Some finite groups with nontrivial centers which are Galois groups

Group Theory ◽  
1989 ◽  
pp. 87-110 ◽  
Keyword(s):  
Finite Groups ◽  
Galois Groups

scholarly journals FINITE GROUPS AS GALOIS GROUPS OF FUNCTION FIELDS WITH INFINITE FIELD OF CONSTANTS

2010 ◽  
Vol 88 (3) ◽  
pp. 301-312
Author(s):  
C. ÁLVAREZ-GARCÍA ◽  
G. VILLA-SALVADOR
Keyword(s):  
Finite Groups ◽  
Function Field ◽  
Function Fields ◽  
Galois Groups ◽  
Infinite Field ◽  
Separable Extension ◽  
Separable Extensions

AbstractLetE/kbe a function field over an infinite field of constants. Assume thatE/k(x) is a separable extension of degree greater than one such that there exists a place of degree one ofk(x) ramified inE. LetK/kbe a function field. We prove that there exist infinitely many nonisomorphic separable extensionsL/Ksuch that [L:K]=[E:k(x)] andAutkL=AutKL≅Autk(x)E.

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 Global and local properties of finite groups with only finitely many central units in their integral group ring

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andreas Bächle ◽  
Mauricio Caicedo ◽  
Eric Jespers ◽  
Sugandha Maheshwary
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Galois Groups ◽  
Group Rings ◽  
Integral Group ◽  
Open Problems ◽  
Integral Group Rings ◽  
Local Properties ◽  
Global And Local ◽  
The Impact

Abstract The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group, we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same; in particular, the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.

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.

scholarly journals Solvable-by-finite groups as differential Galois groups

2002 ◽  
Vol 11 (3) ◽  
pp. 403-423 ◽  
Author(s):  
Claude Mitschi ◽  
Michael F. Singer
Keyword(s):  
Finite Groups ◽  
Galois Groups

scholarly journals One-Valued Mappings of Groups into Fields

1953 ◽  
Vol 6 ◽  
pp. 41-52 ◽  
Author(s):  
Katsuhiko Masuda
Keyword(s):  
Finite Groups ◽  
Galois Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Galois Fields ◽  
Necessary And Sufficient

The aim of this article is to investigate algebraic nature of systems of one-valued mappings of given group into given field and to apply it to the theory of Galois algebras and duality of compact T0-groups. The results obtained in the following are those; factor systems of Galois algebras with finite Galois groups are defined without any restrictions on the orders of Galois groups and the coefficient fields, a necessary and sufficient condition for them to be associated with Galois fields is obtained, dualities of finite groups are obtained very simply without any restrictions for coefficient field of representations, and Tannaka’s duality of compact T0-groups is proved without the use of the compactness of Tannaka representation groups of representations of compact T- groups and the use of Kampen’s theorem.

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