scholarly journals Torsion groups of Mordell curves over cubic and sextic fields

2021 ◽  
Vol 99 (3-4) ◽  
pp. 275-297
Author(s):  
Pallab Kanti Dey ◽  
Bidisha Roy
Keyword(s):  
Torsion Groups

scholarly journals Locally soluble groups with min-n.

1974 ◽  
Vol 17 (3) ◽  
pp. 305-318 ◽  
Author(s):  
H. Heineken ◽  
J. S. Wilson
Keyword(s):  
Soluble Group ◽  
Free Groups ◽  
Torsion Group ◽  
Minimal Condition ◽  
Normal Subgroups ◽  
Soluble Groups ◽  
Isomorphism Classes ◽  
Torsion Groups ◽  
Torsion Free ◽  
General Method

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.

scholarly journals On the normaliser problem for G-adapted group rings of torsion groups

2003 ◽  
Vol 67 (1) ◽  
pp. 171-176 ◽  
Author(s):  
Yuanlin Li
Keyword(s):  
Abelian Subgroup ◽  
Torsion Group ◽  
Group Rings ◽  
Torsion Groups

In this note, we prove that if a torsion group G has an Abelian subgroup B such that G/B is Abelian and R is a G-adapted ring with the property that R (G/B) has only trivial units then G has the normaliser property in RG.

scholarly journals Torsion-groups of abelian coverings of links

1982 ◽  
Vol 271 (1) ◽  
pp. 143-143 ◽  
Author(s):  
John P. Mayberry ◽  
Kunio Murasugi
Keyword(s):  
Torsion Groups

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

Some algebraic algebras with Engel unit groups

2019 ◽  
pp. 2150010
Author(s):  
M. H. Bien ◽  
M. Ramezan-Nassab
Keyword(s):  
Multiplicative Group ◽  
Subnormal Subgroup ◽  
Group Algebras ◽  
Algebraic Structures ◽  
Locally Finite ◽  
Division Rings ◽  
Torsion Groups ◽  
Skew Group Algebras ◽  
Subnormal Subgroups

In this paper, we study some algebras [Formula: see text] whose unit groups [Formula: see text] or subnormal subgroups of [Formula: see text] are (generalized) Engel. For example, we show that any generalized Engel subnormal subgroup of the multiplicative group of division rings with uncountable centers is central. Some of algebraic structures of Engel subnormal subgroups of the unit groups of skew group algebras over locally finite or torsion groups are also investigated.

On Pure-High Subgroups of Abelian Groups

1974 ◽  
Vol 17 (4) ◽  
pp. 479-482 ◽  
Author(s):  
K. Benabdallah
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Torsion Groups ◽  
Mixed Groups

Fuchs, in [3], problem 14, proposes the study of pure-high subgroups of an abelian group. In this paper we show that in abelian torsion groups, pure-high subgroups are also high. A natural problem arises, that of characterizing the pure-absolute summands. We show that this concept is the same as absolute summands in torsion groups, but that it is more general in mixed abelian groups. There is a definite connection between the existence of pure TV-high subgroups and the splitting of mixed groups. The notation is that of [3].

scholarly journals The group of homomorphisms of abelian torsion groups

1979 ◽  
Vol 2 (1) ◽  
pp. 69-80
Author(s):  
M. W. Legg
Keyword(s):  
Natural Manner ◽  
Complete Set ◽  
Torsion Groups

LetGandAbe abelian torsion groups. In[5], R. S. Pierce develops a complete set of invariants forHom(G, A). To compute these invariants he introduces, and uses extensively, the group of small homomorphisms ofGintoA. Also, using some of Pierce's methods, Fuchs characterizes this group in [1]. Our purpose in this paper is to characterizeHom(G, A)in what seems to be a more natural manner than either of the treatments just mentioned.

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