Some infinite groups with a prescribed system of complemented infinite Abelian subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

Download Full-text

Infinite Groups with Complemented Non-Abelian Subgroups

Ukrainian Mathematical Journal ◽

10.1007/s11253-015-1096-y ◽

2015 ◽

Vol 67 (4) ◽

pp. 506-514

Author(s):

P. P. Baryshovets

Keyword(s):

Infinite Groups ◽

Abelian Subgroups

Download Full-text

Infinite Abelian Subgroups of Some Infinite Groups

Journal of the London Mathematical Society ◽

10.1112/jlms/s1-39.1.240 ◽

1964 ◽

Vol s1-39 (1) ◽

pp. 240-244 ◽

Cited By ~ 2

Author(s):

C. R. Kulatilaka

Keyword(s):

Infinite Groups ◽

Abelian Subgroups

Download Full-text

Groups isomorphic to their non-nilpotent subgroups

Glasgow Mathematical Journal ◽

10.1017/s0017089500032572 ◽

1998 ◽

Vol 40 (2) ◽

pp. 257-262 ◽

Cited By ~ 2

Author(s):

Howard Smith ◽

James Wiegold

Keyword(s):

Finite Groups ◽

Proper Subgroup ◽

Complete Classification ◽

Finitely Generated ◽

Infinite Groups ◽

Finite Case ◽

Locally Nilpotent ◽

Infinite Case ◽

Natural Generalisation ◽

Abelian Subgroups

We were concerned in [12] with groups G that are isomorphic to all of their non-abelian subgroups. In order to exclude groups with all proper subgroups abelian, which are well understood in the finite case [7] and which include Tarski groups in the infinite case, we restricted attention to the class X of groups G that are isomorphic to their nonabelian subgroups and that contain proper subgroups of this type; such groups are easily seen to be 2-generator, and a complete classification was given in [12, Theorem 2] for the case G soluble. In the insoluble case, G/Z(G) is infinite simple [12; Theorem 1], though not much else was said in [12] about such groups. Here we examine a property which represents a natural generalisation of that discussed above. Let us say that a group G belongs to the class W if G is isomorphic to each of its non-nilpotent subgroups and not every proper subgroup of G is nilpotent. Firstly, note that finite groups in which all proper subgroups are nilpotent are (again) well understood [9]. In addition, much is known about infinite groups with all proper subgroups nilpotent (see, in particular, [8] and [13] for further discussion) although, even in the locally nilpotent case, there are still some gaps in our understanding of such groups. We content ourselvesin the present paper with discussing finitely generated W-groups— note that a W-group is certainly finitely generated or locally nilpotent. We shall have a little more to say about the locally nilpotent case below.

Download Full-text

Infinite groups with many complemented subgroups

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry ◽

10.1007/s13366-021-00597-w ◽

2021 ◽

Author(s):

M. Ferrara ◽

M. Trombetti

Keyword(s):

Infinite Groups ◽

Abelian Subgroups

AbstractThis paper has two souls. On one side, it is a survey on (infinite) groups in which certain systems of subgroups are complemented (like for instance the abelian subgroups). On another side, it provides generalizations and new, easier proofs of some (un)known results in this area.

Download Full-text

On the structure of groups whose non-abelian subgroups are subnormal

Open Mathematics ◽

10.2478/s11533-014-0444-8 ◽

2014 ◽

Vol 12 (12) ◽

Author(s):

Leonid Kurdachenko ◽

Sevgi Atlıhan ◽

Nikolaj Semko

Keyword(s):

Finite Groups ◽

Infinite Groups ◽

Locally Finite ◽

Locally Finite Groups ◽

Abelian Subgroups

AbstractThe main aim of this article is to examine infinite groups whose non-abelian subgroups are subnormal. In this sense we obtain here description of such locally finite groups and, as a consequence we show several results related to such groups.

Download Full-text

Coverings of Groups by Abelian Subgroups

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-081-1 ◽

1978 ◽

Vol 30 (5) ◽

pp. 933-945 ◽

Cited By ~ 5

Author(s):

V. Faber ◽

R. Laver ◽

R. McKenzie

Keyword(s):

Set Theory ◽

Point Of View ◽

Infinite Groups ◽

Partition Relations ◽

Vertex Set ◽

Abelian Subgroups

Paul Erdôs has suggested an investigation of infinite groups from the point of view of the partition relations of set theory. In particular, he suggested that given a group G, one considers the graph T with vertex set G whose edges are the pairs ﹛g, h﹜ which do not commute.

Download Full-text