# Direct decomposition theory under near-isomorphism for a class of infinite rank torsion-free abelian groups

Ekaterina Blagoveshchenskaya and Lutz H. Strüngmann
From the journal Journal of Group Theory

## Abstract

Near-isomorphism is known as the right concept for classification theorems in the theory of almost completely decomposable groups. As a natural generalization the authors extended in [6] the notion of near-isomorphism to Abelian groups of arbitrary rank. In this article we investigate non-isomorphic direct decompositions of a class of infinite rank torsion-free abelian groups which were defined in [4] as special epimorphic images of so-called almost rigid groups. A complete classification of such decompositions up to near-isomorphism is given.

## 1 Introduction

In the theory of Abelian groups we know many examples where a classification of groups up to some reasonable variant of isomorphism has successfully been obtained. In particular, a characterization by numerical invariants has been of interest. For instance, completely decomposable groups (direct sums of subgroups of the rationals) were classified up to group isomorphism by so-called types and ranks of their homogeneous components. In the torsion case and later on in the mixed case Ulm’s invariants and Warfield’s invariants were used to classify totally projective Abelian p-groups and Warfield groups up to isomorphism. A very good source on these results are the two volumes by Fuchs [7] and Loth [9]. Focusing on the class of torsion-free Abelian groups however, it is in general very difficult (and very often impossible) to obtain such a classification. A very good example where a modified notion of isomorphism led to success in this respect is the class of almost completely decomposable groups (acd-groups) which are finite extensions of completely decomposable groups of finite rank (see Mader [10]). It turned out that for acd-groups the concept of near-isomorphism was more suitable than classical isomorphism. Recall that two torsion-free abelian groups X and Y of finite rank are nearly-isomorphic, in symbols XnrY, if and only if for every prime p there is a monomorphism φp:XY such that Y/φp(X) is finite with zero p-torsion part. Near-isomorphism is indeed an equivalence relation (so XnrY implies YnrX) and from many perspectives almost completely decomposable groups under near-isomorphism are very attractive, in particular, concerning their direct decompositions and the links with their endomorphism rings: nearly-isomorphic acd-groups have isomorphic regulators, nearly isomorphic endomorphism rings (considered as Abelian groups), and sometimes even allow a sort of Baer–Kaplansky theorem. Finally, their direct decompositions in the case of so-called crq-groups (i.e. finite cyclic extensions of completely decomposable groups) can be classified up to near-isomorphism in terms of the so-called type-invariants (see the standard book by Mader [10] on acd-groups and the references in there).

There are several ways to extend the class of almost completely decomposable groups to infinite rank, e.g., by looking at continuous increasing chains of acd-groups or by considering torsion-free Abelian groups of infinite rank with all their finite rank pure fully invariant subgroups isomorphic to almost completely decomposable groups – called local acd-groups. In each case it is also desirable to have an appropriate notion of near-isomorphism. Such concept was introduced by the authors in [6, Definition 2.3, Proposition 3.4] by defining two torsion-free Abelian groups X and Y of arbitrary rank to be nearly-isomorphic (XnrY) if for every prime p there exist monomorphisms ηp:XY and μp:YX such that

1. Y/Xηp and X/Yμp are torsion,

2. (Y/Xηp)p=0=(X/Yμp)p,

3. for all finite rank pure subgroups XX and YY the quotients (Xηp)*Y/Xηp and (Yμp)*X/Yμp are finite.

It was shown in [6] that this notion extends the classical notion of near-isomorphism for finite rank groups (forced by condition (3) above) and for completely decomposable groups it is equivalent to isomorphism as one should expect. Thus keeping the name near-isomorphism is justified.

Our goal in this paper is to classify up to near-isomorphism all possible direct decompositions of countable rank torsion-free Abelian groups from a special class defined in [4]. The first results in this direction were obtained in [3] and later on in [4] for a class 𝔗 of Abelian groups where a group X in 𝔗 is generated by a completely decomposable subgroup R(X)=τTcr(X)CτX of X – the so-called regulator – with critical typeset Tcr(X) an antichain and τ-homogeneous completely decomposable pure subgroups CτX of X such that the quotient X/R(X) is a direct sum of bounded p-groups. Moreover, the relations defining X with respect to R(X) are of a finite character (see [4] for further details). A complete classification up to near-isomorphism was obtained in the series of papers [1, 3, 6] for different subclasses of 𝔗 assuming that the ranks of the homogeneous components CτX of R(X) are finite (this class was denoted by 𝔗 in [4]): the class of rigid groups, i.e. groups X𝔗 such that the rank of any homogeneous component is one, and the class of almost rigid groups, i.e. groups X𝔗 with homogeneous components of finite rank, the critical typeset Tcr(X) is countable and all primary components of X/R(X) are cyclic groups. In fact, almost rigid groups have a decomposition into a direct sum of a completely decomposable group and a rigid summand from X𝔗 which explains the name almost rigid. In particular, it was shown in [3] and [1, Theorem 3.4] that for countable rank groups from the class 𝔗, in particular for almost rigid groups, a classical theorem by Arnold can be extended: If X and Y are nearly-isomorphic and X=X1X2, then there is also a decomposition Y=Y1Y2 such that X1nrY1 and X2nrY2 (for the reader’s convenience we should say that the class of countable rank groups from 𝔗 was denoted by 𝒞 in the paper [1] published in Russian). Moreover, rigid groups and also almost rigid groups from 𝔗 were classified up to near-isomorphism by their regulators and certain numerical invariants mτ (see [3] and [4]) and a decomposition theory was constructed in [3] in terms of the numerical invariants as a development of the combinatorial approach invented for some almost completely decomposable groups in [5] and [10, Chapter 13]. Finally, the authors and Göbel passed to certain epimorphic images of rigid groups and also almost rigid groups, called proper B(1) asd-groups and proper B(1) alr-groups, which are obtained by factoring out a subgroup K of an (almost) rigid group X. The subgroup K is of the form K=iωK(Ai)li, where K(Ai)=τTiατaτ and the regulator of X satisfies

R(X)=iωAi=iωτTiτaτ

(for a more detailed definition see Definition 2.2 below). The integers li, αi and the sets of types TiTcr(X) are called a triple of partitions, coefficients, and parameters for X. For proper (1) asd-groups and proper (1) alr-groups a classification up to near-isomorphism is available (see [4]) in the sense that two such groups G=X/KX and H=Y/KY are nearly-isomorphic if and only if their pre-images X and Y are nearly-isomorphic and the triples of partitions, coefficients, and parameters for X and Y coincide.

In this paper we extend the numerical approach for (almost) rigid groups to the proper (1) alr-groups using the near-isomorphism invariants mτ from [4]: Two proper 𝔅(1)alr-groups B and C are nearly-isomorphic if and only if R(B)R(C) and mτ(B)=mτ(C) for each critical type τ𝔗 (Theorem 3.3). Moreover, two classification theorems up to near-isomorphism of direct decompositions of proper (1) alr-groups are obtained by carrying over the near-isomorphism classifications for almost rigid groups to the class of proper (1) alr-groups (see Lemma 4.4 and Theorem 4.6).

The main result of the present paper is the direct decomposition classification in Theorem 4.6 in terms of the numerical invariants for proper (1) alr-groups. These groups have many common features with almost rigid groups. However, the big difference is that they are torsion extensions of direct sums of strongly indecomposable groups while almost rigid groups are torsion extensions of completely decomposable groups.

## 2 Preliminaries and notation

All definitions and group properties, which form the basis of our considerations, can be found in the two standard books by Fuchs [7] and Mader [10]. The latter contains special notions of the theory of almost completely decomposable groups which we will use and extend in this paper to some classes of infinite rank groups X of ring type, i.e. groups having critical typesets Tcr(X) consisting only of idempotent types τ. Such idempotent types can be represented by characteristics with only 0s and s as entries, see [10, p. 13], [7, Section 85]). We write τ(p)= if τ has as entry at position p and τ(p)=0 for the other p. Then τa is a τ-homogeneous group, containing the element a, that is divisible only by primes p with τ(p)=.

Moreover, the groups X under consideration are block-rigid (i.e. with antichains as their critical typesets) and, therefore, for any τTcr(X) the fully invariant pure subgroup X(τ) is a homogeneous group of type τ denoted by Xτ, and which will be called a τ-homogeneous component of X. As for other notation, we use Y*X to indicate that Y is a pure subgroup of the torsion-free group X, and Tp for the p-component in the primary decomposition of a torsion group T into primary summands. In most cases we say “group” for a torsion-free abelian group X because it is our main object and denote the subgroup Y of Y such that qY=X by Xq with natural q.

We now recall the basic definitions needed in the sequel. Definition 2.1 in [4] of the class 𝔗 with the restrictions before Observation 2.2 in [4] lead to the following definition.

Definition 2.1

A torsion-free abelian group X belongs to the class 𝔗 if there exists a completely decomposable subgroup R(X)=τTcr(R(X))CτX of X such that the following conditions are satisfied:

1. 𝔗=Tcr(R(X)) is an antichain of idempotent types,

2. R(X)τ:=CτX*X is a finite rank pure and τ-homogeneous completely decomposable subgroup of X for all τ𝔗,

3. X/R(X)=pPXTpX for some set of primes PX and pγp-bounded p-groups TpX,

4. for every pPX the set {qPX:[TpX][TqX]} is finite; here, [TpX] is the minimal subset 𝔗p𝔗 such that

TpX((τ𝔗pCτX)*+R(X))/R(X).

We will be interested in factor groups of almost rigid groups, a subclass of the class 𝔗. Recall from [4] that a group X𝔗 is called an almost rigid group (alr-group) if the set 𝔗 is countable and all primary components TpX of X/R(X) are cyclic groups. Knowing their classification from [3] we will use the almost rigid groups as preimages of the groups, which are to be classified here up to near-isomorphism. The following recalls [4, Definitions 5.1, 5.3]:

Definition 2.2

Let X𝔗 be an almost rigid group and let (ατ:τ𝔗) be integers such that the following conditions hold:

1. R(X)=iωAili (where li) with Ai=τTiτaτ, 𝔗=iωTi the countable disjoint union of finite subsets Tcr(Ai)=Ti𝔗 and rkAi=|Ti|2 for any iω,

2. for any iω and any τTi there exists a prime p such that τ(p)= and σ(p) for all στ, σTi,

3. if iω and |Ti|1, then τσ,τTiτ= for any σTi,

4. if iω and |Ti|1, then ατ(τTi) is not p-divisible if σ(p)= for some σTi; if |Ti|=1, then ατ=0 with Ti={τ},

5. if iω and |Ti|1, then gcd({ατ:τσ,τTi})=1 for any σTi,

6. |[TpX]Ti|1 for any pPX and iω,

7. if iω and |[TpX]Ti|=1 for some pPX, then each ατ is relatively prime to p for any τTi,

8. if iω and |[TpX]Ti|=1, then τ(p) for any τTi.

Let KX be defined as follows:

K=iωKli(Ai)XwithK(Ai)=τTiατaτAi.

The partition (Ti:iλ) and the two sets of integers (ατ:τ𝔗), (li:iω) will be called a triple of partition, coefficients and parameters for the group X. Moreover, the group B=X/K is called a proper B(1)alr-group and the canonical epimorphism ϕ:XB=X/K will be called a regular representation of the proper 𝔅(1)alr-group B. The regulator of B is defined as R(B)=R(X)ϕ.

Note that for simplicity we use Kli(Ai) to denote the direct sum (K(Ai))li of li copies of K(Ai) in the above Definition 2.2.

Since we intend to classify proper 𝔅(1)alr-groups in terms of numerical invariants we need to use the invariants mτ(X), τTcr(X), of almost rigid groups defined in [3, Definition 3.2] and [4, Definition 5.4].

Recall that for an almost rigid group X𝔗 there are elements upX such that

pPup+R(X)=X/R(X)with|up+R(X)|=pγp

for some integers γp since the primary components of X/R(X) are assumed to be cyclic. Writing

pγpup=τ𝔗uτp with uτpR(X)τ

we have

pγpup¯=τ𝔗uτp¯,uτp¯R(X)τ/pγpR(X)τ,

where ¯:R(X)R(X)/pγpR(X) is the canonical epimorphism. The numerical invariants mτ(X) of X were then defined as

mτ(X)=pP|uτp¯|for τ𝔗.

The appearance of these numbers for proper 𝔅(1)alr-groups will be discussed in detail below, but it was already shown in [4, Definition-Lemma 5.6] that conditions (6) to (8) of Definition 2.2 can be reformulated using the numerical invariants mτ:

Lemma 2.3

Conditions (6)–(8) of Definition 2.2 are equivalent to the following ones:

1. if there exists iω with τTi and σTi, then gcd(mτ(X),mσ(X))=1,

2. if there exists iω with τTi and σTi, then ατ is relatively prime to mσ(X),

3. if there exists i with τTi and σTi, then τ(p) for any prime divisor p of mσ(X).

The basis of our desired decomposability criterion for proper 𝔅(1)alr-groups will be the corresponding decomposability criterion [3, Theorem 5.2] for almost rigid groups.

Theorem 2.4

## Theorem 2.4 ([3, Theorem 5.2])

Let X be an almost rigid group with Tcr(X)=T and mτ(X)=mτ. Then there exists a decomposition X=iIXi into almost rigid summands with products mτ(X)=iImτi, where mτi=mτ(Xi) if and only if the following compatibility conditions hold:

1. The numbers mτi and mσj are relatively prime if ij and τ,σTcr(X).

2. |{iI:mτi>1}|rk(X(τ)) for any τTcr(X).

In order to carry over this decomposition result for almost rigid groups to their images, the proper 𝔅(1)alr-groups, we will need the so-called admissible decompositions for almost rigid groups (see [4, Definitions 5.8, 5.9] ):

Definition 2.5

Let X𝔗 be an almost rigid group that has a decomposition X=jIXj. If Xj (jI) are rigid groups, having regulators isomorphic to direct sums of groups from the set {Ai,iω}, then this decomposition will be called an admissible decomposition of X.

Among all the admissible decompositions of almost rigid groups there are special ones, determined uniquely up to near-isomorphism and called main decompositions:

Definition 2.6

Let X𝔗 be an almost rigid group. An admissible decomposition X=XF, where X is a rigid group satisfying:

1. Tcr(X)Ti if and only if mτ(X)1 for some τTi,

2. mτ(X)=mτ(X) for all τTcr(X)

will be called a main admissible decomposition of X.

It will be shown next that we are interested only in admissible decompositions of an almost rigid group X. In fact, given X and a proper 𝔅(1)alr-group B=X/K that has a decomposition, then our strategy is to see that there is an admissible decomposition of X such that the direct summands can be considered as preimages of the summands of B. More precisely, we need some restrictions on the admissible decompositions of X which will lead us to the notion of K-admissible decompositions in the later sections.

## 3 Near-isomorphism of 𝔅(1)alr-groups in terms of the invariants

Our aim in this section is to introduce numerical invariants for proper 𝔅(1)alr-groups and to obtain a near-isomorphism criterion for these groups formulated in terms of these invariants. This will be done in complete analogy to [4, Theorem 5.15], which is the main result of that paper.

Let X𝔗 be an almost rigid group and B=X/K a proper 𝔅(1)alr-group with ϕ:XB=X/K the regular presentation of B. For simplicity we denote P=PX in (3) of Definition 2.1. It is almost evident and easy to recall from [4, equations (2), (3), and (12)] that there exists a uniquely determined factor primary representation of X in the following form:

(3.1)X=pPXpwithXp=XR(X)pγp𝔗 and Xp/R(X)TpX.

This implies the factor primary representation of B:

B=X/K=pPBpwithBp=Xpϕ=Xp/K.

Besides Xp with pP, we introduce the set of finite rank fully invariant subgroups of X

Xp=[TpX]*Xp.

Their images Bp=Xpϕ are isomorphic to Xp and they are block-rigid crq-groups with primary regulator quotient. It follows that there exist elements upX such that Xp=R(Xp),up with |Xp/R(Xp))|=pγp and, clearly,

mτ(Xp)=mτ(Xp)=mτ(Bp)

with mτ(Xp)=1 if τTcr(Xp).

Recall from [4, Theorem 5.13] the following

Theorem 3.1

Let ϕ:XB be a regular representation of a B(1)alr-group B with the triple (Ti:iλ), (ατ:τT) and (li:iω) of partition, coefficients and parameters for the group X. Then there exists a main admissible decomposition X=XF with mτ(X)=mτ(X) and the corresponding decomposition B=BH such that Xϕ=B, Fϕ=H.

In [4, Definition 5.14] this fact motivated a new definition as follows,

Definition 3.2

Given a proper 𝔅(1)alr-group B, a decomposition B=BH with a rigid 𝔅(1)alr-group B such that B/R(B)B/R(B) will be called a main decomposition of B.

Now, if B=BH is the main decomposition of the proper 𝔅(1)alr-group B, induced by a main admissible decomposition X=XF, which means that B=Xϕ and H=Fϕ, then let

X=pPXpwithXp=XR(X)pγp and Xp/R(X)TpX

be the factor primary representation of the rigid group X, see (3.1).

Clearly,

mτ(X)=pPmτ(Xp)=pPmτ(Xp′′)

with finite rank rigid crq-subgroup

Xp′′=(τ[Tp]X(τ))*Xp

of X for each pP.

Let B=pPBp with Bp=BR(B)pγp. Define in B uniquely determined crq-subgroups Bp′′=Xp′′ϕXp′′, then mτ(Xp)=mτ(Xp′′)=mτ(Bp′′), τ𝔗.

We may define the set of numbers

mτ(B)=pPmτ(Bp′′)

with the property mτ(B)=mτ(X), τ𝔗. Furthermore, mτ(X)=mτ(X) and we are allowed to put

(3.2)mτ(B)=mτ(B)=mτ(X).

Recall again from [3, Theorem 4.12] that two almost rigid groups X and Y are nearly-isomorphic if and only if their regulators are isomorphic, in symbols R(X)R(Y), and mτ(X)=mτ(Y) for each critical type τ𝔗 and using the fact that by [4, Theorem 5.15] two proper 𝔅(1)alr-groups are nearly-isomorphic if and only if their preimages in regular representations are nearly-isomorphic and the triples of partition, coefficients and parameters coincide for them we immediately obtain the following near-isomorphism criterion:

Theorem 3.3

Let B and C be proper B(1)alr-groups. Then BnrC if and only if R(B)R(C) and mτ(B)=mτ(C) for each critical type τT.

Having this result we are permitted to call the numbers mτ(B) not only type-invariants of a proper 𝔅(1)alr-group B, but also near-isomorphism invariants of B, τ𝔗. The same terminology is appropriate for almost rigid groups, which are preimages of 𝔅(1)alr-groups.

## 4 Direct decompositions of 𝔅(1)alr-groups

We are now approaching our main result which gives a combinatorial decomposability criterion for proper 𝔅(1)alr-groups based on that for almost rigid groups, see Theorem 2.4. In connection with Definition 2.2 we introduce some more definitions. Recall that a proper 𝔅(1)alr-group was defined as the quotient of an almost rigid group X and some subgroup K of the following form:

K=iωKli(Ai)XwithK(Ai)=τTiατaτAi,

where R(X)=iωAili with Ai=τTiτaτ.

Definition 4.1

Any set of elements (Aili)K={(aτn)n=1,,liAili:τTi} will be called a K-basis of Aili if the following hold:

1. Aili(τ)=nliτaτn with τTi,

2. Kli(Ai)={τTiατaτn:n=1,,li}.

Definition 4.2

Automorphisms ΨAut(X) will be called K-basic automorphisms of X if for any iω they act on each τ-homogeneous fully invariant subgroup Aili(τ) as multiplication by the same matrix Di (|detDi|=1) with respect to a K-basis {(aτn)n=1,,liAili:τTi}.

As was promised above we strengthen the definition of admissible decompositions as follows, see Definition 2.5.

Definition 4.3

Let B be a proper 𝔅(1)alr-group and let ϕ:XB be its regular representation. We say that X=jJXj is a K-admissible decomposition of X if it is admissible and obtained by a K-basic automorphism of X.

At this point we would like to remark that the K-admissibility of a decomposition X=jJXj of the almost rigid group X is equivalent to the existence of the corresponding induced decomposition K=jJKj with KjXj (see Definitions 4.1 and 4.2). It then follows that the proper 𝔅(1)alr-group B=X/K is also decomposable as B=jJXj/Kj. This is the reason why we needed the K-admissible automorphisms and this leads to the following very natural

Lemma 4.4

## Lemma 4.4 (Decomposability Criterion 1)

Let B be a proper B(1)alr-group and ϕ:XB its regular representation with the triple (Ti:iλ), (ατ:τT) and (li:iω) of partition, coefficients and parameters for the group X. Then one has B=B1B2 if and only if there exists a K-admissible decomposition X=X1X2 such that Bj=Xjϕ for j=1,2.

## Proof.

The sufficiency of the condition is trivial by the remark above.

Let B=B1B2. This implies the existence of the corresponding decomposition of its regulator, fully invariant subgroup 𝒜=G1G2 with G1=𝒜B1 and G2=𝒜B2. Then we have B/𝒜=B1/G1B2/G2 and B1 and B2 are also 𝔅(1)alr-groups. We have B/𝒜(X/K)/(R(X)/K)X/R(X) which is isomorphic to pPTpX. Then there exists the partition P=P1P2 such that B1/G1pP1TpX and B2/G2pP2TpX.

For the same reason we have the corresponding decompositions of the groups 𝒜(Ti) with critical typesets Ti, which are direct sums of li copies of strongly indecomposable groups Ai/K(Ai), see [7, Section 92] and [8, Section 3.3]. Namely, 𝒜(Ti)=Gi1Gi2 with Gi1=𝒜(Ti)G1 and Gi2=𝒜(Ti)G2 for each iω since 𝒜(Ti)=τTi𝒜(τ) are fully invariant in 𝒜, see [3, Introduction]. Denote 𝒜i=𝒜(Ti) and let 𝒜i(τ) be its fully invariant τ-homogeneous subgroup with τTi. Then 𝒜i(τ)=(𝒜i(τ)B1)(𝒜i(τ)B2) with one of the summands allowed to be zero. Recall that 𝒜i(τ)Ai(τ), which implies the corresponding decomposition R(X)=R1R2 of the regulator of X with G1=R1ϕ and G2=R2ϕ.

Clearly, B1=(G1)*B, B2=(G2)*B and X=X1X2 is a K-admissible direct decomposition such that X1=(R1)*X, X2=(R2)*X and X1ϕ=B1, X2ϕ=B2. ∎

Since the number of direct summands of B is at most countable (because it is not greater than the rank of the group), a trivial induction leads to the following.

Corollary 4.5

Let B be a proper B(1)alr-group and let ϕ:XB be its regular representation with the triple (Ti:iλ), (ατ:τT) and (li:iω) of partition, coefficients and parameters for the group X. Then B=jJBj if and only if there exists a K-admissible decomposition X=jJXj such that Bj=Xjϕ, jJ.

Now we need some preparation for transforming the obtained decomposability criterion into the one based on the numerical invariants of 𝔅(1)alr-groups which will be our main result.

Let M denote the set enumerating the groups Ai to write iM for our convenience. Then R=R(X)=iMAili, see Definition 2.2. For any filtration M=kIMk with MkMk+1 and M0= there exists a corresponding filtration X=kIXk with the ascending chain of fully invariant pure subgroups

Xk=(R(Xk))*X

such that R(Xk)=iMkAili and R(X)=kIR(Xk). Note that all Xk are crq-groups. Without loss of generality assume that it is a special filtration characterized by the condition that mσ(X) and mτ(X) are relatively prime for any σTcr(Xk-1) and τ𝔗Tcr(Xk).

For any MM introduce a K-basis of iMAli as iM(Ali)K, see Definition 4.1.

Denote Tk=Tcr(Xk), kI. Recall from [3, Lemma 4.6] that

mτ(Xk)=mτ(X)

if τTk-1. Moreover, near-isomorphism invariants of the groups Xk can be calculated on the basis of [3, Proposition 2.11], namely,

(4.1)mτ(Xk)=mτ(X)gcd(mτ(X),Qk+1/Qk)

with Qk=expXk/R(Xk), kI.

If X=X1X2, then

(4.2)Xk=Y1kY2kwithY1k=X1Xk and Y2k=X2Xk,kI,

as Xk is fully invariant in X. Moreover, near-isomorphism invariants of Y1k and Y2k can also be calculated in the following way:

mτ(Y1k)=mτ(X1)gcd(mτ(X1),Q1k+1/Q1k)withQ1k=expX1k/R(X1k),kI,

and

mτ(Y2k)=mτ(X2)gcd(mτ(X2),Q2k+1/Q1k)withQ2k=expX2k/R(X2k),kI,

if we naturally denote

X1k=((iMkAili)Xk)*X1 and X2k=((iMkAili)Xk)*X2.

In the proof of the next theorem we need to construct decompositions of the proper 𝔅(1)alr-group B=X/K with type invariants induced by the decomposition of the almost rigid group X=X1X2. So, we will say that the invariants mτ(Y1k) and mτ(Y2k) are predicted (or even a decomposition of Xk is predicted in the sense that its regulator and near-isomorphism invariants are known).

The next ingredient of the algorithm will be the extension of a decomposition (4.2) from Xk to Xk+1, which means that Y1k+1Xk-1=Y1kXk-1 and Y2k+1Xk-1=Y2kXk-1.

Following [2] we say that a direct sum of strongly indecomposable groups is a strongly decomposable group. It is associated with completely decomposable groups which are direct sums of rank-one groups.

Finally, we have approached the main result summarizing the above discussion. Let

X=kIXk

be a special filtration of X with an ascending chain of fully invariant pure subgroups Xk satisfying the following condition: mσ(X) and mτ(X) are relatively prime for any σTcr(Xk-1) and τ𝔗Tcr(Xk).

The following main Theorem 4.6 is a generalization of the corresponding Theorem 5.2 from [3] on almost rigid groups to the class of proper 𝔅(1)alr-groups. Its proof is quite technical because it needs a double induction. First, we restrict ourselves to the case of two summands and induct on kI. Then we induct on the number of the summands.

Theorem 4.6

## Theorem 4.6 (Decomposability Criterion 2)

Let B be a proper B(1)alr-group with mτ=mτ(B). Then there exists a decomposition

(4.3)B=jJBjC

into rigid summands Bj with mτj=mτ(Bj) and strongly decomposable C if and only if the following hold:

1. mτ=jJmτj for any τ𝔗,

2. gcd(mτj,mσk)=1 if jk,

3. |{j:τTimτj>1}|li, iω, jJ.

## Proof.

Let ϕ:XB be the regular representation of B. On the basis of Decomposability Criterion 1 (Lemma 4.4) we have that the decomposition

B=jJBjC

with mτj=mτ(Bj) satisfying (1)–(3) exists if and only if there is a K-admissible decomposition

X=jJXjH

with mτj=mτ(Xj), Bj=Xjϕ and C=Hϕ, see (3.2).

Note that the required admissible decomposition (4.3) of X with conditions (1)–(3) is one of the decompositions described in the decomposability criteria [3, Theorem 5.2]. Then we need only to show that it can be obtained by a K-basic automorphism of X. Moreover, the necessity of these conditions also follows from the mentioned criteria.

To prove that conditions (1)–(3) are sufficient for the existence of the required decomposition of B we take its main decomposition B=BH, which exists by Theorem 3.1, and obtain a decomposition (4.3) on this basis. Without loss of generality assume that lcmτTimτ0 for any i with TiTcr(B). Since there exists a corresponding admissible main decomposition

(4.4)X=XH

with Xϕ=B and Hϕ=F, we need to obtain from (4.4) by a K-basic automorphism an admissible decomposition

(4.5)X=jJXjH

into rigid alr-groups Xj of rank more than one and mτ(Xj)=mτj. We may restrict ourselves to the situation when lcmτTimτ(Xj)1 for each i satisfying TiTcr(Xj) with jJ.

More precisely, our purpose is to get the corresponding decomposition

R(X)=(jJRj)HwithXj=(Rj)*X.

We may also assume that H=0. If H0, then H contains a completely decomposable summand H′′H, therefore H′′ and H can be removed from (4.4) and (4.5) accordingly and in this case for any iM there exists τTi such that mτ(X)1. Moreover, without loss of generality assume that there is no non-trivial partition M=M1M2 such that gcd(mτ(B),mσ(B))=1 whenever τiM1Ti and σjM2Tj. Otherwise group B is a direct sum of proper 𝔅(1)alr-groups with independent direct decomposition constructions. It follows from (3.2) that the same connection condition should be assumed for group X, that is for any partition M=M1M2 there exist τiM1Ti and σjM2Tj such that mτ(X) and mσ(X) are not relatively prime.

So, we need to have the necessary decompositions of alr-groups.

Case I. Let us first consider the case |J|=2, that is B=B1B2 and li2 for all iM. By Lemma 4.4 we need a decomposition X=X1X2 with not necessarily indecomposable rigid summands X1,X2 and invariants mτ1=mτ(X1), mτ2=mτ(X2), τ𝔗. For the corresponding decomposition of the regulator R(X)=R1R2 such that X1=(R1)*X and X2=(R2)*X we have that for each τ𝔗 the following holds:

R(τ)=R1(τ)R2(τ)

with X1=(τ𝔗R1(τ))*X and X2=(τ𝔗R2(τ))*X (remark that R1(τ) or R2(τ) is allowed to be zero). We are concentrated on the admissible decompositions of X, therefore, if Rj(τ)0 for some τ𝔗Ti then Rj(σ)0 for any σ𝔗Ti with iM, j=1,2.

We now intend to obtain by a K-basic automorphism such a decomposition

(4.6)X=X1X2

from the main decomposition (4.4) with mτ1=mτ(X1) , mτ2=mτ(X2).

Clearly there exists a K-basis R(X)K={aτn:n=1,2} of R(X) such that

(4.7)X=τTcr(X)aτ1*XandH=τTcr(X)aτ2

with aτ2=0 if τTcr(H), see (4.4).

Let X1 be an arbitrary fully invariant pure subgroup of X with |M1|>1. We are now concentrated on X2 with the regulator R(X2)=iM2Aili. Recall that M1M2.

Since X2 is fully invariant in X, the above decomposition X=X1X2, which is to be constructed, reflects on the group X2 in the way that it must have the corresponding decomposition X2=Y12Y22 with Y12=X1X2, Y22=X2X2. Similarly, X3=Y13Y23 with Y13=X1X3, Y23=X2X3 etc.

Recall from [3, Lemma 4.6] that we have mτ(X2)=mτ(X) if τT1, and mτ(X3)=mτ(X) if τT2.

Our first object now is to get the necessary decomposition of X2. To this end, Let Q=expX2/R(X2), Q0=expX1/R(X1) and Q~=expX3/R(X3). These numbers can be viewed as Q=lcmτT2mτ(X2), Q0=lcmτT1mτ(X1) and Q~=lcmτT3mτ(X3). We also have that

mτ(X2)=mτ(X3)gcd(mτ(X3),Q~/Q)=mτ(X)gcd(mτ(X),Q~/Q),

see [3, Proposition 2.11] and (4.1). It is clear that

Q=Q1Q2

with Q1=expY12/R(Y12), Q2=expY22/R(Y22), and also

Q~=Q~1Q~2

with Q~1=expY13/R(Y13), Q~2=expY23/R(Y23). Evidently, gcd(Q~1,Q~2) is equal to 1, see conditions (1)–(2) for mτ(Xj)=mτ(Bj).

By number theory there exist relatively prime integers μ1 and μ2 such that μ1Q2+μ2Q1=1, that is

(4.8)μ1Q21(modQ1),μ2Q11(modQ2).

Moreover, we can find integers u and v such that u(μ1Q2)Q2-v(μ2Q1)Q1=1.

Determine a new K-basis of R(X2) as follows:

(4.9)bτ1=μ1Q2aτ1+vQ1aτ2andbτ2=μ2Q1aτ1+uQ2aτ1,τTcr(X2).

For each τTcr(X2) the matrix of linear transformation of X2(τ)=X(τ) is

D=(μ1Q2μ2Q1vQ1uQ2)withdetD=1.

If li=1 for some i, that is aτ2=0 for all τTi, then mτ(X2) coincides with mτ(Y12) or mτ(Y22) and we take accordingly

bτ1=aτ1,bτ2=0orbτ2=aτ1,bτ1=0,withτT2=Tcr(X2),

see (4.8). Therefore

R(X2)=τT2(τbτ1τbτ2).

Denote Y12=(τT2τbτ1)*X and Y22=(τT2τbτ2)*X. By construction,

[Y12:R(Y12)]=Q1and[Y22:R(Y22)]=Q2,

then [Y12Y22:R(X2)]=Q and X2=Y12Y22. Hence the K-basic automorphism Ψ of X, which acts on each τ-homogeneous component of rank 2 as multiplication by the matrix D with respect to {aτ1,aτ2} and is identity, if aτ2=0, led us to the predicted K-admissible decomposition of X2 determined by (4.6).

The next purpose is to get the admissible decomposition of X3=Y13Y23 determined by (4.6) which preserves the decomposition of X1 in the following sense:

(4.10)X1Y13=X1Y12andX1Y23=X1Y22.

Let α and β be some integers such that

α(μ1Q~2)-β(μ2Q~1)=1.

Put -z=α(μ1Q~2) and y=β(μ2Q~1) we have -y-z=1 and also

1+y=-αμ1Q~2and1+z=-βμ2Q~1.

It is important that the two numbers 1+y and 1+z are relatively prime and gcd(1+y,μ2Q1)=1 as well as gcd(1+z,μ1Q2)=1 by construction as Q1|Q~1 and Q2|Q~2. Then the numbers μ1Q2(1+y) and μ2Q1(1+z) are relatively prime and there exist integers u~, v~ such that the matrix

D~=(μ1Q2(1+y)μ2Q1(1+z)v~Q~1u~Q~2)hasdetD~=1

and

(4.11)Q~1|y,Q~2|z.

Now we are able to construct the required decomposition of X3. Determine a K-basis of R(X3) as follows:

bτ1=μ1Q2(1+y)aτ1+v~Q~1aτ2andbτ2=μ2Q1(1+z)aτ1+u~Q~2aτ1

for each τTcr(X3)Tcr(X1).

If li=1 for some i, that is aτ2=0 for τTi, then mτ(X3) coincides with mτ(Y13) or mτ(Y23) and we take accordingly

bτ1=aτ1,bτ2=0orbτ2=aτ1,bτ1=0,τTcr(X3)Tcr(X1).

Simultaneously, bτ1 and bτ2 are left the same as in (4.9) if τTcr(X1) by the condition (4.11) associated with matrices D and D~.

We have

R(X3)=τT3(τbτ1τbτ2).

Denote Y13=(τT3τbτ1)*X and Y23=(τT3τbτ2)*X.

Since the elements of the first lines of the matrices D and D~ are congruent modulo Q1 and Q2 accordingly by (4.11) as Q1|Q1~ and Q2|Q2~, we obtain that [Y13:R(Y13)]=Q~1 and [Y23:R(Y23)]=Q~2, which means

[Y13Y23:R(X3)]=Q~andX3=Y13Y23.

Hence the K-basic automorphism Ψ of X3, which acts on each τ-homogeneous component of rank 2 as multiplication by the matrix D or D~ depending on τTcr(X1) or τTcr(X3)Tcr(X1) with respect to {aτ1,aτ2} and it is identity, if aτ2=0, led us to the required admissible decomposition of X3 determined by (4.6).

Continuing the process we take the groups X2, X3 and X4 and consider them as X1, X2 and X3 respectively to apply for them the above construction. Instead of D we take the constructed matrix D~, the numbers Q~1 and Q~2 will serve as Q1 and Q2, the numbers v~ and u~ will serve as v and u and the numbers μ1Q2(1+y)/Q~2 and μ2Q1(1+z)/Q~1 will work as the new μ1 and μ2 accordingly. Due to the above connection restriction we get the required decomposition of Aili for any iM in finitely many steps.

Case II. Consider the general case |J|>2, which implies for each iM that li can be any finite positive integer. Under the condition H=0 we have to construct a decomposition

(4.12)X=jJXj

into rigid summands of rank more than one with a set of near-isomorphism invariants satisfying conditions (1)–(3) on the basis of the main K-admissible decomposition (4.4). Without loss of generality assume that the rigid summands Xj are listed in (4.12) in the following order: (X1,,Xj2) are all the summands having non-zero intersection with X2, ((X1,,Xj2),,Xj3) are all the summands having non-zero intersection with X3 and so on. Assume that j2>2 (we are able to choose X2 satisfying this condition, otherwise the decomposition has been already constructed in Case I). The case jk+1=jk is allowed for some (or even all) kI.

As above, {aτn:τTi,n=1,,li,iM} is a K-basis of R(X). It follows from the decomposition in (4.12) that lijk if iMk because, by construction, Xk(X1Xjk). Let ki be the minimal natural number such that iMki (equivalently, ki is the natural number satisfying iMkiMki-1 or, the same, TiTcr(Xki)Tcr(Xki-1)). Then we may rewrite the K-basis of each Aili, iM, in the following way:

(Aili)K={aτn:τTi,iMkMk-1,n=1,,jk,k1with aτn=0 if and only if TiTcr(Xn)=}.

First we should concentrate on X1, X2 and X3 again as in Case I. The main decomposition of X2 determined by the main admissible decomposition (4.4) is

X2=X2(τTi,TiT1,n=2,,j1τaτn)(τTi,TiT2T1,n=2,,j2τaτn)

with X2=XX2, see (4.7). Denote the groups Yjk=XjXk (jJ, k3) with the predicted invariants, which are under construction for the required decomposition (4.12).

Let us take a subgroup Z12 of X2 with homogeneous components of rank not greater than 2, which is

(4.13)Z12=X2(τTi,TiT1τaτ2)(τTi,TiT2T1τaτ2),

and construct its K-admissible decomposition Z12=Y12Z12 with the rigid summand Z12, uniquely determined up to near-isomorphism by its near-isomorphism invariants

mτ(Z12)=mτ(X2)mτ(Y12)

under the restriction in (4.13) that aτ2=0 if mτ(Z12)=0, see Case I.

The connection condition provides that a special numbering the homogeneous components Ai, iM, yields that mτ(X3) and mσ(X3) are not relatively prime for some τT2, σT3T2. Again using Case I we extend the obtained decomposition of Z12 to Z13, which is

Z13=X3(τT1τaτ2)(τT2T1τaτ2)(τT3T2τaτ2)

with X3=XX3 and pure fully invariant subgroup Z12. Then it follows that Z13=Y13Z13 with the rigid summand Y13 having the predicted invariants, and rigid summand Z13, uniquely determined up to near-isomorphism by its regulator and the numbers

mτ(Z13)=mτ(X3)mτ(Y13),τT3.

It is important that X1Y13=X1Y12 and X1Z13=X1Z12 (in particular, the possible equalities Y13=Y12 or Z13=Z12 do not create any problems), see (4.10).

Since j2>2, we have a subgroup of X2 which is

Z22=Z12(τT1τaτ3)(τT2T1τaτ3)

and construct its K-admissible decomposition Z22=Y22Z22 with the predicted Y22 and rigid summand Z22 satisfying

mτ(Z22)=mτ(Z22)mτ(Y22),τT2.

Extending this decomposition to the group

Z23=Z13(τT1τaτ3)(τT2T1τaτ3)(τT3T2τaτ3),

we obtain Z23=Y23Z23 with predicted Y23 and

mτ(Z23)=mτ(X3)mτ(Y13)mτ(Y23),τT3,

satisfying X1Y23=X1Y22 and X1Z23=X1Z22 (as above, the case Y23=Y22 is obviously allowed).

Continuing this process in (j2-1) steps we obtain a decomposition

X3=j<j2Yj3Zj23G

with the predicted invariants of rigid groups Yj3 with j<j2, rigid Zj23 satisfying

mτ(Zj23)=mτ(X3)j<j2mτ(Yj3)

and completely decomposable G with T2Tcr(G)=. Note that by construction

(4.14)X2=j<j2Yj2Yj22withYj22=Zj23X2.

Moreover, Yj3X1=Yj2X1 for each j<j2 and Zj23X1=Yj22X1. If j2=j3, then Zj23=Yj23=Xj2X3. If j2<j3, then the summands Yj3 of X3 with j2<jj3 have zero intersection with X2 and can be obtained on the basis of the main decomposition Zj23G as above for the group X2, see (4.14). Finally, we have

(4.15)X3=jj3Yj3

The main idea of the described decomposition process is that we are able to get a predicted decomposition of X2 and then to extend this to X3 so that the corresponding decomposition of the regulator iM1Aili of X1 would not be changed.

Case III. Our object now is extending the decomposition (4.15) from X3 to X4. Again, we start with the main decomposition

X4=X4TiT4,τTi,n=2,,liτaτn

having the rigid summand X4=XX4.

We take the groups X2, X3 and X4 instead of X1, X2 and X3 and apply for them the above construction which preserves the decomposition of the regulator iM2Aili of X2 obtained previously.

Inductively, due to the above connection restriction we get the required decomposition of each Aili, iM, in finitely many steps. The proof is completed. ∎

Remark 4.7

(a) It follows from Theorem 3.3 that conditions (1)–(3) determine the direct summands Bj,jJ, of group X up to near-isomorphism, see (4.5).

(b) In the proof under the natural assumption H=0 we have the equalities in the condition (3) by construction, as the rank of an arbitrary homogeneous component is equal to the number of direct summands having non-trivial intersection with this component. However, the presence of rank-one summands of H leads to the required inequalities in the general case.

It is routine to prove the following theorem.

Theorem 4.8

## Theorem 4.8 (Indecomposability Criterion)

A proper B(1)alr-group B with

Tcr(B)=iMTi

is indecomposable if and only if it is rigid and its near-isomorphism invariants mτ(B) satisfy the connection condition: for any partition M=M1M2 there exist τiM1Ti and σjM2Tj such that mτ(B) and mσ(B) are not relatively prime

In memory of our good friend and colleague Rüdiger Göbel

Communicated by Evgenii I. Khukhro

Funding statement: The first named author is grateful to the German Academic Exchange Service (DAAD) for their support of this research in 2014.

## Acknowledgements

The authors are thankful to the referee for her/his many helpful comments.

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