# A Distinguished Subgroup of Compact Abelian Groups

^{1}

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^{4}

^{*}

## Abstract

**:**

## 1. Introduction

**$\delta $–subgroups**enter into the Resolution Theorem for compact abelian groups ([1], Theorem 8.20, p. 420, see also Section 6). The duals of the short exact sequences $\Delta \rightarrowtail G\twoheadrightarrow T$ where G is a compact group, $\Delta $ is a $\delta $–subgroup of G and thus T is a torus, are precisely the exact sequences $F\rightarrowtail A\twoheadrightarrow D$ where A is a discrete group, F is a free subgroup of A and D is a torsion group. This suggests the study of the

**full free subgroups**F of A, i.e., the free subgroups of A with torsion quotient. Let $\mathcal{F}\left(A\right)$ denote the family of all full free subgroups of A and let $\mathcal{D}\left(G\right)$ denote the family of all $\delta $–subgroups of the compact group G. In Theorem 1, a comprehensive description of $\mathcal{F}\left(A\right)$ is established, and by duality a similarly comprehensive description of $\mathcal{D}\left(G\right)$ is obtained (Theorem 6). In fact, there is an anti-isomorphism of semi-lattices $\delta :\mathcal{F}\left(A\right)\to \mathcal{D}\left(G\right)$ where $G={A}^{\wedge}$ (Theorem 5).

- (FD1)
- It contains $tor\left(G\right)$, is dense in G, and $G/\mathbf{\Delta}\left(G\right)$ is torsion-free and divisible (Theorem 6(2),(4),(6) and Theorem 10(2)).
- (FD2)
- If G is not totally disconnected, then $\mathbf{\Delta}\left(G\right)$ is a proper subgroup of G, and hence is not locally compact (Proposition 6(1)).
- (FD3)
- $\mathbf{\Delta}\left(G\right)$ is zero-dimensional (Theorem 19), and contains every closed totally disconnected subgroup of G (Proposition 5).
- (FD4)
- Fat Delta is a functorial subgroup in the sense that for any morphism $f:G\to H$ we have $f[\mathbf{\Delta}(G\left)\right]\subseteq \mathbf{\Delta}\left(H\right)$ (Corollary 3, Proposition 10(1)), moreover $f[\mathbf{\Delta}(G\left)\right]=\mathbf{\Delta}\left(H\right)$ if f is surjective (Proposition 10(2)).
- (FD5)
- The Fat Delta of a product is the product of the Fat Deltas of the factors (Theorem 10(4), Proposition 10(4)).
- (FD6)
- If $G={A}^{\wedge}$ is a compact group, then $\mathbf{\Delta}\left(G\right)=Hom(A,\mathbb{Q}/\mathbb{Z})$ (see Theorem 10(1) for a more rigorous formulation).
- (FD7)
- $\mathbf{\Delta}\left(G\right)$ determines G up to topological isomorphism (Theorem 12).

## 2. Notation and Background

- The category $\mathrm{AG}$ of discrete abelian groups with morphisms algebraic homomorphisms, ≅ denoting isomorphism in this category, also called algebraic isomorphism;
- $\mathrm{TAG}$ is the category of topological abelian groups with morphisms continuous algebraic homomorphisms, ${\cong}_{\mathrm{t}}$ denoting isomorphism in this category;
- $\mathrm{LCA}$ is as usual the full subcategory of $\mathrm{TAG}$ consisting of locally compact Hausdorff groups.

**torus**is a topological group isomorphic with a power ${\mathbb{T}}^{\mathfrak{m}}$ where $\mathfrak{m}$ is any cardinal.

**divisible hull**of A if D is divisible and A is an essential subgroup of D, equivalently, if $D/A$ is a torsion group and ${\u2a01}_{p\in \mathbb{P}}D\left[p\right]\subseteq A$. Divisible hulls exist for any group and divisible groups are direct sums of copies of $\mathbb{Q}$ and of $\mathbb{Z}\left({p}^{\infty}\right)$, $p\in \mathbb{P}$ ([17], p. 136).

**zero-dimensional**if G has a base of clopen sets. Clearly, every linearly topologized group is zero-dimensional and every zero-dimensional group is totally disconnected. Recall that a group is linearly topologized if it possesses a neighborhood basis at 0 consisting of subgroups.

**Lemma**

**1**

**.**Let G be a locally compact abelian group. Then G is totally disconnected if and only if G is zero-dimensional.

**precompact**if its completion is compact. It is a well-known and deep fact that a topological abelian group G is precompact if and only if the topology of G is generated by its continuous characters, which means that the characters $\chi \in {G}^{\wedge}$ separate the points of G and the injective (continuous) diagonal map $G\to {\prod}_{\chi \in {G}^{\wedge}}\chi \left[G\right]\le {\mathbb{T}}^{{G}^{\wedge}}$ is an embedding ([3]).

**Proposition**

**1.**

**Proof.**

**proper**if $\alpha $ is open onto its range. A short exact sequence $K\rightarrowtail G\twoheadrightarrow H$ is

**proper**if both maps are proper. Embeddings of subgroups are examples of proper monomorphisms, and proper epimorphisms are quotient maps. For a subgroup H of an abelian group G, we denote by $H\stackrel{ins}{\rightarrowtail}G$ the inclusion homomorphism, a proper map.

**Proposition**

**2.**

**Proof.**

**Lemma**

**2.**

- (1)
- Suppose that φ is continuous and K is dense in G. Then H is indiscrete.
- (2)
- Suppose that φ is an open map and H is indiscrete. Then K is dense in G.
- (3)
- Suppose that H is indiscrete and $cHom(G,H)$ is endowed with the compact-open topology. Then $cHom(G,H)$ is indiscrete.

**Proof.**

## 3. The Meet Semi-Lattice $\mathcal{F}\left(A\right)$ of Full Free Subgroups in AG

**rank**of A is the dimension of $\mathbb{Q}{A}_{0}$: $rk\left(A\right):=rk\left({A}_{0}\right):={dim}_{\mathbb{Q}}\left(\mathbb{Q}{A}_{0}\right)$.

**rank of**A by ${rk}_{p}\left(A\right):={dim}_{\mathbb{Z}/p\mathbb{Z}}\left(A\left[p\right]\right)$.

**Lemma**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Lemma**

**4.**

- (1)
- $\{{a}_{i}\mid i\in I\}$ is a linearly independent set in A if and only if $\{{a}_{i}+tor\left(A\right)\mid i\in I\}$ is a linearly independent set in ${A}_{0}$. Moreover, $\{{a}_{i}\mid i\in I\}$ is maximal linearly independent if and only if $\{{a}_{i}+tor\left(A\right)\mid i\in I\}$ is maximal linearly independent.
- (2)
- If $\{{a}_{i}\mid i\in I\}$ is a (maximal) linearly independent set in A and $\forall \phantom{\rule{0.166667em}{0ex}}i\in I:{t}_{i}\in tor\left(A\right)$, then $\{{a}_{i}+{t}_{i}\mid i\in I\}$ is a (maximal) linearly independent subset of A.
- (3)
- Every linearly independent set extends to a maximal linearly independent set. In particular, every torsion-free element in A is contained in a maximal linearly independent subset.
- (4)
- If $\{{a}_{i}\mid i\in I\}$ is a maximal linearly independent subset of A, then $F={\u2a01}_{i\in I}\mathbb{Z}{a}_{i}$ is a full free subgroup of A. Conversely, if $F={\u2a01}_{i\in I}\mathbb{Z}{a}_{i}$ is a full free subgroup of A, then $\{{a}_{i}\mid i\in I\}$ is a maximal linearly independent subset of A.
- (5)
- If $F\in \mathcal{F}\left(A\right)$, then ${F}_{0}\cong F$ and ${A}_{0}/{F}_{0}\cong A/(F\oplus tor\left(A\right))$, ${F}_{0}\in \mathcal{F}\left({A}_{0}\right)$, and ${\phi}_{0}^{-1}\left[{F}_{0}\right]=F\oplus tor\left(A\right)$.
- (6)
- Given ${F}_{0}\in \mathcal{F}\left({A}_{0}\right)$, there exists $F\in \mathcal{F}\left(A\right)$ such that ${\phi}_{0}\left[F\right]={F}_{0}$ and ${\phi}^{-1}\left[{F}_{0}\right]=F\oplus tor\left(A\right)$. If $F,{F}^{\prime}\in \mathcal{F}\left(A\right)$ and ${F}_{0}={F}_{0}^{\prime}$, then there is $\phi \in Hom(F,tor(A\left)\right)$ such that ${F}^{\prime}=\{\phi \left(x\right)+x\mid x\in F\}$. Note that $Hom(F,tor\left(A\right))\cong tor{\left(A\right)}^{rk\left(A\right)}$
- (7)
- A maximal linearly independent subset of ${A}_{0}$ is a $\mathbb{Q}$–basis of $\mathbb{Q}{A}_{0}$.
- (8)
- If $\{{v}_{i}\mid i\in I\}$ is a $\mathbb{Q}$–basis of $\mathbb{Q}{A}_{0}$, then there exist positive integers ${m}_{i}$ such that $\forall \phantom{\rule{0.166667em}{0ex}}i\in I:{m}_{i}{v}_{i}\in A$ and $F={\u2a01}_{i\in I}\mathbb{Z}\left({m}_{i}{v}_{i}\right)$ is a full free subgroup of A.

**Proof.**

**Theorem**

**1.**

- (1)
- Let $F,{F}^{\prime}\in \mathcal{F}$. Then $F\cap {F}^{\prime}\in \mathcal{F}$.
- (2)
- If $F\in \mathcal{F}$, ${F}^{\prime}\le F$ and $F/{F}^{\prime}$ is a torsion group, then ${F}^{\prime}\in \mathcal{F}$.
- (3)
- If $F\in \mathcal{F}$, then $\forall \phantom{\rule{0.166667em}{0ex}}m\in \mathbb{N}:mF\in \mathcal{F}$ and ${\bigcap}_{m}mF=\left\{0\right\}$.
- (4)
- $\bigcap \mathcal{F}=\left\{0\right\}$. If $A\ne tor\left(A\right)$ then $\bigcup \mathcal{F}=A\backslash tor\left(A\right)$ and $\sum \mathcal{F}=A$.
- (5)
- $\mathcal{F}$ is a meet semi-lattice with meet ∩.
- (6)
- Let $F\in \mathcal{F}$. Then $\forall \phantom{\rule{0.166667em}{0ex}}m\in \mathbb{N}:{m}_{A}^{-1}F={F}^{\prime}\oplus A\left[m\right]$ for some ${F}^{\prime}\in \mathcal{F}$, and $({F}^{\prime}\oplus A\left[m\right])/F=(A/F)\left[m\right]$. If A is torsion-free, then ${m}_{A}^{-1}F\in \mathcal{F}$.

**Proof.**

**Remark**

**1.**

**Proposition**

**3.**

**Proof.**

**Theorem**

**2.**

- (a)
- (([20], Theorem 2.2) and its proof) Given a free subgroup F of A with $rk\left(F\right)=rk\left(A\right)$, there is a second free subgroup ${F}_{1}$ such that $A=F+{F}_{1}$.
- (b)
- ([20], Corollary 3.5) There exists a full free subgroup ${F}_{0}$ of A such that $A/{F}_{0}$ is divisible (A is “quotient divisible”).

**Definition**

**1.**

**$\mathcal{F}$-summable**if for any ${F}_{1},{F}_{2}\in \mathcal{F}\left(A\right)$ also ${F}_{1}+{F}_{2}\in \mathcal{F}\left(A\right)$.

**Theorem**

**3.**

**Proof.**

## 4. The Semi-Lattices $\mathcal{F}\left(A\right)$ and $\mathcal{D}\left(G\right)$

**Lemma**

**5.**

**Proof.**

**annihilator ${X}^{\perp}$ of $X\subseteq G$ in ${G}^{\wedge}$**by ${X}^{\perp}:=({G}^{\wedge},X)=\{\chi \in {G}^{\wedge}\mid \chi \left[X\right]=0\}$ while for $Y\subseteq {G}^{\wedge}$, we define ${Y}^{\perp}=\{g\in G\mid \forall \phantom{\rule{0.166667em}{0ex}}\rho \in Y:\rho \left(g\right)=0\}$

**Lemma**

**6.**

**Proof.**

**Theorem**

**4**

**.**Let $G\in \mathrm{LCA}$. Then $H\mapsto {H}^{\perp}=({G}^{\wedge},H)$ with ${H}^{\perp \perp}=H$, is a lattice anti-isomorphism between $Lat\left(G\right)$ and $Lat\left({G}^{\wedge}\right)$. In particular, $H\subseteq K$ if and only if ${K}^{\perp}\subseteq {H}^{\perp}$.

**Theorem**

**5.**

- $\mathcal{D}\left(G\right)$ is a join semi-lattice with join +.
- $\forall \phantom{\rule{0.166667em}{0ex}}F,{F}_{1},{F}_{2}\in \mathcal{F}\left(A\right):\delta \left(F\right)={F}^{\perp}\in \mathcal{D}\left(G\right)$; if ${F}_{1}\subseteq {F}_{2}$, then $\delta \left({F}_{2}\right)\subseteq \delta \left({F}_{1}\right)$; $\delta ({F}_{1}\cap {F}_{2})=\delta \left({F}_{1}\right)+\delta \left({F}_{2}\right)$.

**Proof.**

**Definition**

**2.**

**Theorem**

**6.**

- (1)
- $\mathcal{D}$ is a join semi-lattice with join +. Hence, $\mathbf{\Delta}\left(G\right)=\bigcup \mathcal{D}$.
- (2)
- $\mathbf{\Delta}\left(G\right)$ is dense in G, while $\bigcap \mathcal{D}=\left\{0\right\}$ if $c\left(G\right)\ne \left\{0\right\}$, otherwise $\bigcap \mathcal{D}=G$.
- (3)
- Let $\Delta =\delta \left(F\right)$ and ${\Delta}^{\prime}=\delta \left({F}^{\prime}\right)$ and assume that $\Delta \subseteq {\Delta}^{\prime}$. Then ${F}^{\prime}\subseteq F$ and ${\Delta}^{\prime}/\Delta {\cong}_{\mathrm{t}}{(F/{F}^{\prime})}^{\wedge}$.
- (4)
- If $\Delta \in \mathcal{D}$, then ${m}_{G}^{-1}\Delta \in \mathcal{D}$ for any $m\in \mathbb{N}$. Hence, $tor(G/\Delta )\subseteq \Delta /\Delta $ and $tor\left(G\right)\subseteq \mathbf{\Delta}\left(G\right)$.
- (5)
- Let $\Delta \in \mathcal{D}$ and $m\in \mathbb{N}$. Then there is ${\Delta}^{\prime}\in \mathcal{D}$ such that $m\Delta ={\Delta}^{\prime}\cap mG$. If A is torsion-free, then $m\Delta \in \mathcal{D}$.
- (6)
- $G/\mathbf{\Delta}\left(G\right)$ is torsion-free.

**Proof.**

**Proposition**

**4.**

**Proof.**

- $\Delta :={\psi}^{\wedge}[{\left(\right)}^{\frac{A}{F}}\wedge ]$
- ${G}_{0}:={\phi}_{0}^{\wedge}\left[{A}_{0}^{\wedge}\right]$
- ${\Delta}_{0}:=({\phi}_{0}^{\wedge}\circ {\psi}_{0}^{\wedge})[{\left(\right)}^{\frac{{A}_{0}}{{F}_{0}}}\wedge ]\subseteq G$

**Theorem**

**7.**

- (1)
- ${G}_{0}={\phi}_{0}^{\wedge}\left[{A}_{0}^{\wedge}\right]=tor{\left(A\right)}^{\perp}$ coincides with the 0–component $c\left(G\right)$ of G.
- (2)
- $c\left(G\right)$ is divisible and so, algebraically, $G\cong c\left(G\right)\oplus {T}^{\wedge}$ and $G{\cong}_{\mathrm{t}}c\left(G\right)\oplus {T}^{\wedge}$ if and only if A splits, i.e., $A\cong {A}_{0}\oplus T$.
- (3)
- ${\Delta}_{0}=\Delta \cap c\left(G\right)$. Thus $\mathcal{D}(c(G\left)\right)=\{D\cap c(G)\mid D\in \mathcal{D}(G\left)\right\}$, $\mathbf{\Delta}(c(G\left)\right)=\mathbf{\Delta}\left(G\right)\cap c\left(G\right)$, and $\Delta (c(G\left)\right)$ is closed in $\mathbf{\Delta}\left(G\right)$.
- (4)
- $G=\Delta +c\left(G\right)$ and $\mathbf{\Delta}\left(G\right)=\Delta +\mathbf{\Delta}(c(G\left)\right)$.
- (5)
- $c\left(G\right)/{\Delta}_{0}{\cong}_{\mathrm{t}}G/\Delta $ and $\Delta /{\Delta}_{0}{\cong}_{\mathrm{t}}G/c\left(G\right){\cong}_{\mathrm{t}}{T}^{\wedge}$.
- (6)
- With the established notation $\mathbf{\Delta}(c(G\left)\right)$ is divisible and hence algebraically a direct summand of $\mathbf{\Delta}\left(G\right)$.
- (7)
- There is a topological isomorphism $tor{\left(A\right)}^{\wedge}{\cong}_{\mathrm{t}}\frac{\Delta}{{\Delta}_{0}}\to \frac{\mathbf{\Delta}\left(G\right)}{\mathbf{\Delta}(c(G\left)\right)}$.

**Proof.**

**Theorem**

**8.**

- (1)
- Suppose that D is a closed subgroup of G such that $G/D$ is a torus. Then there exists $\Delta \in \mathcal{D}\left(G\right)$ such that $D\cap \Delta \cap c\left(G\right)=0$. In particular, for every $\Delta \in \mathcal{D}\left(G\right)$, there is ${\Delta}^{\prime}\in \mathcal{D}\left(G\right)$ such that $\Delta \cap {\Delta}^{\prime}\cap c\left(G\right)=0$.
- (2)
- There exists $\Delta \in \mathcal{D}\left(G\right)$ such that ${\Delta}_{0}=\Delta \cap c\left(G\right)\in \mathcal{D}(c\left(G\right))$ is torsion-free.

**Proof.**

**Corollary**

**2.**

- (1)
- Suppose that D is a subgroup of G such that $G/D$ is a torus. Then there exists a subgroup ${D}^{\prime}$ of G such that $D\cap {D}^{\prime}=0$ and $G/{D}^{\prime}$ is a torus. In particular, for every $\Delta \in \mathcal{D}\left(G\right)$, there is ${\Delta}^{\prime}\in \mathcal{D}\left(G\right)$ such that $\Delta \cap {\Delta}^{\prime}=0$.
- (2)
- There exists a torsion-free $\Delta \in \mathcal{D}\left(G\right)$.

**Theorem**

**9.**

**Proof.**

**Remark**

**2.**

**free-reduced**if $Hom(A,\mathbb{Z})=\left\{0\right\}$, equivalently, if A has no free direct summands. Evidently, A is free-reduced if and only if $G={A}^{\wedge}$ is torus-free. For a compact group G, let $\mathcal{T}\left(G\right)=\{T\mid T\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{torus}\phantom{\rule{4.pt}{0ex}}\mathrm{subgroup}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}G\}$.

**Remark**

**3.**

## 5. The Fat Delta of Compact Groups

- $\mathbf{\Delta}\left(G\right)=\sum \mathcal{D}\left(G\right)=\bigcup \mathcal{D}\left(G\right)$,
- $\mathbf{\Delta}\left(G\right)$ is dense in G,
- $tor\left(G\right)\subseteq \mathbf{\Delta}\left(G\right)$ and $G/\mathbf{\Delta}\left(G\right)$ is torsion-free.
- $\Delta (c(G\left)\right)$ is divisible.

**Lemma**

**7.**

- (1)
- If G is totally disconnected, then so is H.
- (2)
- If G is a torus, then so is H.

**Proof.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

- (1)
- $\mathbf{\Delta}\left(G\right)$ is zero-dimensional, in particular totally disconnected (Theorem 19). Consequently, if G is not totally disconnected, then $G\ne \mathbf{\Delta}\left(G\right)$ and hence $\mathbf{\Delta}\left(G\right)$ is not a locally compact subgroup of G.
- (2)
- Any countable extension of $\mathbf{\Delta}\left(G\right)$ is zero-dimensional (in particular totally disconnected) as well (Proposition 11).

#### $\Delta \left({A}^{\wedge}\right)=Hom(A,\mathbb{Q}/\mathbb{Z})$

**Lemma**

**8.**

- (1)
- Suppose that H is a topological subgroup of K such that for all $f\in cHom(G,K)$ we have $f\left[G\right]\subseteq H$. Let $ins:H\to K$ be the insertion. Then ${ins}_{*}:cHom(G,H)\to cHom(G,K):{ins}_{*}\left(f\right)=ins\circ f$ is a topological isomorphism.
- (2)
- Suppose that $H\stackrel{\alpha}{\rightarrowtail}K\stackrel{\beta}{\twoheadrightarrow}L$ is a short exact sequence in $\mathrm{TAG}$, α is proper, and G is some other topological group. Then$$cHom(G,H)\stackrel{{\alpha}_{*}}{\rightarrowtail}cHom(G,K)\stackrel{{\beta}_{*}}{\to}cHom(G,L),\phantom{\rule{4.pt}{0ex}}where\phantom{\rule{4.pt}{0ex}}{\alpha}_{*}\left(f\right)=\alpha \circ f,\phantom{\rule{4pt}{0ex}}{\beta}_{*}\left(f\right)=\beta \circ f,$$
- (3)
- Let $A\stackrel{\alpha}{\rightarrowtail}B\stackrel{\beta}{\twoheadrightarrow}C$ be a short exact sequence of discrete groups and let G be a divisible topological group. Then$$cHom(C,G)\stackrel{{\beta}^{*}}{\rightarrowtail}cHom(B,G)\stackrel{{\alpha}^{*}}{\twoheadrightarrow}cHom(A,G),\phantom{\rule{4.pt}{0ex}}where\phantom{\rule{4.pt}{0ex}}{\beta}^{*}\left(f\right)=f\circ \beta ,\phantom{\rule{4pt}{0ex}}{\alpha}^{*}\left(f\right)=f\circ \alpha ,$$
- (4)
- For a discrete torsion group $T={\u2a01}_{p\in \mathbb{P}}{tor}_{p}\left(T\right)$, we have $cHom(T,\mathbb{Q}/\mathbb{Z}){\cong}_{\mathrm{t}}{T}^{\wedge}$, the topological isomorphism being ${ins}_{*}$, and ${T}^{\wedge}{\cong}_{\mathrm{t}}{\prod}_{p\in \mathbb{P}}{\left({tor}_{p}\left(T\right)\right)}^{\wedge}$ where ${\left({tor}_{p}\left(T\right)\right)}^{\wedge}{\cong}_{\mathrm{t}}$$Hom({tor}_{p}\left(T\right),\mathbb{Z}\left({p}^{\infty}\right))$.

**Proof.**

**Lemma**

**9.**

**Proof.**

- (1)
- By standard discrete homological algebra the diagram is commutative and rows and columns are exact.
- (2)
- All the domains of the Hom groups carry the discrete topology, hence $cHom=Hom$ in all cases.
- (3)
- All Hom groups in the diagram carry the compact-open topology. It follows from Lemma 8(2) that all the maps ${(\xb7)}_{*}$ are continuous. It follows from Lemma 8(3) that all the maps ${(\xb7)}^{*}$ are continuous.
- (4)
- By Lemma 8(1) the left most ${ins}_{*}$ is a topological isomorphism.
- (5)
- By Lemma 8(2) columns 2 and 3 are exact in $\mathrm{TAG}$.
- (6)
- By Lemma 8(3) the three rows are exact in $\mathrm{TAG}$.
- (7)
- The situation of Lemma 9 matches the top part of (5) and we conclude that ${ins}^{*}:Hom(A,\mathbb{Q}/\mathbb{Z})\to Hom(F,\mathbb{Q}/\mathbb{Z})$ is a quotient map. It is easy to see that ${ins}^{*}:Hom(A,\mathbb{R}/\mathbb{Q})\to Hom(F,\mathbb{R}/\mathbb{Q})$ is an isomorphism. Since both groups are indiscrete (by Lemma 2(3)), this is a topological isomorphism.

**Theorem**

**10.**

- (1)
- $\mathbf{\Delta}\left(G\right)={ins}_{*}[Hom(A,\mathbb{Q}/\mathbb{Z})]\subseteq G$ where $ins:\mathbb{Q}/\mathbb{Z}\to \mathbb{T}$.
- (2)
- $G/\mathbf{\Delta}\left(G\right)\cong {\mathbb{R}}^{\mathfrak{m}}$. Algebraically, $c\left(G\right)=\mathbf{\Delta}(c(G\left)\right)\oplus K$ where $K\cong {\mathbb{R}}^{\mathfrak{m}}$.
- (3)
- If ${G}_{i}={A}_{i}^{\wedge}$ where ${A}_{i}\in \mathrm{AG}$$(i\in I)$, then $\Delta \left({\prod}_{i\in I}{G}_{i}\right){\cong}_{\mathrm{t}}{\prod}_{i\in I}\Delta \left({G}_{i}\right)$.

**Proof.**

**Corollary**

**3.**

- (1)
- Let G and H be compact abelian groups and $g\in cHom(G,H)$. Then $g(\mathbf{\Delta}(G\left)\right)\subseteq \mathbf{\Delta}\left(H\right)$; in particular $\mathbf{\Delta}\left(G\right)$ is fully invariant in G and if $G\le H$, then $\mathbf{\Delta}\left(G\right)\le \mathbf{\Delta}\left(H\right)$.
- (2)
- Let $G,H,K$ be compact abelian groups. Suppose that $G\rightarrowtail H\twoheadrightarrow K$ is a short exact sequence in $\mathrm{TAG}$. Then $\mathbf{\Delta}\left(G\right)\rightarrowtail \mathbf{\Delta}\left(H\right)\twoheadrightarrow \mathbf{\Delta}\left(K\right)$ is a short exact sequence in $\mathrm{TAG}$.

**Proof.**

**Proposition**

**7.**

**Proof.**

**Theorem**

**11**

**.**Let $G,C$ be Hausdorff abelian groups, assume that C is complete, H is a dense subgroup of G. Then every morphism $f:H\to C$ has a unique extension $\overline{f}:G\to C$.

**Theorem**

**12.**

**Proof.**

- (1)
- D is totally disconnected and zero-dimensional (Proposition 11).
- (2)
- The completion $\widehat{D}$ of D is compact (i.e., D is precompact), $D=\Delta \left(\widehat{D}\right)$ and $tor\left(D\right)=tor\left(\widehat{D}\right)$.
- (3)
- D contains a directed family $\mathcal{D}$ of compact totally disconnected subgroups such that $D=\bigcup \mathcal{D}$.
- (4)
- $D\in \mathrm{LCA}$ if and only if $\widehat{D}$ is totally disconnected, and, if so, $D=\widehat{D}$ is compact.
- (5)
- D is totally minimal (Theorem 19).

**solenoid**is a compact connected group of dimension 1, i.e., the dual of a torsion-free group of rank 1. To do so, we will use a simple result on divisible hulls of discrete groups and Lemma 10 on divisible hulls of certain products of groups.

**Lemma**

**10.**

- (1)
- $D\left(X\right)$ is a divisible hull of X,
- (2)
- $D\left(X\right)={\prod}_{p\in P}^{\mathrm{loc}}({D}_{p},{X}_{p}):=\{\left({d}_{p}\right)\in D\mid {d}_{p}\in {X}_{p}\phantom{\rule{4.pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}almost\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}p\in P\}$,
- (3)
- $D\left(X\right)/X\cong {\u2a01}_{p\in P}{D}_{p}/{X}_{p}$.

**Proof.**

**rank-one groups**for short, are discussed and classified in ([17], Chapter 12, Section 1). These are exactly the groups isomorphic with additive subgroups of $\mathbb{Q}$ containing $\mathbb{Z}$. Types are equivalence classes $\left[{\left({h}_{p}\right)}_{p\in \mathbb{P}}\right]$ of “height sequences” ${\left({h}_{p}\right)}_{p\in \mathbb{P}}$ where $0\le {h}_{p}\le \infty $. Two height sequences are equivalent if they differ only at finitely many places where both sequences have finite entries. For the precise definition of type see Lemma 11(1) or ([17], p. 409, 411).

**Lemma**

**11.**

- (1)
- Let $\mathbb{Z}\le A\le \mathbb{Q}$. Then there exist values ${h}_{p}$ such that$$A=\left(\right)open="\langle "\; close="\rangle ">\frac{1}{{p}^{{h}_{p}}}\mathbb{Z}\mid p\in \mathbb{P},0\le {h}_{p}\le \infty ,\frac{A}{\mathbb{Z}}\cong {\u2a01}_{p\in \mathbb{P}}\mathbb{Z}\left({p}^{{h}_{p}}\right),\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}tp\left(A\right)=\left[{\left({h}_{p}\right)}_{p\in \mathbb{P}}\right].$$For ${\mathbb{P}}_{\infty}:=\{p\mid {h}_{p}=\infty \}$, one has $p\in {\mathbb{P}}_{\infty}$ if and only if $pA=A$.
- (2)
- Let $\Sigma ={A}^{\wedge}$ and $\Delta :={(A/\mathbb{Z})}^{\wedge}$. Then (with a harmless identification) $\Delta \in \mathcal{D}(\Sigma )$, and $\Delta {\cong}_{\mathrm{t}}{\prod}_{p\in \mathbb{P}}\widehat{\mathbb{Z}}\left({p}^{{h}_{p}}\right)$ where $\widehat{\mathbb{Z}}\left({p}^{\infty}\right)={\widehat{\mathbb{Z}}}_{p}$ is the group of p-adic integers and $\widehat{\mathbb{Z}}\left({p}^{{h}_{p}}\right)=\mathbb{Z}\left({p}^{{h}_{p}}\right)$ is the cyclic group of order ${p}^{{h}_{p}}$ for ${h}_{p}<\infty $. Furthermore, Σ and $\mathbf{\Delta}(\Sigma )$ are divisible, $\mathbf{\Delta}(\Sigma )/\Delta \cong \mathbb{Q}/\mathbb{Z}$, and $tor(\Sigma )\subseteq \mathbf{\Delta}(\Sigma )$.
- (3)
- $Soc(\Sigma )={\u2a01}_{p\notin {\mathbb{P}}_{\infty}}\mathbb{Z}\left(p\right)$ and $tor(\Sigma )={\u2a01}_{p\notin {\mathbb{P}}_{\infty}}\mathbb{Z}\left({p}^{\infty}\right)$.

**Proof.**

**Example**

**1.**

- (1)
- For a first concrete example, let ${A}_{1}={\sum}_{p\in \mathbb{P}}\frac{1}{p}\mathbb{Z}$ and ${\Sigma}_{1}={A}_{1}^{\wedge}$. Then $tp\left({A}_{1}\right)=\left[(1,1,\dots )\right]$, and $\mathbf{\Delta}\left({\Sigma}_{1}\right)$ is the divisible hull of $\Delta ={\prod}_{p\in \mathbb{P}}\mathbb{Z}\left(p\right)$ and $\mathbf{\Delta}\left({\Sigma}_{1}\right)={\prod}_{p\in \mathbb{P}}^{\mathrm{loc}}(\mathbb{Z}\left({p}^{\infty}\right),\mathbb{Z}\left(p\right))$.
- (2)
- Next let ${A}_{2}=\mathbb{Q}$. Then $tp\left({A}_{2}\right)=\left[(\infty ,\infty ,\dots )\right]$, ${\Sigma}_{2}={\mathbb{Q}}^{\wedge}$ is torsion-free, $\Delta ={\prod}_{p\in \mathbb{P}}{\widehat{\mathbb{Z}}}_{p}$ and $\mathbf{\Delta}\left({\Sigma}_{2}\right)$ is the divisible hull of Δ, so $\mathbf{\Delta}\left({\Sigma}_{2}\right)={\prod}_{p\in \mathbb{P}}^{\mathrm{loc}}({\widehat{\mathbb{Q}}}_{p},{\widehat{\mathbb{Z}}}_{p})$ where ${\widehat{\mathbb{Q}}}_{p}=\frac{1}{{p}^{\infty}}{\widehat{\mathbb{Z}}}_{p}$ is the additive group of p-adic numbers.
- (3)
- For ${A}_{3}=\mathbb{Z}$, $tp\left({A}_{3}\right)=\left[(0,0,\dots )\right]$, ${\Sigma}_{3}={\mathbb{Z}}^{\wedge}=\mathbb{T}$, $\Delta =\left\{0\right\}$, but $Soc\left({\Sigma}_{3}\right)={\u2a01}_{p\in \mathbb{P}}\mathbb{Z}\left(p\right)\subseteq \mathbf{\Delta}\left({\Sigma}_{3}\right)$, $\mathbf{\Delta}\left({\Sigma}_{3}\right)$ is the divisible hull of $Soc\left({\Sigma}_{3}\right)$, so $\mathbf{\Delta}\left({\Sigma}_{3}\right)={\u2a01}_{p\in \mathbb{P}}\mathbb{Z}\left({p}^{\infty}\right)=\mathbb{Q}/\mathbb{Z}=tor\left({\Sigma}_{3}\right)$.

**Proof.**

**Theorem**

**13.**

**Proof.**

**Remark**

**4.**

## 6. Resolutions

**Definition**

**3.**

- (i)
- ([1], Proposition 7.38(i), p. 374) $\mathfrak{L}\left(G\right)=\mathfrak{L}\left(c\right(G\left)\right)$ and $\mathfrak{L}$ commutes with products, i.e., $\mathfrak{L}\left({\prod}_{i}{G}_{i}\right){\cong}_{\mathrm{t}}{\prod}_{i}\mathfrak{L}\left({G}_{i}\right)$.
- (ii)
- ([1], Proposition 7.38(ii), p. 374) If $\phi :G\to H$ is a morphism in $\mathrm{TAG}$, then $\mathfrak{L}\left(\phi \right)$ is injective, whenever $Ker\phi $ is totally disconnected;
- (iii)
- ([1], Corollary 8.19, p. 419) if G is a compact group and $\Delta \in \mathcal{D}\left(G\right)$ with $G/\Delta ={\mathbb{T}}^{\mathfrak{m}}$, then, with $\phi :G\to G/\Delta $, $\mathfrak{L}\left(\phi \right):\mathfrak{L}\left(G\right)\to \mathfrak{L}(G/\Delta )={\mathbb{R}}^{\mathfrak{m}}$ is a topological isomorphism. The last equality is in fact a topological isomorphism obtained as composition of two others. The first one is the isomorphism $\mathfrak{L}\left({\mathbb{T}}^{\mathfrak{m}}\right){\cong}_{\mathrm{t}}\mathfrak{L}{\left(\mathbb{T}\right)}^{\mathfrak{m}}$ from (i). The second one is $\mathfrak{L}\left(\mathbb{T}\right){\cong}_{\mathrm{t}}\mathbb{R}$, that can be obtained from the obvious equality $\mathfrak{L}\left(\mathbb{T}\right)={\mathbb{R}}^{\wedge}$, by letting $\rho :\mathbb{R}\to \mathfrak{L}\left(\mathbb{T}\right):\rho \left(r\right)\left(x\right)=rx+\mathbb{Z}$ for $r,x\in \mathbb{R}$.

**Proposition**

**8**

**.**For a compact abelian group G there is a compact zero-dimensional subgroup Δ of G such that the homomorphism

- (1)
- φ is continuous, surjective, and open, i.e., is a quotient morphism.
- (2)
- $Ker\left(\phi \right)$ is algebraically and topologically isomorphic to $\Gamma :={exp}^{-1}[\Delta ]$, and Γ is a closed totally disconnected subgroup of $\mathfrak{L}\left(G\right)$. In particular, it does not contain any nonzero vector spaces.
- (3)
- $\phi \left[\right\{0\}\times \mathfrak{L}(G\left)\right]=exp\left[\mathfrak{L}\right(G\left)\right]$ is dense in $c\left(G\right)$, the identity component of G.

**Theorem**

**14**

**.**Let G be a compact abelian group of finite dimension $n:=dim\left(G\right)$. For $\Delta \in \mathcal{D}\left(G\right)$ define $\phi :\Delta \times \mathfrak{L}\left(G\right)\to G$ by $\phi (d,\chi )=d+exp\left(\chi \right)$ for $(d,\chi )\in \Delta \times \mathfrak{L}\left(G\right)$. Then:

- (1)
- φ is surjective, continuous, and open.
- (2)
- $\Gamma :=Ker\left(\phi \right)=\{(-exp\left(\chi \right),\chi )\mid \chi \in {exp}^{-1}[\Delta ]\}$. The projection $\Delta \times \mathfrak{L}\left(G\right)\to \mathfrak{L}\left(G\right)$ maps Γ isomorphically onto ${exp}^{-1}[\Delta ]$, so $\Gamma {\cong}_{\mathrm{t}}{exp}^{-1}[\Delta ]$. Furthermore, ${exp}^{-1}[\Delta ]$ is a closed totally disconnected subgroup of $\mathfrak{L}\left(G\right)$.
- (3)
- $\mathfrak{L}\left(G\right){\cong}_{\mathrm{t}}{\mathbb{R}}^{n}$, in particular ${dim}_{\mathbb{R}}\left(\mathfrak{L}\left(G\right)\right)=n$;
- (4)
- $\Gamma {\cong}_{\mathrm{t}}{\mathbb{Z}}^{n}$ where ${\mathbb{Z}}^{n}$ carries the discrete topology, i.e., the subspace topology in ${\mathbb{R}}^{n}$.
- (5)
- $exp\left[{exp}^{-1}[\Delta ]\right]=\Delta \cap a\left(G\right)$ is dense in Δ.

**Proof.**

**Remark**

**5.**

**Theorem**

**15**

**.**Let G be a compact abelian group and $\mathbf{\Delta}\left(G\right)=\bigcup \mathcal{D}\left(G\right)$. Then

- (1)
- the map $\phi :\mathbf{\Delta}\left(G\right)\times \mathfrak{L}\left(G\right)\to G:\phi \left(\right(d,\chi \left)\right)=d+exp\left(\chi \right)=d+\chi \left(1\right)$ is surjective, continuous, and open;
- (2)
- $\Gamma :=Ker\left(\phi \right)=\{(exp\left(\chi \right),-\chi )\mid \chi \in {exp}^{-1}[\mathbf{\Delta}\left(G\right)]\}{\cong}_{t}{exp}^{-1}[\mathbf{\Delta}\left(G\right)]\subset \mathfrak{L}\left(G\right)$ is torsion-free and φ induces an isomorphism $(\mathbf{\Delta}\left(G\right)\times \mathfrak{L}\left(G\right))/\Gamma {\cong}_{\mathrm{t}}G$;
- (3)
- If G is connected of finite dimension $dim\left(G\right)=n$, then $\Gamma {\cong}_{\mathrm{t}}{\mathbb{Q}}^{n}$.
- (4)
- $exp\left[{exp}^{-1}[\mathbf{\Delta}\left(G\right)]\right]=a\left(G\right)\cap \mathbf{\Delta}\left(G\right)$ is dense in G.

**Proof.**

**Example 2.**

- (a)
- For the solenoid, $\mathbb{T}={\mathbb{Z}}^{\wedge}$ there is an isomorphism $\rho :\mathbb{R}\to \mathfrak{L}\left(G\right)$ and $exp\left(\rho \right(r\left)\right)=r+\mathbb{Z}$, where $r\in \mathbb{R}$, by Definition 3(iii). Since $\mathbf{\Delta}\left(T\right)=tor\left(\mathbb{T}\right)=\mathbb{Q}/\mathbb{Z}$, we obtain the canonical resolution $\phi :\mathbb{Q}/\mathbb{Z}\times \mathbb{R}\to \mathbb{T}:\phi \left(\right(a+\mathbb{Z},r\left)\right)=(a+\mathbb{Z})+(r+\mathbb{Z})=a+r+\mathbb{Z}$ with $\Gamma =\left\{\right(r+\mathbb{Z},-r)\mid r\in \mathbb{Q}\}$ and evidently $\Gamma {\cong}_{\mathrm{t}}\mathbb{Q}$.
- (b)∧
- For the solenoid ${\Sigma}_{2}={\mathbb{Q}}^{\wedge}$ from Example 1 (2) $\Delta \left({\Sigma}_{2}\right)$ is the divisible hull of its delta subgroup $\Delta ={\prod}_{p}{\widehat{\mathbb{Z}}}_{p}$. Moreover, $\mathfrak{L}\left({\Sigma}_{2}\right){\cong}_{\mathrm{t}}\mathbb{R}$ (see Remark 5(c)). Theorem 15 gives the canonical resolution $\phi :\Delta \left({\Sigma}_{2}\right)\times \mathbb{R}\to {\Sigma}_{2}$ with $\Gamma =ker\phi {\cong}_{\mathrm{t}}\mathbb{Q}$ as in (a) and $a\left({\Sigma}_{2}\right)\cap \Delta \left({\Sigma}_{2}\right)\cong \mathbb{Q}$ dense in ${\Sigma}_{2}$.Denote by $\tilde{\mathbb{Q}}$ the group $\Delta \left({\Sigma}_{2}\right)$ equipped with the finer topology obtained by taking Δ as an open topological subgroup of $\tilde{\mathbb{Q}}$. Then $\tilde{\mathbb{Q}}$ is a locally compact ring and $\mathbb{A}:=\tilde{\mathbb{Q}}\times \mathbb{R}$ is the adele ring of $\mathbb{Q}$. Composing φ with the identity $\mathbb{A}\to \Delta \left({\Sigma}_{2}\right)\times \mathbb{R}$ we obtain a continuous surjective homomorphism $\phi :\mathbb{A}\to {\Sigma}_{2}$ which is again open by the Open Mapping Theorem (as $\mathbb{A}$ is σ-compact). Hence, ${\Sigma}_{2}$ is a quotient of $\mathbb{A}$.

## 7. Fat Delta through the Looking Glass of Quasi-Torsion Elements

#### 7.1. Quasi-Torsion Elements

**Definition 4**

**Example**

**3.**

**Remark**

**6.**

**Proposition**

**9.**

- 1
- If $x\in G$, then $x\in td\left(G\right)$ if and only if there exists a continuous homomorphism $f:(\mathbb{Z},{\nu}_{\mathbb{Z}})\to G$ with $f\left(1\right)=x$;
- 2
- $td\left(G\right)$ is a subgroup of G containing every compact totally disconnected subgroup of G;
- 3
- If G is complete (in particular, locally compact), then $td\left(G\right)$ coincides with the union of all compact, totally disconnected subgroups of G.

**Proof.**

**Theorem**

**16.**

**Proposition 10.**

- (1)
- ([3], Theorem 4.1.7(a)) If $f:G\to H$ is a continuous homomorphism of topological abelian groups, then $f[td(G\left)\right]\subseteq td\left(H\right)$, i.e., td is a functorial subgroup; in particular $td\left(G\right)$ is fully invariant in G.
- (2)
- ([27], Theorem 11) If G and H in (1) are compact and f is surjective, then $f[td(G\left)\right]=td\left(H\right)$.
- (3)
- (4)

**Remark**

**7.**

- (a)
- Items (1), (3) and (4) follow from Proposition 9 and reinforce Corollary 3(1) by showing that td is a functorial subgroup in the
**larger**category $\mathrm{TAG}$. - (b)
- In (2) “compact” cannot be replaced by “locally compact” (take $G=\mathbb{R}$, $H=\mathbb{T}$ and f the canonical quotient map, then $td\left(\mathbb{R}\right)=\left\{0\right\}$, while $td\left(\mathbb{T}\right)=\mathbb{Q}/\mathbb{Z}\ne \left\{0\right\}$).
- (c)
- Item (4) reinforces Theorem 10(3) showing that it remains valid in the
**larger**category $\mathrm{TAG}$.

**Proposition**

**11.**

**Proof.**

**Claim 1.**

**Proof.**

#### 7.2. The Subgroup $td\left(G\right)$ of Compact Groups and Minimality

**Definition**

**5.**

- (a)
**minimal**if every continuous isomorphism $f:G\to H$ onto a Hausdorff topological group H is open.- (b)
**totally minimal**if G satisfies the (full) Open Mapping Theorem, i.e., every continuous homomorphism $f:G\to H$ onto a Hausdorff topological group H is open.

**Definition**

**6**

**.**A subgroup H of a topological abelian group G is

**totally dense**if $\overline{N\cap H}=N$ for every closed subgroup N of G.

**Definition**

**7**

**Theorem**

**17.**

**Theorem**

**18**

**Proof.**

**Proposition**

**13.**

**Proof.**

**Theorem**

**19.**

**Proof.**

**Corollary**

**4.**

**Proof.**

#### 7.3. Sylow Subgroups of $td\left(G\right)$ for $G\in \mathrm{TAG}$

**Definition**

**8**

**topologicalp-Sylow subgroup**of G. We shall also keep this terminology when G is not necessarily profinite. Clearly, ${H}_{p}={G}_{p}\cap H$ for a subgroup H of G.

**quasi-p-torsion**in [4]) such that $\langle x\rangle $ is either a cyclic p-group, or $\langle x\rangle $ is isomorphic to $\mathbb{Z}$ equipped with the p-adic topology.

**Theorem**

**20**

**.**For a compact abelian group G and every prime p the subgroup ${td}_{p}(c\left(G\right))$ is dense in $c\left(G\right)$. In particular, the following conditions are equivalent:

- (1)
- G is totally disconnected;
- (2)
- $td\left(G\right)=G$, i.e., $td\left(G\right)$ is compact;
- (3)
- ${td}_{p}\left(G\right)$ is compact for every prime p;
- (4)
- ${td}_{p}\left(G\right)$ is compact for some prime p;
- (5)
- ${td}_{p}\left(G\right)$ is closed in G for some prime p (equivalently, for all primes p);
- (6)
- the topology induced from G on $wtd\left(G\right)={\u2a01}_{p\in \mathbb{P}}{td}_{p}\left(G\right)$ coincides with the topology induced by the product topology of ${\prod}_{p\in \mathbb{P}}{td}_{p}\left(G\right)$.

**Definition**

**9**

**Theorem**

**21**

- (1)
- $wtd\left(G\right)$ is torsion;
- (2)
- $Soc\left(G\right)$ is topologically essential;
- (3)
- G contains copies of the p-adic integers ${\widehat{\mathbb{Z}}}_{p}$ for no prime p;
- (4)
- $n=dim\left(G\right)<\infty $ and for every continuous surjective homomorphism $f:G\to {\mathbb{T}}^{n}$ we have $Kerf={\prod}_{p}{B}_{p}$, where each ${B}_{p}$ is a (bounded) compact p-group;
- (5)
- $n=dim\left(G\right)<\infty $ and there exists a homomorphism $f:G\to {\mathbb{T}}^{n}$ as in (3);
- (6)
- $wtd\left(G\right)\cong {(\mathbb{Q}/\mathbb{Z})}^{n}\times {\u2a01}_{p\in \mathbb{P}}{B}_{p}$ algebraically, where each ${B}_{p}$ is a (bounded) compact p-group;
- (7)
- A is
**strongly non-divisible**, i.e., all non-trivial quotients of A are non-divisible; - (8)
- every proper subgroup of A is contained in some maximal subgroup of A;
- (9)
- A admits a surjective homomorphism $A\to \mathbb{Z}\left({p}^{\infty}\right)$ for no prime p;
- (10)
- $n=rk\left(A\right)<\infty $ and $A/F\cong \u2a01{T}_{p}$, where each ${T}_{p}$ is a bounded p-group, for every $F\in \mathcal{F}\left(A\right)$;
- (11)
- $n=rk\left(A\right)<\infty $ and there exists $F\in \mathcal{F}\left(A\right)$ as in (10).

**Corollary**

**5.**

**almost countable**if it is the completion of countable minimal abelian group. This class of compact groups was described by Prodanov [12] as follows: a compact abelian group G is almost countable if and only if $n=dim\left(G\right)<\infty $ and there exists a homomorphism $f:G\to {\mathbb{T}}^{n}$ such that $Kerf={\prod}_{p}({\widehat{\mathbb{Z}}}_{p}^{{e}_{p}}\times {F}_{p})$, where ${F}_{p}$ is a finite p group and ${e}_{p}\in \{0,1\}$ for every prime p. These are the compact abelian groups G such that $td\left(G\right)$ has a countable essential subgroup.

**Example**

**4.**

- (1)
- G is an exotic torus if and only if ${\mathbb{P}}_{\infty}=\varnothing $ (i.e., $tp\left(A\right)$ has no entries ∞).
- (2)
- It follows from (1) that there are $\mathfrak{c}$ many pairwise non-isomorphic connected one-dimensional exotic tori G; they all have $wtd\left(G\right)\cong \mathbb{Q}/\mathbb{Z}$, according to Corollary 5. Nevertheless, for these exotic tori G the subgroups $wtd\left(G\right)$ remain pairwise non isomorphic (since, similarly to Theorem 12, if $wtd\left(G\right){\cong}_{t}wtd\left(H\right)$, then $G{\cong}_{t}H$ for every pair of compact abelian groups $G,H$).
- (3)
- According to Theorem 13, if G is an exotic torus, then $Soc\left(G\right)$ is dense in G if and only if ${\mathbb{P}}_{0}$ is infinite (see ([32], Proposition 2.5) for a more general result in the case of connected exotic tori of arbitrary dimension). According to Theorem 21, in this case, $Soc\left(G\right)$ is the smallest dense topologically essential subgroups of G.
- (4)
- The second assertion in (3) is related to the following more general fact proved in ([33], Theorem 5.1) justifying the interest in dense socles: a connected compact abelian group G contains a smallest dense topologically essential (i.e., smallest dense minimal) subgroup of G if and only if G is an exotic torus with dense $Soc\left(G\right)$.

## 8. Final Comments and Open Problems

**Problem**

**1.**

**Example**

**5.**

**Problem**

**2.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Dikranjan, D.; Lewis, W.; Loth, P.; Mader, A.
A Distinguished Subgroup of Compact Abelian Groups. *Axioms* **2022**, *11*, 200.
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Dikranjan D, Lewis W, Loth P, Mader A.
A Distinguished Subgroup of Compact Abelian Groups. *Axioms*. 2022; 11(5):200.
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