Indecomposable Groups Research Articles

On subgroups lifting modulo central commutant

We consider a finitely generated group G with the commutant of odd order \(p_1^{n_1 } \ldots p_s^{n_s } \) located at the center and prove that there exists a decomposition of G/G′ into the direct product of indecomposable cyclic groups such that each factor except at most n l + ... + n s factors lifts modulo commutant.

Acylindrical accessibility for groups

We define the notion of acylindrical graph of groups of a group. We bound the combinatorics of these graphs of groups for f.g. freely indecomposable groups. Our arguments imply the finiteness of acylindrical surfaces in closed 3-manifolds [Ha], finiteness of isomorphism classes of small splittings of (torsion-free) freely indecomposable hyperbolic groups as well as finiteness results for small splittings of f.g. Kleinian and semisimple discrete groups acting on non-positively curved simply connected manifolds. In order to get our accessibility for f.g. groups we generalize parts of Rips' analysis of stable actions of f.p. groups on real trees to f.g. groups. The concepts we present play an essential role in constructing the canonical JSJ decomposition ([Se1],[Ri-Se2]), in obtaining the Hopf property for hyperbolic groups [Se2], and in our study of sets of solutions to equations in a free group [Se3].

The direct decompositions of a group 𝐺 with 𝐺/𝐺’ finitely generated

We consider the class C \mathcal {C} which consists of the groups M M with M / M ′ M/M’ finitely generated which satisfy the maximal condition on direct factors. It is well known that any C \mathcal {C} -group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any C \mathcal {C} -group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for C \mathcal {C} -groups, a slightly different notion of decomposition, that we call J J -decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of J J -decompositions, several generalizations of results of R. Hirshon. For instance, we have Z × G ≅ Z × H \mathbb {Z} \times G \cong \mathbb {Z} \times H for any groups G G , H H which satisfy M × G ≅ M × H M \times G \cong M \times H for a C \mathcal {C} -group M M .

Quasi-isomorphism invariants for two classes of finite rank Butler groups

A complete set of numerical quasi-isomorphism invariants is given for a class of torsion-free abelian groups containing all groups of the form G [ A ] \mathcal {G}[\mathcal {A}] , where A = ( A 1 , … , A n ) \mathcal {A} = ({A_1}, \ldots ,{A_n}) is an n n -tuple of subgroups of the additive rationals and G [ A ] \mathcal {G}[\mathcal {A}] is the cokernel of the diagonal embedding ⋂ A i → ⊕ A i \bigcap {{A_i} \to \oplus {A_i}} . This classification and its dual include, as special cases, earlier classifications of strongly indecomposable groups of the form G [ A ] \mathcal {G}[\mathcal {A}] and their duals.

On the quasi-splitting of exact sequences

Perhaps the most significant difference between the theories of torsion and torsion-free abelian groups is the large number of splitting results in the former, and the lack of such in the latter. For instance, Szele’s theorem [F, Vol. 1, Proposition 27.11 has no obvious equivalent in the theory of torsion-free abelian groups, although it is one of the widely used results in the discussion of torsion and mixed groups: Let B g QJI Z/p’7 be a subgroup of an abelian group G. Then B is a direct summand of G if and only if B np”G = 0. The author was able to extend Szele’s theorem to valuated p-groups in [A2, Satz 6.81 using methods similar to those which proved useful in the discussion of Baer’s lemma in [AL], [A3], [A4]. It is the goal of this paper to prove a result which can be considered the equivalent of Szele’s theorem in the quasi-category of torsion-free abelian groups. Before we can describe the result, it is necessary to introduce some notation. We consider an abelian group A. The class of A-projective groups of finite A-rank is obtained by closing {A} with respect to direct summands and finite direct sums. If G is an abelian group, then the A-radical of G, which is denoted by R,(G), is R,(G) = n {ker 41 #E Hom(G, A)). Finally, consider modules B and C over a ring R. A homomorphism ~1: B + C quasi-splits if there are a non-zero integer n and a map /3~Hom,(C, B) with /?cr=n.id,, where id, denotes the identity map on B. If the additive group of B is torsion-free, then a quasi-splitting homomorphism is one-to-one [Jl, 52, Re]. We say that an abelian group A has the radical-splitting property if every exact sequence 0 + PA G of torsion-free abelian groups, in which P is a quasi-summand of an A-projective group of finite A-rank and u(P) n R,(G) = 0, quasi-splits. Szele’s original result guarantees the splitting of the sequence in the case in which A z Z/p’7 for some n < w. Moreover, if A z Z/p”Z, then R,(G) = p”G. Theorem 3.1 of this paper establishes that every strongly indecomposable group of finite rank has the

Direct decompositions of torsion-free Abelian groups of finite rank

It is proved that ifr1,r2, ...,rs;l1,l2, ...,lt are the ranks of the indecomposable summands of two direct decompositions of a torsion-free Abelian group of finite rank and if s0 is the number of units among the numbers ri, while t0 is the number of units among the numbers lj, thenri ⩽n - t0,lj⩽n−s0 for all i, j. Moreover, if for some i we have ri=n−t0, then among the lj's only one term is different from 1 and it is equal to n−t0; similarly if lj=n−s0 for some j. In addition, a construction is presented, allowing to form, from several indecomposable groups, a new group, called a flower group, and it is proved that a flower group is indecomposable under natural restrictions on its defining parameters.

Indecomposable met-Abelian groups

We prove a theorem which solves problem 7.5 in the “Kurosh Tetrad.” In the present article we describe periodic two-step solvable indecomposable groups [see [1], question 7.5)]. We recall that a group is said to be indecomposable if each two of its proper subgroups generate a proper subgroup. The class of groups under consideration is rather broad: It contains, for example, the groups constructed by Heineken and Meldrum, but, as is well-known [3–6], there is an uncountable pairwise non-isomorphic family of such groups.

On uniqueness of direct decompositions of groups into directly indecomposable factors

In this paper, we give a necessary and sufficient condition for certain groups G to allow an essentially unique direct decomposition into non-trivial directly indecomposable groups. For any such G we show that if G has at least two essentially distinct direct decompositions into non-trivial directly indecomposable groups, then we can find finitely generated free abelian groups F and F 1 and a directly indecomposable group B for which 1. (a) B × F is a direct factor of G × F 1 and 2. (b) B × F has at least two essentially distinct direct decompositions into non-trivial directly indecomposable factors. The torsion free rank of F 1 may be expressed in terms of certain invariants of G.

FINITE AUTOMORPHISM GROUPS OF TORSION-FREE ABELIAN GROUPS OF FINITE RANK

Abelian torsion-free groups of finite rank with finite automorphism groups are considered as rigid extensions of a system of strongly indecomposable groups Aj, j = 1, ..., k, of finite rank and having finite automorphism groups, by a finite p-group P. Such groups are called (A, p)-groups. The author introduces for (A, P)-groups the concept of (A, P)-type, which represents a choice of k integer matrices. A complete description of (A, P)-groups is given by means of (A, P)-types. Using this description, a series of problems on finite groups of automorphisms of torsion-free abelian groups of finite rank are solved. Furthermore, it is shown that the actual solution of any one of these problems comes down to a question of the consistency of a system of equations of the first degree modulo pt, where pt is the maximal order of elements of P. Bibliography: 11 titles.

Some properties of direct products

always implies C NN D. The classical theorem of Remak 1161 states that every finite group G, G # 1 can be expressed as a direct product of non trivial directly indecomposable groups in essentially one way. That is, in any two such decompositions, there is a one to one correspondence between the terms in each decomposition with corresponding terms isomorphic. Consequently cancellation in finite groups G is a trivial matter. A natural question is that of determining conditions to guarantee the cancellability of a group M. Over the last few decades various authors have considered this problem. A brief browsing through [4 J, for example, reveals concern for these questions in abelian groups. Related studies have been made on refinements of direct decompositions and attempts to obtain certain invariants. A partial listing (by no means complete) of these works is included in the reference of [lo]. These results are too numerous to elaborate on in this introduction. However, perhaps one can get some inkling of the inherent difficulties by considering the following sample results:

A presentation for the stabiliser of an element in a free product

Let G = ∗ i ϵ l G i be the free product of finitely many indecomposable groups none of which are infinite cyclic. With natural assumptions on the factors G i , the stabiliser in the automorphism group Aut G of any element of G is finitely presented.

Direct decompositions of torsion-free Abelian groups of finite rank

A certain wide new class of torsion-free Abelian groups of finite rank, which admits an exhaustive description of all its decompositions into direct sums of indecomposable summands, is constructed. It is proved that if n=r1+...+rs=l 1+...+l t are two decompositions of the number n into a sum of natural numbers such that all the terms of the first decomposition do not exceed n−t+1 while all the terms of the second decomposition do not exceed n−s+1, then there exists an Abelian group of rank n which admits a decomposition into a direct sum of indecomposable groups of ranks zl,..., zs and a decomposition into a direct sum of indecomposable groups of ranks ll,..., lt.

A cancellation criterion for finite-rank torsion-free abelian groups

In this paper, a necessary ring-theoretical criterion is given for a finite-rank torsion-free abelian group to have the cancellation property. This generalizes results obtained by L. Fuchs and F. Loonstra [5] for the rank-one case and resolves the cancellation problem for strongly indecomposable groups.

The number of indecomposable terms in direct decompositions of groups with the maximal condition

If nt and n are positive integers, m > 1, n > 1, then there exists a finitely generated torsion free nilpotent group which can be expressed as a direct product of n directly indecomposable groups and as a direct product of m directly indecomposable groups such that no factor in the first decomposition is isomorphic to a factor in the second [ 11. This leads us to ask: If a group G satisfies the maximal condition for normal subgroups, what conditions will guarantee that any two direct decompositions of G into nontrivially directly indecomposable factors have the same number of factors? If G has the above property we call G a UNIF group ( a group with a unique number of indecomposable factors in direct products). The object of this paper is to study the UNIF property. In Section 3 we show that every group G with the maximal condition for normal subgroups which fails to have the UNIF property gives rise to a directly indecomposable homomorphic image M such that for some finitely generated free abelian group F, M x F is not a UNIF group. Section 4 contains a refinement of this result. In Section 5 we use the results of Sections 3 and 4 to show that a group G which obeys the maximal condition for normal subgroups has the UNIF property if either of the following holds:

Boolean powers: direct decomposition and isomorphism types

We determine properties of Boolean powers of groups and other algebraic structures, and we generalize Jónsson’s theorem on Boolean powers of centerless, directly indecomposable groups. We show that every nonabelian, finitely generated group has 2 ℵ 0 {2^{{\aleph _0}}} nonisomorphic countable Boolean, and hence subcartesian, powers. We show that nonabelian groups G G such that either (i) G G is not the central product of two nonabelian groups or (ii) every pair of nontrivial normal subgroups of G G intersect nontrivially yield nonisomorphic Boolean powers with respect to nonisomorphic Boolean algebras.

Thick Groups and Essentially Finitely Indecomposable Groups

In this article, we examine the relationship between the notions of thick groups and that of essentially finitely indecomposable (e.f.i.) groups. These groups are indecomposable objects in the quotient category of the category of abelian groups induced by the Serre-class of bounded groups.

On constructing indecomposable groups in L

On constructing indecomposable groups in L

Torsion-free Abelian groups with cyclic p-basic subgroups

We investigate the structure of indecomposable torsion-free Abelian groups all of whose p-basic subgroups are cyclic, and also the structure of the groups and rings of endomorphisms of such groups. We prove the existence of a torsion-free Abelian group of countable rank with cyclic p-basic subgroups which has no indecomposable nonzero direct summands.

On $4$-manifolds cross $I$

In this note we set forth conditions under which for a given 4- manifold M there exist a countably infinite number of 4-manifolds Mi such that irxifiaMi) are distinct indecomposable groups and each Mixl is homeomorphic with M X I.

Almost Completely Decomposable Torsion Free Abelian Groups

A finite rank torsion free abelian group $G$ is almost completely decomposable if there exists a completely decomposable subgroup $C$ with finite index in $G$. The minimum of $[G:C]$ over all completely decomposable subgroups $C$ of $G$ is denoted by $i(G)$. An almost completely decomposable group $G$ has, up to isomorphism, only finitely many summands. If $i(G)$ is a prime power, then the rank 1 summands in any decomposition of $G$ as a direct sum of indecomposable groups are uniquely determined. If $G$ and $H$ are almost completely decomposable groups, then the following statements are equivalent: (i) $G \oplus L \approx H \oplus L$ for some finite rank torsion free abelian group $L$. (ii) $i(G) = i(H)$ and $H$ contains a subgroup $G’$ isomorphic to $G$ such that $[H:G’]$ is finite and prime to $i(G)$. (iii) $G \oplus L \approx H \oplus L$ where $L$ is isomorphic to a completely decomposable subgroup with finite index in $G$.