On Abelian Groups Close to E-Solvable Groups

The concept of autonilpotent groups was introduced by Moghaddam and Parvaneh in 2010 which is related to concepts of absolute centre and autocommutator subgroups that were first proposed by Hegarty in 1994 and 1997 respectively. In the present article we give some useful properties of such groups. In the first section, we define a subgroup GK_n for a given group G, as GK_n=〈 [g ,α^n ] ┤| g∈ G,α∈Aut(G) 〉 which is characteristic subgroup of G . In this section, based on the induction method, we obtain the structure of GK_n as 〈 [ g ,α ]^n ┤|g∈ G,α∈ Aut(G) 〉, 〈 [ g ,α ]^n [g,α,α]^(n(n-1)/2) ┤|g∈ G,α∈ Aut(G) 〉 and 〈 (∏_(i=0)^(n-1)▒〖[g,α] [g,α,α]^i) [g,α,α,α]^((n(n-1)(n-2))/6) 〗┤|g∈ G,α∈ Aut(G) 〉 in autonilpotent groups of classes 2, 3 and 4 respectively. In second section, n-fixed group is introduced as a group G with GK_n=1 for some n∈N. Then among other results of n-fixed groups, we characterize some n-fixed groups. Based on the role of the absolute centre subgroup in the structure of the group, we prove a non-trivial finite group G is isomorphic to Z_2 iff G is 1-fixed group also we show that abelian group G is p -fixed autonilpotent group iff it is isomorphic to G≅ C_(2^k ) where 1≤ k≤ 3

Let C be an Abelian group. An Abelian group A from a class X of Abelian groups is said to be CH-definable in X if, for any group B ∈ X, the isomorphism Hom(C,A) ≅ Hom(C,B) implies that A ≅ B. If every group from X is CH-definable in X, then X is called an CH-class. In this paper, we study conditions under which a class of completely decomposable torsion-free Abelian groups is an CH-class, where C is a vector group.

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

Let $$\Lambda$$ be a class of Abelian groups. A group $$A\in\Lambda$$ is said to be determined by its endomorphism semigroup $$E^\star(A)$$ in the class $$\Lambda$$ if every isomorphism $$E^\star(A)\cong E^\star(B)$$ , where $$B\in\Lambda$$ , implies the isomorphism $$A\cong B$$ . The paper describes those Abelian groups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ of completely decomposable quotient divisible Abelian groups which are determined by their endomorphism semigroups in the class $$\mathscr Q\mathscr D_{\mathrm{cd}}$$ .

Let C be an Abelian group. A class X of Abelian groups is called a CE•H-class if, for every groups A, B ∈ X, the isomorphisms E•(A) ≅ E•(B) and Hom(C, A) ≅ Hom(C, B) imply the isomorphism A ≅ B. In the paper, necessary and sufficient conditions on a completely decomposable torsion-free Abelian group C are described under which a given class of torsion-free Abelian groupsisa CE•H-class.

Let G be a completely decomposable torsion-free Abelian group and G= ⊕ Gi, where Gi is a rank 1 group. If there exists a strongly constructive numbering ν of G such that (G,ν) has a recursively enumerable sequence of elements gi ∈ Gi, then G is called a strongly decomposable group. Let pi, i∈ω, be some sequence of primes whose denominators are degrees of a number pi and let \(\mathop \oplus \limits_{i \in \omega } Q_{Pi} \). A characteristic of the group A is the set of all pairs ‹ p,k› of numbers such that \(p_{i_1 } = ... = p_{i_k } = p\) for some numbers i1,...,ik. We bring in the concept of a quasihyperhyperimmune set, and specify a necessary and sufficient condition on the characteristic of A subject to which the group in question is strongly decomposable. Also, it is proved that every hyperhyperimmune set is quasihyperhyperimmune, the converse being not true.

The splitting length of a mixed Abelian group G is defined as the smallest positive integer n such that \( \mathop \otimes \limits^n G \) splits. The task of determining the splitting length of mixed Abelian groups was formulated by Irwin, Khabbaz, and Rayna. In this paper, a criterion for determining whether \( \mathop \otimes \limits^n G \) splits for countable mixed Abelian groups G of torsion-free rank 1 is found.

The hopficity property in a well-known class of Abelian groups—completely decomposable torsion-free groups—as well as in a class of modules over generalized matrix rings is studied. Examples of non-Hopfian completely decomposable torsion-free group are constructed.

First, we give a necessary and sufficient condition for a torsion-free rank-one subgroup of an arbitrary abelian group to be purifiable in a given group and show that all pure hulls of a purifiable torsion-free rank-one subgroup are isomorphic. Next, we show that if a $T(G)$-high subgroup A of an abelian group $G$ is purifiable in $G$, then there exists a subgroup $T'$ of $T(G)$ such that $G=H\oplus T'$ for every pure hull $H$ of $A$ in $G$. An abelian group $G$ is said to be a strongly ADE decomposable group if there exists a purifiable $T(G)$-high subgroup of $G$. We present an example $G$ such that not all $T(G)$-high subgroups of a strongly ADE decomposable group G are purifiable in $G$. Moreover, we characterize the strongly ADE decomposable groups of torsion-free rank 1. Finally, we use previous results to give a necessary and sufficient condition for an abelian group of torsion-free rank 1 to be splitting.

In the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter), an embedding problem for torsion-free Abelian groups was considered. It was shown for a large class of such groups, including the class of all bounded extensions of completely decomposable groups, that any member of the class can be purely embedded into some completely decomposable group. Moreover, an algorithm was given that determines explicitly the pure embedding and the completely decomposable overgroup. We continue the approach from the work of Herden and Strüngmann (In Models, modules and Abelian groups (2008) 169–186 de Gruyter) improving the algorithm and extending the main theorem to a broader class of torsion-free Abelian groups including some Hawaiian groups from the article of Mader and Strüngmann (J. Algebra 229 (2000) 205–233) and thus complementing the main result from the article of Strüngmann (Proc. Amer. Math. Soc. 137 (2009) 3657–3668). A byproduct and starting point for this generalization will be a discussion of the following group class problem: Which groups $G$ have the property that for any cardinal $\kappa$ any subgroup $U$ of the direct sum $G^{(\kappa)}$ is the kernel of some endomorphism of $G^{(\kappa)}$?

Умножение на абелевой группе G - это гомоморфизм $$\mu: G\otimes G\rightarrow G$$. Абелева группа G называется MT-группой, если любое умноженеие на ее периодической части однозначно продолжается до умножения на G. MT-группы изучались во многих работах по теории аддитивных групп колец, но вопрос об их строении остается открытым. В настоящней работе для MT-группы G рассматривается сервантная вполне характеристическая подгруппа $$G^*_\Lambda$$, одно из основных свойств которой заключается в том, что подгруппа $$\bigcap\limits_{p \in \Lambda (G)}pG^*_\Lambda$$ является ниль-идеалом в любом кольце с аддитивной группой G (здесь $$\Lambda(G)$$ - множество всех простых чисел p, для которых p-примарная компонента группы G отлична от нуля). Показано, что для любой MT-группы G либо $$G=G^*_\Lambda$$, либо факторгруппа $$G/G^*_\Lambda$$ несчетна.

Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsion-free abelian groups. For a special class of almost completely decomposable groups we determine the possible ranks of indecomposable groups and show that the possible ranks are realized by indecomposable groups in the class.

Introduction. In [7] and [8] we have determined the isomorphisms of the endomorphism rings of two special classes of torsion-free modules. In [7] the modules were free, over principal left ideal domains, and the isomorpnism was always induced by a semi-linear transformation of the modules. In [8] the ring of scalars was a complete discrete valuation ring, but no further restriction was placed on the modules. Except for the special case when both modules are divisible, the same result holds as in [7]. In each case a key point in the argument is that a certain indecomposable direct summand can be shown to be cyclic. Even for ordinary torsion-free abelian groups, the abundance of the indecomposable ones makes it impossible to prove that isomorphism of the endomorphism rings implies isomorphism of the underlying groups. We show first that for any cardinal number c, there exist homogeneous completely decomposable groups G and H of rank c which are not isomorphic, but have isomorphic rings of endomorphisms. This makes it clear that strong restrictions must be placed on the class of groups considered if it is hoped that the ring isomorphism is to be induced by a group isomorphism. In particular, homogeneity coupled with separability or complete decomposability is not sufficient. We are, however, able to generalize the results of [7] to a class of what we call locally free modules A over principal left ideal domains F. A locally free abelian group (F is the ring of integers) is just a homogeneous separable group of null type [6, p. 208]. This latter class of groups includes the complete direct sum of infinite cyclic groups, as well as all its subgroups [4, p. 176]. For example, the set of all bounded sequences of integers is such a group. We also determine (Theorem B) which semi-linear transformations induce the same ring isomorphisms. The results are similar to the vector space case considered in [2], and apply to the endomorphism rings of locally free modules as well as the torsion-free modules of [8].

(1984). The theory of separable mixed abelian groups. Communications in Algebra: Vol. 12, No. 15, pp. 1813-1834.

We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups A and B for which the homomorphism group Hom(A, B) is fully idempotent. We show that if B is a torsion group or a mixed split group and if at least one of the groups A or B is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups A or B is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups Hom(A, A) comes down to reduced mixed groups A with dense elementary torsion part.