scholarly journals A Survey on Just-Non-𝔛Groups

2010 ◽  
Vol 2010 ◽  
pp. 1-8
Author(s):  
Daniele Ettore Otera ◽  
Francesco G. Russo
Keyword(s):  
Present Note ◽  
Topological Groups ◽  
Quotient Groups ◽  
Proper Quotient

Let𝔛be a class of groups. A group which does not belong to𝔛but all of whose proper quotient groups belong to𝔛is called just-non-𝔛group. The present note is a survey of recent results on the topic with a special attention to topological groups.

scholarly journals Sequential completeness of quotient groups

2000 ◽  
Vol 61 (1) ◽  
pp. 129-150 ◽  
Author(s):  
Dikran Dikranjan ◽  
Michael Tkačenko
Keyword(s):  
Abelian Group ◽  
Direct Products ◽  
Topological Groups ◽  
Complete Group ◽  
Sequential Completeness ◽  
Algebraic Properties ◽  
Countable Compactness ◽  
Quotient Groups

We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

Groups with Hypercyclic Proper Quotient Groups

2003 ◽  
Vol 55 (4) ◽  
pp. 566-575
Author(s):  
L. A. Kurdachenko ◽  
P. Soules
Keyword(s):  
Quotient Groups ◽  
Proper Quotient

scholarly journals Quotients of Strongly Realcompact Groups

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
L. Morales ◽  
M. Tkachenko
Keyword(s):  
Topological Group ◽  
Closed Subgroup ◽  
Topological Groups ◽  
Quotient Groups

AbstractA topological group is strongly realcompact if it is topologically isomorphic to a closed subgroup of a product of separable metrizable groups. We show that if H is an invariant Čech-complete subgroup of an ω-narrow topological group G, then G is strongly realcompact if and only if G/H is strongly realcompact. Our proof of this result is based on a thorough study of the interaction between the P-modification of topological groups and the operation of taking quotient groups.

Finitep-Groups All of Whose Proper Quotient Groups are Abelian or Inner-Abelian

2010 ◽  
Vol 38 (8) ◽  
pp. 2797-2807 ◽  
Author(s):  
Qinhai Zhang ◽  
Lili Li ◽  
Mingyao Xu
Keyword(s):  
Quotient Groups ◽  
Proper Quotient

scholarly journals On varieties of metabelian p-groups, and their laws

1967 ◽  
Vol 7 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Warren Brisley
Keyword(s):  
Critical Group ◽  
Proper Factor ◽  
Simple Basis ◽  
Quotient Groups ◽  
Proper Quotient

The purpose of this article is to present some results on varieties of metabelian p-groups, nilpotent of class c, with the prime p greater than c. After some preliminary lemmas in § 3, it is established in § 4, Theorem 3, that there is a simple basis for the laws of such a variety, and this basis is explicitly stated. This allows the description of the lattice of such varieties, and in § 5, Theorem 4, it is shown that each such variety has a two-generator member which generates it; this is established by the help of Theorem 5, which states that each critical group is a two-generator group, and Theorem 6, which gives explicitly the varieties generated by the proper subgroups, by the proper quotient groups, and by the proper factor groups of such a critical group.

scholarly journals Hausdorffness in varieties of topological groups

1998 ◽  
Vol 57 (1) ◽  
pp. 147-151
Author(s):  
Carolyn E. McPhail
Keyword(s):  
Proper Class ◽  
Topological Groups ◽  
Variety Of Groups ◽  
Quotient Groups

A variety of topological groups is a class of (not necessarily Hausdorff) topological groups closed under the operations of forming subgroups, quotient groups and arbitrary products. It is well known that the class of groups underlying the topological groups contained in any variety of topological groups is a variety of groups. Much work on topological groups is restricted to Hausdorff topological groups and so it is relevant to know if the class of groups underlying Hausdorff topological groups in is a variety of groups. It is shown that this is not always the case. Indeed it is proved that this is not the case for an important proper class of varieties of topological groups.

scholarly journals Generating varieties of topological groups

1973 ◽  
Vol 18 (3) ◽  
pp. 191-197 ◽  
Author(s):  
M. S. Brooks ◽  
Sidney. A. Morris ◽  
Stephen. A. Saxon
Keyword(s):  
General Theorem ◽  
Topological Groups ◽  
Cartesian Products ◽  
Quotient Groups

Recently several papers on varieties of topological groups have appeared. In this note we investigate the question: if Ω is a class of topological groups, what topological groups are in the variety V(Ω) generated by Ω that is, what topological groups can be “manufactured” from Ω using repeatedly the operations of taking subgroups, quotient groups and arbitrary cartesian products? We seeka general theorem which will be useful for investigating V(Ω) for well-known classesΩ.

Sign in / Sign up

Export Citation Format

Share Document