Finite Subgroups in Integral Group Rings

1996 ◽  
Vol 48 (6) ◽  
pp. 1170-1179 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Stanley O. Juriaans
Keyword(s):  
Solvable Groups ◽  
Nilpotent Groups ◽  
Fourth Power ◽  
Group Rings ◽  
Integral Group ◽  
Frobenius Groups ◽  
Integral Group Rings ◽  
Finite Subgroups ◽  
Generalized Quaternion

AbstractA p-subgroup version of the conjecture of Zassenhaus is proved for some finite solvable groups including solvable groups in which any Sylow p-subgroup is either abelian or generalized quaternion, solvable Frobenius groups, nilpotent-by-nilpotent groups and solvable groups whose orders are not divisible by the fourth power of any prime.

scholarly journals Integral group rings of solvable groups with trivial central units

2018 ◽  
Vol 30 (4) ◽  
pp. 845-855 ◽  
Author(s):  
Andreas Bächle
Keyword(s):  
Solvable Group ◽  
Group Ring ◽  
Solvable Groups ◽  
Integral Group Ring ◽  
Finite Solvable Group ◽  
Group Rings ◽  
Integral Group ◽  
Frobenius Groups ◽  
Integral Group Rings ◽  
Rational Groups

Abstract The integral group ring {\mathbb{Z}G} of a group G has only trivial central units if the only central units of {\mathbb{Z}G} are {\pm z} for z in the center of G. We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

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.

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