scholarly journals Subgroups generated by images of endomorphisms of Abelian groups and duality

2018 ◽  
Vol 21 (5) ◽  
pp. 885-900 ◽  
Author(s):  
Grigore Călugăreanu ◽  
Andrey R. Chekhlov ◽  
Piotr A. Krylov
Keyword(s):  
Proper Subgroup ◽  
Abelian Groups ◽  
Pure Subgroup ◽  
Simple Groups

Abstract A subgroup H of a group G is called endo-generated if it is generated by endo-images, i.e. images of endomorphisms of G. In this paper we determine the following classes of Abelian groups: (a) the endo-groups, i.e. the groups all of whose subgroups are endo-generated; (b) the endo-image simple groups, i.e. the groups such that no proper subgroup is an endo-image; (c) the pure-image simple, i.e. the groups such that no proper pure subgroup is an endo-image; (d) the groups all of whose endo-images are pure subgroups; (e) the ker-gen groups, i.e. the groups all of whose kernels are endo-generated. Some dual notions are also determined.

An inverse theorem for additive bases

2016 ◽  
Vol 12 (06) ◽  
pp. 1509-1518 ◽  
Author(s):  
Yongke Qu ◽  
Dongchun Han
Keyword(s):  
Abelian Group ◽  
Proper Subgroup ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Regular Sequence ◽  
Inverse Theorem ◽  
Additive Basis

Let [Formula: see text] be a finite abelian group of order [Formula: see text], and [Formula: see text] be the smallest prime dividing [Formula: see text]. Let [Formula: see text] be a sequence over [Formula: see text]. We say that [Formula: see text] is regular if for every proper subgroup [Formula: see text], [Formula: see text] contains at most [Formula: see text] terms from [Formula: see text]. Let [Formula: see text] be the smallest integer [Formula: see text] such that every regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] forms an additive basis of [Formula: see text], i.e. [Formula: see text]. Recently, [Formula: see text] was determined for many abelian groups. In this paper, we determined [Formula: see text] for more abelian groups and characterize the structure of the regular sequence [Formula: see text] over [Formula: see text] of length [Formula: see text] and [Formula: see text].

Abelian Groups in Which Every α-Pure Subgroup is β-Pure

1973 ◽  
Vol 25 (3) ◽  
pp. 560-566 ◽  
Author(s):  
J. Douglas Moore ◽  
Edwin J. Hewett
Keyword(s):  
General Problem ◽  
Abelian Groups ◽  
Pure Subgroup ◽  
Ordinal Numbers ◽  
Special Cases

The determination of the abelian groups in which every neat subgroup is pure is a relatively routine exercise (see [6]). There are numerous problems of this type; for example, the determination of the groups in which every pure subgroup is isotype or the groups in which every subgroup is isotype. These are all special cases of the general problem of determining the abelian groups in which every α-pure subgroup is β-pure for arbitrary ordinal numbers α and β. The solution of this general problem is the object of this paper.

Equivalent Abelian Groups

1957 ◽  
Vol 9 ◽  
pp. 291-297 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Natural Number ◽  
Abelian Groups ◽  
Pure Subgroup ◽  
Torsion Free

Throughout this note all groups are abelian, written additively. We refer to Kurosh (8; 9) for notation, terminology and theorems used without reference. We recall the notion of a serving subgroup (or pure subgroup) of a group . This is a subgroup in which for every natural number n every equation nx = s, s ∊ can be solved provided that it can be solved in . If is torsion-free, “linearly closed” subgroups coincide with serving subgroups and is a serving subgroup if and only if / is torsionfree.

Scott heights of abelian groups

1994 ◽  
Vol 59 (4) ◽  
pp. 1351-1359 ◽  
Author(s):  
Mark E. Nadel
Keyword(s):  
Abelian Groups ◽  
Pure Subgroup ◽  
The Other ◽  
Complete Theory ◽  
Primary Motivation ◽  
Direct Sum ◽  
Ordinal Measure ◽  
Free Abelian Groups ◽  
Torsion Free ◽  
Relevant Material

The Scott height of a structure gives ordinal measure of the inhomogeneity of the structure. The Scott specturm of a collection of structures is the set of Scott heights of structures in the collection. We will recall the precise definitions of these and related concepts in the next section. The reader thoroughly unfamiliar with these notions may want to skip ahead before reading the rest of this Introduction.In [11] it is shown that every model of the complete theory of (, +, 1), where, as usual, denotes the integers, is ℵ0-homogeneous, and therefore has Scott height at most ω. On the other hand, a footnote in [1] gives a model of the theory of (, +) which is not ℵ0-homogeneous, while in [11] such a model is described which can be expanded to a model of the theory of (, +, 1). However, since it is also true that any model of the theory of (, +) is isomorphic to a subgroup, in fact a pure subgroup, of a direct sum of and a torsion-free divisible group, it is easy to see that any such model must be ≡∞ω to a model of cardinality at most and so must have Scott height below .After having recalled the relevant material about Scott heights in §2, we will review the situation for torsion abelian groups in §3. In §4 we shall produce torsion-free abelian groups of high Scott height. It is the proof of Theorem 15 that was our primary motivation in writing this paper.

On the weakly monomial subgroups of finite simple groups

2019 ◽  
Vol 18 (02) ◽  
pp. 1950037
Author(s):  
Shuqin Dong ◽  
Hongfei Pan ◽  
Feng Tang
Keyword(s):  
Finite Group ◽  
Proper Subgroup ◽  
Maximal Subgroups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a finite group. A proper subgroup [Formula: see text] of [Formula: see text] is said to be weakly monomial if the order of [Formula: see text] satisfies [Formula: see text]. In this paper, we determine all the weakly monomial maximal subgroups of finite simple groups.

On Quasi-Essential Subgroups of Primary Abelian Groups

1970 ◽  
Vol 22 (6) ◽  
pp. 1176-1184 ◽  
Author(s):  
Khalid Benabdallah ◽  
John M. Irwin
Keyword(s):  
Abelian Groups ◽  
Pure Subgroup ◽  
Negative Integer ◽  
Primary Group ◽  
Prime Integer ◽  
Image Position

All groups considered in this paper are abelian. A subgroup N of a group G is said to be a quasi-essential subgroup of G if G = 〈H, K〉 whenever H is an N-high subgroup of G and K is a pure subgroup of G containing N. We started the study of such subgroups in [5]; in particular, we characterized subsocles of a primary group which were both quasi-essential and centres of purity. In this paper we show that quasi-essential subsocles of a primary group are necessarily centres of purity answering thus in the affirmative a question raised in [5].We obtain the following theorem: A subsocle S of a p-group G is quasi-essential if and only if either S ⊂ G1or (pnG)[p] ⊃ S ⊃ (pn+1G)[p] for some non-negative integer n. The notation is that of [1]. If G is a group, thenwhere p is a prime integer.

Finite groups with a unique non-abelian proper subgroup

2019 ◽  
Vol 18 (11) ◽  
pp. 1950210
Author(s):  
Bijan Taeri ◽  
Fatemeh Tayanloo-Beyg
Keyword(s):  
Finite Groups ◽  
Proper Subgroup ◽  
Abelian Groups

In this paper we classify finite non-abelian groups having a unique non-abelian proper subgroup.

scholarly journals Annihilator equivalence of torsion-free abelian groups

1992 ◽  
Vol 52 (1) ◽  
pp. 119-129
Author(s):  
P. Schultz ◽  
C. Vinsonhaler ◽  
W. J. Wickless
Keyword(s):  
Structural Properties ◽  
Equivalence Relation ◽  
Abelian Groups ◽  
Equivalence Classes ◽  
Pure Subgroup ◽  
The Other ◽  
Closure Properties ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe define an equivalence relation on the class of torsion-free abelian groups under which two groups are equivalent ifevery pure subgroup of one has a non-zero image in the other, and each has a non-zero image in every torsion-free factor of the other.We study the closure properties of the equivalence classes, and the structural properties of the class of all equivalence classes. Finally we identify a class of groups which satisfy Krull-Schmidt and Jordan-Hölder properties with respect to the equivalence.

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