## 1 Introduction

Following [7, 10], we say that a *ball structure* is a triple *X*, *P* are non-empty sets, and for all *X* which is called a *ball of radius* α around *x*. It is supposed that *X* is called the *support* of *P* is called the *set of radii*.

Given any

A ball structure *ballean* if

- •for any
, there exist$\alpha ,\beta \in P$ such that, for every${\alpha}^{\prime},{\beta}^{\prime}\in {P}^{\prime}$ ,$x\in X$ $B(x,\alpha )\subseteq {B}^{*}(x,{\alpha}^{\prime}),{B}^{*}(x,\beta )\subseteq B(x,{\beta}^{\prime}),$ - •for any
, there exists$\alpha ,\beta \in P$ such that, for every$\gamma \in P$ ,$x\in X$ $B(B(x,\alpha ),\beta )\subseteq B(x,\gamma ),$ - •for any
, there exists$x,y\in X$ such that$\alpha \in P$ .$y\in B(x,\alpha )$

We note that a ballean can be considered as an asymptotic counterpart of a uniform space, and could be defined [11] in terms of the entourages of the diagonal *coarse structure*. For categorical look at the balleans and coarse structures as “two faces of the same coin” see [3].

Let *mapping* if, for every

A bijection *asymorphism* between *f* and *asymorphic*.

Let *Y* of *X* defines a *subballean**Y* of *X* is called *large* if *X* and *coarsely equivalent* if there exist large subsets

We recall [8] that an ideal *G* is a *group ideal* if *G* and denote by *e* is the identity of *G*.

For an infinite group *G* and an infinite cardinal

We say that two groups *G* and *H* are *-asymorphic* (*-coarsely equivalent*) if the balleans

In the case *G* and *H* are *finitarily asymorphic* and *finitarily coarsely equivalent* respectively. We note that finitely generated groups are finitarily coarsely equivalent if and only if *G* and *H* are quasi-isometric [5, Chapter 4].

A classification of countable locally finite groups (each finite subset generates finite subgroup) up to finitary asymorphisms is obtained in [6] (cf. [7, p. 103]).

*Two countable locally finite groups *

- (i)
*for every finite subgroup*, , there exists a finite subgroup$F\subset {G}_{1}$ *H*of such that${G}_{2}$ is a divisor of$\left|F\right|$ $\left|H\right|$ - (ii)
*for every finite subgroup**H*of , there exists a finite subgroup${G}_{2}$ *F*of such that${G}_{1}$ is a divisor of$\left|H\right|$ .$\left|F\right|$

It follows that there are continuum many distinct types of countable locally finite groups and each group is finitarily asymorphic to some direct sum of finite cyclic groups.

The following coarse classification of countable Abelian groups is obtained in [2].

*Two countable Abelian groups are finitarily coarsely equivalent if and only if the torsion-free ranks of G and H coincide and G and H are either both finitely generated or infinitely generated.*

In particular, any two countable torsion Abelian groups are finitely coarsely equivalent.

The referee suggested to mention [1, 4, 12] in connection to the coarse classification of Abelian groups, as well as the asymptotic dimension and its connection to the Hirsh length.

This note is motivated by the following result [9, Theorem 3]. Let *G*, *H* be groups of cardinality γ, *G* and *H* are γ-asymorphic.

What happens if γ is singular? Are *G* and *H*

## 2 Theorems

*Let G be an Abelian group of cardinality γ, *

*Let G be an Abelian group of cardinality γ, *

The referee asks: “The ‘huge jump’ from Abelian to free (non-Abelian) groups in Theorem 2 gives rise to the natural question of what one can do for groups close to being Abelian (as done in [1] in the case of finitary coarse equivalence).”

In this line, we conjecture that Theorem 1 remains true if *G* satisfies the following condition:

## 3 Proofs

We choose inductively an increasing chain of subgroups *G* such that

- (1)
,${G}_{0}=\left\{e\right\}$ ,$G={\bigcup}_{\alpha <\gamma}{G}_{\alpha}$ - (2)
for all${G}_{\alpha}\subset {G}_{\beta}$ ,$\alpha <\beta <\gamma $ - (3)
for every limit ordinal${G}_{\beta}={\bigcup}_{\alpha <\beta}{G}_{\alpha}$ ,$\beta <\gamma $ - (4)
for every$|{G}_{\alpha +1}:{G}_{\alpha}|={\aleph}_{0}$ .$\alpha <\gamma $

For each

We take an arbitrary element

By the construction, the representation is unique, so we can denote

We identify

We show that the bijection *f* is an asymorphism between the balleans

To show that

and it is suffices to note that

The verification that *f* is a *Y* the smallest subgroup of *G* containing *F* and such that if *Y* can be obtained in the following way. For a subset *A* of *G*, we denote by *A* and

To conclude the proof, we take an arbitrary

We are going to get a contradiction assuming only that there is a *G* is written additively.
In view of Theorem 1, we may suppose that *G* is a group of exponent 2.
Since *f* is a

We note that the family

- (5)
for each$\left|{K}_{y}\right|<\delta $ .$y\in G$

If

We denote *f* is a *Y* of

We consider *g*, and for subset *S* of

Now we show how to find

- (6)
,$alpf\left(a\right)\setminus alp{K}_{b}\ne \varnothing $ - (7)
.$alpf\left(b\right)\setminus alp{K}_{a}\ne \varnothing $

Since

and find

We put

- (8)
,$f\left(b\right)=f\left(a+\left(a+b\right)\right)\in {K}_{a}z$ - (9)
.$f\left(a\right)=f\left(b+\left(a+b\right)\right)\in {K}_{b}z$

We take *u* occurs in *z* before *v*, then, by (9),
*v*. If *v* occurs in *z* before *u*, then, by (8), *u*.
∎

We thank the referee for constructive comments, corrections and additional references.

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