(A) := AoC2, the so-called dihedral exten-sion of A is a trivial locally compact near abelian group. It is abelianif and only if A has exponent 2, that is, is algebraically a vector spaceover GF(2). If H is an inductively monothetic locally compact groupwith a morphism α : H → C2, then G = A oα H is a trivial locallycompact abelian group which maps homomorphically into D(A).

Compact near abelian p-groups in the sense of the definition in [11,page 2] are near abelian. Here is an illustrative near abelian p-group,a locally compact version of [11, Theorem 4.21 Case B]:

Example 4.6. Let p be a fixed prime. For every n ∈ N let Gn be eitherCase A: a copy of Qp,Case B: a copy of Zp,and fix an open proper subgroup Cn of Gn. Then Cn ∼= Zp. Themultiplicative subgroup

H := (1 + pkZp,×),

{k = 1 if p > 2

k = 2 if p = 2

}∼= (Zp,+)


of the ring (Zp,+, ·) acts by scalar automorphisms on the direct product∏n∈NGn (see Corollary 2.3). Therefore we can form the local product

A :=∏loc

n∈N(Gn, Cn) (see Theorem 2.1) and get H to act on A by scalarautomorphisms (see Corollary 2.4). Therefore we obtain a periodicnear abelian p−group G := A o H with base group A and compactscaling group H; the base group A and thus the group G is noncompactin both Case A and Case B.

In the preceding examples, the base group A was noncompact locallycompact while the scaling group was compact. The following exampleof a locally compact group illustrates the reverse situation:

Example 4.7. We let A =∏

p Zp with p ranging through all primenumbers. Let

Hp := (1 + pkZp,×),

{k = 1 if p > 2

k = 2 if p = 2

}∼= (Zp,+),

Then each Hp acts by scalar automorphisms on Zp, and so H :=∏

pHp

acts by scalar automorphisms on A =∏

p Zp, componentwise. Then

AoH ∼=∏p

(Zp oHp)

is a compact near abelian group. If Kp = (1 + pk+1Zp,×), then

Hp/Kp = Cp is cyclic of order p, and L :=∏loc

p (Hp, Kp) as a locally

compact noncompact inductively monothetic group (see Example 3.9above), and ε : L→ H induced by the inclusion is an injective contin-uous morphism with dense image. Accordingly, G = AoL is a locallycompact noncompact near abelian group with a compact base group∼= A and a noncompact scaling group ∼= L which is mapped injectivelyand continuously into the compact near abelian group A o H withdense image by the morphism idA×ε.

Here is an example of a locally compact abelian group G with a basegroup A for which G/A is an infinite torsion group:

Example 4.8. Choose primes pi and qi such that pi ≡ 1 (mod qi). Thenthere is ri ∈ Zqi such that rpii = 1. Set A :=

∏i∈N Zqi and let a

generator bi of Z(pi) act trivially on Zqj if j 6= i and as zbi := zri .Forming the discrete direct sum H :=

⊕i∈N Z(pi) we can extend the

actions of the bi to a scalar action of H. The semidirect product G :=AoH has a torsion scaling group.

Finally we provide an example with G/A torsion free:


Example 4.9. Let p be an odd prime and recall thatSAut(Zp) ∼= Zp × Z(p− 1).

Let H = pF× denote the set of all rational numbers relatively prime top (see Proposition 3.11). By Proposition 3.11 (ii) every element in His a unit in the ring Zp. Put A := Zp and equip H with the discretetopology. Then G := AoH has torsion free discrete countable scalinggroup.

4.3. The structure of near abelian groups. Our observations inthe two preceding sections allow us to draw some conclusions in view ofwhat we saw in Theorems 3.6 and 3.10 on the structure of inductivelymonothetic groups and their faithful actions on locally compact abeliangroups. We keep in mind that we have a good understanding of locallycompact abelian periodic groups due to Braconnier’s Theorem 2.1 ontheir primary decomposition. It will be efficient to distinguish betweenthe nontrivial and the trivial situation. We keep in mind that the twosituations are mutually exclusive.

Theorem 4.10. (First Structure Theorem of Nontrivial Near AbelianGroups) Let G be a nontrivial locally compact near abelian group witha base group A. Then the following statements hold:

(1) A is periodic and contains G′ 6= {1}.(2) G/A is isomorphic to one of the following groups:

(2a) a subgroup of the discrete group Q,(2b) a subgroup of the discrete group Q/Z,(2c) an infinite compact monothetic group,

(2d) a local product M =∏loc

p (Mp, Cp) with the p-primary com-

ponents Mp such that Mp/Cp is finite cyclic or ∼= Z(p∞)(whenever Mp

∼= Qp) and that both∏

pCp and⊕

pMp/Cpare infinite.

(3) Γ = G/CG(A) is isomorphic to one of the following:(3a) a nontrivial proper subgroup of the discrete group Q,(3b) an infinite subgroup of the discrete group Q/Z whose p-

primary components are all finite,(3c) an infinite compact monothetic group,(3d) a noncompact inductively monothetic local product H =∏loc

p (Hp, Kp) where all p-primary factors Hp are compact.

Proof. (1) Since the action of Γ is nontrivial, G is nonabelian, A isperiodic by Lemma 4.2. Since G/A is abelian, G′ ⊆ A.


(2) Since G/A is inductively monothetic by definition, Theorem 3.6applies to it. The connected Case 3.6 (2) cannot apply since by (1)above and Theorem 2.9, SAut(A) is profinite and ψ : G → SAut(A),ψ(g)(a) = gag−1 is not trival by hypothesis.

(3) Here we apply Theorem 3.10 which suffices to conclude the proof.�

As an immediate consequence of Theorem 4.10 and Corollary 3.7 wehave the following

Corollary 4.11. A nontrivial near abelian locally compact group G istotally disconnected, and the following statements are equivalent:

(1) G is periodic.(2) G/A is not isomorphic to a subgroup of the discrete group Q.

The less sophisticated counterpart of Theorem 4.10 is the case of atrivial near abelian group G. The general structure of G/A is stillgiven by Theorem 3.6. Recall that G0 denotes the identity componentof a topological group G and G′ the commutator subgroup.

Theorem 4.12. (Structure Theorem of Trivial Near Abelian Groups)Let G be a trivial but nonabelian locally compact near abelian groupwith a base group A. Then the following statements hold

(1) Γ := G/CG(A) is of order 2.(2) G0 ⊆ G′ ⊆ A and G′ 6= {1}.(3) A is not periodic unless A has exponent 3 or 4.

The following diagram summarizes the situation:

inductively monothetic

G∣∣∣ }∼= {1,−1}

CG(A)∣∣∣ }inductively monothetic

A∣∣∣G′∣∣∣G0∣∣∣{1},

[arbitrarily nonperiodic]


where the property in square brackets is invalid if the exponent of A is3 or 4.

Proof. (1) is a direct consequence of the definition of a trivial nearabelian group and the hypothesis, that G is not abelian.(2) By Theorem 3.6, G/A is either zero dimensional or compact con-nected. We claim the former holds: If G/A{1,−1}, which factorsthrough G/A, would have to be trivial with the consequence that Gwould be abelian, contrary to the hypothesis. Thus G/A is totallydisconnected, that is G0 ⊆ A.Since G/A is abelian by Theorem 3.6, we know that G′ ⊆ A. Inview of the action of G on A by scalar automorphisms implementedby multiplication with {1,−1} when written additively, the subgroup[G,A] generated by all commutators gag−1a−1, a ∈ A, g ∈ G, is infact generated by the elements a−1a−1 = a−2, which themselves forma subgroup. Since the identity component G0 of a locally compactabelian group is divisible (see [10], Corollary 7.58 and Corollary 8.5),every element in G0 is a square and so G0 ⊆ [A,G] ⊆ G′.(3) Since G is trivially near abelian but not abelian, the assertion fol-lows from Theorem 2.8. �

We note that the case that the exponent of A is 2 is excluded bythe hypothesis that G is nonabelian and so G/CG(A) has order 2.There is a subtle difference between this hypothesis and the hypothesisSAut(A) = {idA,− idA} as the latter statement allows for the possibil-ity that − id = id in the case that the exponent of A is 2.The constructions in Example 4.5 indicate that Theorem 4.12 is ratheroptimal. The fact that the two structure theorems cover two disjointsituations show, for instance, that a near abelian group with nonde-generate connected subspaces is necessarily trivial.

There is material in the “other” version of Section 4, whichwould have to be integrated with Section 4 here. In par-ticular I would like to see the profinite structure theoremof near abelian groups. I wonder whether that would bebest presented in a section by itself.


I still think of two, maybe three, papers to be prepared:One on the structure of locally compact near abelian groupsas we know it by now, and another one on how near abeliangroups are related to the lattice theory of subgroups withthe entire hierarchy os stuff that the Russians have alreadydone. The first one would be the material in the paper plusmore material from the old section 4 not included here yet.This separation would allow us a bit more leisure. Thereis a third paper on material that got discarded from [11]because the referee did not find it interesting enough atthe time to be included. I want to have a look at it and seewhether someone might find it interesting in the contextof the structure of profinite p-groups.

5. Notes on the historical background

Material from the Introduction of earlier versions is tem-porarily placed here

Historical notes in [11, Section 8] and [25] observe a series of problemsin the proofs of the original results of Iwasawa [25, Theorem 2.3.1] inthe case of finite groups.Another interesting historical aspect is Strunkov’s article [26] of theyear of 1965. Here he gives a classification of compact groups in whichall closed subgroups are normal. Yet this paper fails to be mentioned in[4] in 1986, where similar results are proved in a complete different way.In fact, it seems that Strunkov’s article was available only in Russianduring the decade after its appearance. On the other hand, [26] isquoted by Kummich (1977, 1979), Kummich and Scheerer (1983) andScheiderer (1984) in German during the next decade (see [5, 11, 13,14, 24]). Again we observe the problem of availability of the originalresults in in the commonly used literature.Even more interesting is the observation of Mukhin’s work from 1967through 1988 [18, 19, 20, 21, 22] in which he classified the structure oflocally compact M-groups, where a topological group G is called an M -group if the lattice C(G) of its closed subgroups satisfies the Dedekindidentity:

〈X, Y 〉 ∩ Z = 〈X, Y ∩ Z〉 for X ⊆ Z, X, Y, Z ∈ C(G).

The classical Iwasawa theorems [25, Theorem 2.3.1] show that a finitep-group is an M -group if and only if it is quasihamiltonian. Mukhin


generalised the notion of M -groups to the locally compact case and de-scribed quasihamiltonian compact p-groups, yet his descriptions don’tprovide many details of their structure, say, in terms of splitting theo-rems. Only recently did we discover his results on this topic. His resultswere available only in Russian, unknown even to Strunkov, and theyremained unnoticed by most of the mathematical community. Still itwas he who has probably made the first attempt of generalizing theIwasawa theory to the context of topological groups. To the best ofour knowledge, Mukhin’s results were not translated into English tothe present day.Mukhin’s approach deals with a notion of rank (see [18, 19, 20]), inwhich an old terminology of Charin and Kurosh is used (see [3, 15]).This terminology was not easy to interpret. Analogies and differencesbetween Mukhin’s results in [21, 22] and near abelian compact p-groupsin [11] are described in the present paper. Roughly speaking, while thenotion of rank in the sense of Mukhin is not very useful for a precisedescription of near abelian compact groups its generalisation to locallycompact groups turns out to be very helpful.

Section 2 of our article presents some preliminary results, which allowus to classify the inductively monothetic groups in Section 3 (see Theo-rem 3.6 below). They are a special type of near abelian groups, studiedin [7, 11] in the compact case. Section 4 contains results of structureabout locally compact near abelian groups, namely Theorems 4.10 and4.12. They generalize (and complete) the preceding classifications in[7, 11]. In order to prove these theorems, we will use a special terminol-ogy for certain factorizations. More precisely, we follow [11] in sayingthat, if G = AH is a near abelian compact p-group, with a closed nor-mal abelian subgroup A and a monothetic compact subgroup H actingon A by scalar multiplication, the group A is an abelian base, while His called a scaling subgroup. Roughly speaking, we will see that thenotion of scaling subgroup will allow us to embed a locally compactnear abelian group in certain semidirect products due to Braconnier[2], which we will study with details. Finally, Section 5 describes someproblems of factorization in the present context of investigation. Hereis also noted that locally compact near abelian groups form a wide classof groups, containing most of the locally compact M -groups and theirgeneralizations, mentioned in [18, 19, 20, 21].

End of notes from old introduction

Recall that Mukhin’s condition D for a topological group G states thatevery monothetic subgroup X and closed subgroups Y and Z of G


with X ≤ Z satisfy the modular law 〈X, Y 〉 ∩ Z = 〈X, Y ∩ Z〉. Using

Mukhin’s notation X ∨ Y := 〈X, Y 〉 this becomes more familiar

(X ∨ Y ) ∩ Z = X ∨ (Y ∩ Z)

In a modular group all closed subgroups X, Y , and Z have to satisfythis modular law. It is clear from this that every modular group isa D-group. A topologically quasihamiltonian group is defined by theproperty that any two closed subgroups X and Y enjoy that XY = Y Xas sets and that XY = {xy | x ∈ X, y ∈ Y } is closed. Mukhin in [20]restricted his investigations to inductively compact groups in order toavoid the appearence of Tarski monsters. The same concern has ledIwasawa to restrict his investigations to locally finite groups – they arenothing but inductively compact groups with respect to the discretetopology.Now the results of Mukhin in Theorem 2.11 of [20] will show that allof the following groups are near abelian.

• Locally compact groups which are topologically quasihamilto-nian groups (tqh). These have been considered in [13, 14] andcompletely described in [20].• Every tqh-group is modular, i.e., its lattice of closed subgroups

is modular.• According to Yu. M. Mukhin in [20], every modular group

belongs to the class D of groups G whose monothetic subgroupsD satisfy the modular law 〈D,X〉 ∩ Y = 〈D,X ∩ Y 〉 for allclosed subgroups X and Y of G such that D ≤ Y holds.

It can be deduced from Mukhin’s structure theorem that these classesof locally compact groups contain each other as follows:

tqh ⊂ modular ⊂ D ⊂ near abelian

When we restrict our attention to locally compact p-groups then it isknown that all four classes coincide unless p = 2. This point illustratesthe relation of Mukhin’s results and ours as displayed in the previoussections.We close the paper by discussing Question 9.32, posed by Yu. N.Mukhin in the Kourovka Notebook [12]:

Question 5.1. For which locally compact groups G is the product ofany closed pair of subgroups A and B a closed subgroup ?

Remark 5.2. Note that a group with this property is by definition topo-logically quasihamiltonian.

The following efforts during the past are known to us:


• A topological group G is called almost connected if the factorgroup G/G0 is compact. In [13], Kummich proves that an al-most connected nonabelian group G is the projective limit offinite quasihamiltonian groups. But then, as Kummich in Fol-gerung 1 observes, such G is pronilpotent, i.e., is the cartesianproduct of its p-Sylow subgroups and each Sylow subgroup is acompact quasihamiltonian p-group. Such groups satisfy a struc-ture theorem analogeous to the one for (locally) finite groups,given by Iwasawa in 1943. This structure theorem has beenproved by Mukhin in [20] and became reproved in [11].• In [13], Kummich already observes that G0 6= 1 implies that G

must be abelian.• Mukhin, in [19], proves that every abelian M -group is topolog-

ically quasihamiltonian. He mentions this fact in his commentaccompanying the question.

Which question?

• Mukhin, in [20], observes that every locally compact M -groupcontaining Z as a discrete subgroup is also a topologically quasi-hamiltonian group.

References

[1] N. Bourbaki, Topologie generale, Chap. 10, Hermann, Paris, 1961.[2] J. Braconnier, Sur les groupes topologiques localement compacts, J. Math.

Pures Appl. (9) 27 (1948). 1–85 [French].[3] V. S. Charin, Groups of finite rank, Ukrainian Math. J. 16 No.2 (1964).[4] P. Diaconis and M. Shahshahani, On square roots of the uniform distribution

on compact groups, Proc. Amer. Math. Soc. 98 (1986), 341–348.[5] A. Di Bartolo, G. Falcone, P. Plaumann and K. Strambach, Algebraic groups

and Lie groups with few factors, Springer, Berlin, 2008.[6] L. Fuchs, Infinite Abelian Groups, II, Academic Press, New York, London,

1973.[7] W. Herfort, Factorizing profinite groups intwo two abelian subgroups, Int. J.

Group Theory 2 (2013), 45–47.[8] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. I Springer, Berlin,

1963.[9] K. H. Hofmann and W. A. F. Ruppert, Lie Groups and Subsemigroups

with Surjective Exponential Function, Mem. Amer. Math. Soc. 130 (1997)viii+174pp.

[10] K. H. Hofmann and S. A. Morris, The Structure of Compact Groups, deGruyter, Berlin, 2013, 3rd Ed., xxii+924 pp.

[11] K. H. Hofmann and F. G. Russo, Near abelian profinite groups, Forum Math.(2013), DOI 10.1515/forum-2012, to appear.


[12] E. I. Khukhro and V. D. Mazurov, Unsolved Problems in Group Theory, TheKourovka Notebook, 17th Edition, Sobolev Institute of Mathematics, Novosi-birsk, 2010.

[13] F. Kummich, Topologisch quasihamiltonische Gruppen, Arch. Math. (Basel)29 (1977), 392–397 [German].

[14] —, Quasinormalitat in topologischen Gruppen, Monat. Math. 87 (1979), 241–245 [German].

[15] A. G. Kurosh, The Theory of Groups, American Mathematical Society, Prov-idence, 1979.

[16] M. D. Levin, The automorphism group of a locally compact abelian group,Acta Mathematica 127 (1971), 259–278

[17] Z. I. Moskalenko, Locally compact abelian groups of finite rank, UkrainianMath. J. 22 (1970), 174–181.

[18] Yu. N. Mukhin, Locally compact groups with distributive lattice of closedsubgroups, Siberian Math. J. 8 (1967), 366–375.

[19] Yu. N. Mukhin, Topological abelian groups with Dedekind lattice of closedsubgroups, Mat. Zametki 8 (1970), 509–519.

[20] Yu. N. Mukhin, Automorphisms that fix the closed subgroups of topologicalgroups, Siberian Math. J. 16 (1975), 1231–1239.

[21] Yu. N. Mukhin and S.P. Khomenko, Monothetic groups and the subgrouplattice, Mat. Zap. Uralskogo Un-ta 6 (1987), 67–79 [Russian].

[22] Yu. N. Mukhin, Topological groups with a Dedekind lattice of closed sub-groups, Mat. Issled. 105 Moduli, Algebry, Topol. (1988), 129–141 [Russian].

[23] L. Ribes and P. Zalesskii, Profinite Groups, Springer, Berlin, 2010.[24] C. Scheiderer, Topologisch quasinormale Untergruppen zusammenhangender

lokalkompakter Gruppen, Monat. Math. 98 (1984), 75–81 [German].[25] R. Schmidt, Subgroup Lattices of Groups, de Gruyter, Berlin, 1994.[26] S. B. Strunkov, Topological hamiltonian groups, Uspehi Mat. Nauk. 20

(1965), 1157–1161 [Russian].[27] R. Winkler, Algebraic Endomorphisms of LCA-Groups that Preserve Closed

Subgroups, Geom. Dedicata 43 (1992) 307–320.


Institute for Analysis and Scientific ComputationTechnische Universitat WienWiedner Hauptstraße 8-10/101Vienna, AustriaE-mail address: [email protected]

Fachbereich MathematikTechnische Universitat DarmstadtSchlossgartenstr. 764289 Darmstadt, GermanyE-mail address: [email protected]

Instituto de MatematicaUniversidade Federal do Rio de JaneiroAv. Athos da Silveira Ramos 149, Centro de Tecnologia, Bloco CCidade Universitaria, Ilha do Fundao, Caixa Postal 6853021941-909, Rio de Janeiro, BrasilandIMPA, Instituto de Matematica Pura e AplicadaEstrada Dona Castorina 110, Jardim Botanico22460-320, Rio de Janeiro, BrasilE-mail address: [email protected]

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