scholarly journals Tripleableness of pro-c-groups

1976 ◽  
Vol 21 (3) ◽  
pp. 299-309 ◽  
Author(s):  
Lim Chong-Keang
Keyword(s):  
Algebraic Number ◽  
Full Subcategory ◽  
Forgetful Functor ◽  
Galois Groups ◽  
Inverse Limits ◽  
Discrete Groups ◽  
Profinite Groups ◽  
Finite Products

Let C be a nontrivial full subcategory of the category F of finite discrete groups and continuous homomorphisms, closed under subobjects, quotient and finite products. We consider the category PC of pro-C-groups and continuous homomorphisms (i.e. inverse limits of C-groups) which forms a variety in category PF of profinite groups and continuous homomorphisms. The study of pro-Cgroups is motivated by their occurrence as Galois groups of filed extensions in algebraic number thory (see Serre (1965)). The purpose of this paper is to study the tripleableness of the forgetful functors from PC to various underlying categories. It is also shown that PC is equivalent to the category of algebras of the theory of the forgetful functor from C to S (the category of sets and mappings).

Profinite Modules

1973 ◽  
Vol 16 (3) ◽  
pp. 405-415
Author(s):  
Gerard Elie Cohen
Keyword(s):  
Finite Group ◽  
General Theory ◽  
Finite Length ◽  
Finite Groups ◽  
Inverse Limit ◽  
Abelian Groups ◽  
Point Of View ◽  
Inverse Limits ◽  
Profinite Groups ◽  
Inverse Systems

An inverse limit of finite groups has been called in the literature a pro-finite group and we have extensive studies of profinite groups from the cohomological point of view by J. P. Serre. The general theory of non-abelian modules has not yet been developed and therefore we consider a generalization of profinite abelian groups. We study inverse systems of discrete finite length R-modules. Profinite modules are inverse limits of discrete finite length R-modules with the inverse limit topology.

scholarly journals Some categorical aspects of the inverse limits in ditopological context

2018 ◽  
Vol 19 (1) ◽  
pp. 101
Author(s):  
Filiz Yildiz
Keyword(s):  
Compatibility Condition ◽  
Natural Transformation ◽  
Inverse Limits ◽  
Infinite Products ◽  
Inverse Systems ◽  
Finite Products

<p>This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category if<strong>PDitop</strong>, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category <strong>Inv<sub>ifPDitop</sub></strong> consisting of the inverse systems constructed by the objects and morphisms of if<strong>PDitop</strong>, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical - functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems - limits constructed in if<strong>PDitop</strong> and the ditopological infinite products are characterized by the finite products via inverse limits.</p>

Effective aspects of profinite groups

1981 ◽  
Vol 46 (4) ◽  
pp. 851-863 ◽  
Author(s):  
Rick L. Smith
Keyword(s):  
Finite Groups ◽  
Inverse Limit ◽  
Commutator Subgroup ◽  
Galois Groups ◽  
Profinite Group ◽  
Profinite Groups ◽  
Frattini Subgroup ◽  
A Finite Group ◽  
Work Done ◽  
General Method

Profinite groups are Galois groups. The effective study of infinite Galois groups was initiated by Metakides and Nerode [8] and further developed by LaRoche [5]. In this paper we study profinite groups without considering Galois extensions of fields. The Artin method of representing a finite group as a Galois group has been generalized (effectively!) by Waterhouse [14] to profinite groups. Thus, there is no loss of relevance in our approach.The fundamental notions of a co-r.e. profinite group, recursively profinite group, and the degree of a co-r.e. profinite group are defined in §1. In this section we prove that every co-r.e. profinite group can be effectively represented as an inverse limit of finite groups. The degree invariant is shown to behave very well with respect to open subgroups and quotients. The work done in this section is basic to the rest of the paper.The commutator subgroup, the Frattini subgroup, thep-Sylow subgroups, and the center of a profinite group are essential in the study of profinite groups. It is only natural to ask if these subgroups are effective. The following question exemplifies our approach to this problem: Is the center a co-r.e. profinite group? Theorem 2 provides a general method for answering this type of question negatively. Examples 3,4 and 5 are all applications of this theorem.

scholarly journals Central extensions and Schur’s multiplicators of Galois groups

1983 ◽  
Vol 90 ◽  
pp. 137-144 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Galois Groups ◽  
Algebraic Number Theory ◽  
Galois Extensions ◽  
Central Extensions ◽  
Global Norms ◽  
Local Norms ◽  
Abelian Fields

When he developed the theory of central extensions of absolute abelian fields in [1], Fröhlich clearly pointed out a role of Schur’s multiplicators of the Galois groups in algebraic number theory. Another role of them was to be well known when the gaps between the everywhere local norms and the global norms of finite Galois extensions were cohomologically described by Tate [10]. The relation of two roles was investigated by Furuta [2], Shirai [9], Heider [3] and others.

scholarly journals Kronecker classes of fields and covering subgroups of finite groups

1994 ◽  
Vol 57 (1) ◽  
pp. 17-34 ◽  
Author(s):  
Cheryl E. Praeger
Keyword(s):  
Group Theory ◽  
Finite Groups ◽  
Algebraic Number ◽  
Number Fields ◽  
Galois Groups ◽  
Algebraic Number Fields ◽  
Covering Property ◽  
Bounded Degree ◽  
Field Extensions ◽  
Equivalent Field

AbstractKronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of bounded degree are explored.

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