Combinatorics and Parallel Structures in Torsion-Free Abelian Group Theory

Combinatorics and parallel structures in torsion-free abelian group theory

Ekaterina Blagoveshchenskaya

St. Petersburg, Russia

1 Let X be a torsion-free abelian group of ﬁnite rank n.

Let {a1,...,an} be a linearly independent system of its element ∼ X and let τi ⊂ Q, with Z ⊂ τi = ai∗ .

Then A = τ1a1 ⊕ ... ⊕ τnan ⊂ X and for any b ∈ X \ A there exist integers α, αi such that αb = α1a1 + ... + αnan with at least two non-zero coeﬃcients αi. Let B =∼ X/A. We may consider a torsion-free abelian group X of ﬁnite rank as an essential extension of a completely decomposable torsion-free group A by means of a torsion group B.

By Deﬁnition, Tcr(A)= {τ1,...,τk} is called the set of critical types of A (repetitions are excluded, k ≤ n). If B is ﬁnite then A is uniquely determined up to isomorphism and

Tcr(A)= Tcr(X). Moreover, there exists just one ”special” fully invariant subgroup of X isomorphic to A, which is called the regulator of X and denoted by R(X). Assume A = R(X).

2 Main Deﬁnitions A torsion-free abelian group is completely decomposable, a CD-group, if it is a direct sum of subgroups of the rationals Q.

An ACD-group X (almost completely decomposable group) is a torsion-free abelian group of ﬁnite rank, that contains a fully invariant completely decomposable group A = R(X), its regulator, for which X/A is a ﬁnite group.

In case of cyclic X/A we say that ACD-group X is a CRQ-group (with cyclic regulator quotient).

3 A.G. Kurosh. Theory of groups, M. Nauka, 1967. L. Fuchs. Inﬁnite Abelian Groups, vol. 1, 2, Academic Press 1970, 1973. D. Arnold. Finite Rank Torsion Free Abelian Groups and Rings, Lecture Notes in Mathematics, vol. 931, Springer Verlag, 1982. A. Mader. Almost completely decomposable abelian groups, Gordon and Breach, Algebra, Logic and Applications Vol. 13, Amsterdam, 2000. P.Krylov, A. Michalev, A. Tuganbaev. Endomorphism rings of abelian groups, Kluwer Academic Publishes, Dordrecht-Boston-London, 2003. E. Blagoveshchenskaya. Almost Completely Decomposable Abelian Groups and their Endomorphism Rings, Mathematics in Polytechnical University, St. Petersburg, 2009.

4 X ∈ ACD:

1) End X ⊂ End A with A = Aτ and rk Aτ ≥ 1. Lτ∈Tcr(X)

2) X/A = Lp∈P Tp, ∼ 3) X = Pp∈P Xp, X/A = Pp∈P Xp/A = Pp∈P Tp with fully invariant, pure subgroups Xp with Xp/A =∼ Tp,

4) End Xp ⊂ End A and End X = Tp∈P End Xp.

More precisely, X Xp = A +( M Aτ )∗(p) τ∈[Tp] with [Tp] the minimal subset of Tcr(X) such that X ( Aτ ) / Aτ =∼ Tp. Lτ∈[Tp] ∗(p) Lτ∈[Tp]

5 Deﬁnition Let G and H be torsion-free abelian groups of ﬁnite rank. Then G and H are called nearly isomorphic (in symbols

G =∼nr H) if and only if for any prime q there are monomorphisms

ηq : G −→ H and ξq : H −→ G such that H/ηq(G) and G/ξq(H) are ﬁnite groups and |H/ηq(G)| and q as well as |G/ξq(H)| and q are relatively prime.

Arnold (1982): If G =∼nr H and G = G1 ⊕ ... ⊕ Gk then there exists a decomposition H = H1 ⊕ ... ⊕ Hk such that

G1 =∼nr H1, ... , Gk =∼nr Hk.

6 This leads to near direct decomposition theory of CRQ-groups with pairwise incomparable critical types (Blagoveshchenskaya and Mader, 1994):

If X = X1 ⊕ X2 then A = A1 ⊕ A2 with A1 ⊂ X1, A2 ⊂ X2.

This implies X/A =∼ X1/A1 ⊕ X2/A2 and mτ (X)= mτ (X1)mτ (X2) for any τ.

Near Baer-Kaplansky Theorem (Blagoveshchenskaya, Ivanov, Schultz (2001)):

Let X,Y ∈ CRQ. Then End(X) =∼ End(Y ) if and only if X =∼nr Y .

Question: What groups of inﬁnite rank admit the similar decomposition theory up to the new (extending from CRQ-class) near-isomorphism equivalence?

7 Near-isomorphism for torsion-free abelian groups is expected: (1) to preserve decomposability properties of groups (in the sense of Arnold’s theorem) (2) to be isomorphism for completely decomposable groups (3) to be traditional near-isomorphism for groups of ﬁnite rank

8 Definition (Blagoveshchenskaya, Strüngmann, Göbel, 2002 – 2007) Let G and H be torsion-free abelian groups. Then G and H are called nearly isomorphic, G =∼nr H if for every integer N there exist monomorphisms ϕN : G → H and ψN : H → G such that

1. H/GϕN and G/HψN are torsion;

2. (H/GϕN )p =0=(G/HψN )p for all primes p dividing N; 3. for every ﬁnite rank pure subgroups G′ ⊆ G and H′ ⊆ H the ′ H ′ ′ G ′ quotients (G ϕN )∗ /G ϕN and (H ψN )∗ /H ψN are ﬁnite.

9 Blagoveshchenskaya, Göbel (2002): consideration of unions of ascending chains of finite rank crq-groups leads to Definition 1 Let T be a set of finitely or countably many pairwise incomparable ring types and suppose that A = Lτ∈T Aτ is a direct sum of τ-homogeneous cd-groups Aτ of finite rank. Then a torsion-free group X is called a local crq-group (X ∈ CRQloc) if the following conditions hold: (1) A ⊆ X

(2) X/A is a direct sum of p-primary cyclic groups Tp of orders αp ep = p for distinct primes p from a set P and αp ∈ N.

(3) Each Aτ , τ ∈ T = Tcr(X), is pure in X.

Deﬁnition 2 Let X ∈ CRQloc. We call X an almost rigid group if it is ﬁnitely presented, that is each τ, τ ∈ T = Tcr(X), belongs to

ﬁnitely many [Tp], p ∈ P .

10 New Deﬁnitions

A group A0 is called a strongly decomposable group (sd-group) if it is a direct sum of strongly indecomposable groups. If a group B contains a strongly decomposable group as a subgroup of ﬁnite index, then B will be called an almost strongly decomposable group (asd-group).

11 L. Fuchs, C. Metelli, D. Arnold, C. Vinsonhailer (1991-1993):

Let A = Aτ with Aτ = τaτ be a rigid completely Lτ∈Tcr(A) decomposable group of ﬁnite rank and let {ατ : τ ∈ Tcr(A)} be integers, for which the following hold,

1. rk A = |Tcr(A)|≥ 3,

2. for any τ ∈ Tcr(A) there is a prime p with τ(p)= ∞ and

σ(p) = ∞ for all σ = τ, σ ∈ Tcr(A), Z 3. Tτ= σ τ = for each σ ∈ Tcr(A),

4. gcd({ατ | τ = σ, τ ∈ Tcr(A)}) = 1 for any σ ∈ Tcr(A),

5. each ατ is not p-divisible if σ(p)= ∞ for some σ ∈ Tcr(A). Let K(A) =∼ Z be a group generated by the element

ατ aτ . Then the strongly indecomposable group Pτ∈Tcr(A) B(A)= A/K(A) is called a proper B(1)-group.

12 Deﬁnition (Blagoveshchenskaya, 2007)

Let X be a rigid crq-group of ring type and let ατ , τ ∈ T , be integers such that the following conditions hold,

1) A = A ⊕ ... ⊕ At is the regulator of X with Ai = τaτ , 1 Lτ∈Ti Tcr(X)= Si=1,...,t Ti, Ti = Tcr(Ai) and Ti ∩ Tj = ∅ if i = j,

2) Let K = K(Ai) ⊂ X with K(Ai)= ατ aτ and Li≤t Pτ∈Ti φ : A −→ A/K.

3) Xpφ =∼ Xp and Xp and Xpφ are fully invariant, pure, p-generated subgroups over A = R(X) and A/K accordingly. The group B = X/K is called a B(1)crq-group. φ : X −→ B = X/K with a rigid crq-group X will be called a regular representation of a B(1)crq-group B.

13 Recall CRQ ⊂ ACD.

⊂ ⊂ CRQ −→ CRQloc −→ ACDloc ↓ (φ) ↓ (φ) ↓ (φ) (1) ⊂ (1) ⊂ (1) B CRQ −→ B CRQloc −→ B ASDloc

Theorem (Blagoveshchenskaya, Str¨ungmann, G¨obel, 2011)

Let X and Y be nearly isomorphic rigid ACDloc-groups (or block-rigid CRQloc-groups) with T = Tcr(X)= Tcr(Y ) and

Φ1 : X −→ B and Φ2 : Y −→ C be regular representations of groups B and C with the partition T = Si∈ω Ti and the same coeﬃcients ατ , τ ∈ T . Then B and C are nearly isomorphic. Corollary Near-isomorphism criteria and decomposition theory of (1) B CRQloc-groups in terms of type invariants mτ derive from the theory of CRQloc-groups.

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