A SURVEY ON FREE SUBGROUPS IN THE GROUP OF UNITS OF GROUP RINGS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350004 ◽  
Author(s):  
JAIRO Z. GONÇALVES ◽  
ÁNGEL DEL RÍO
Keyword(s):  
Finite Groups ◽  
Free Groups ◽  
Group Algebras ◽  
Group Rings ◽  
Integral Group ◽  
Natural Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Survey Results ◽  
Free Subgroups

In this survey we revise the methods and results on the existence and construction of free groups of units in group rings, with special emphasis in integral group rings over finite groups and group algebras. We also survey results on constructions of free groups generated by elements which are either symmetric or unitary with respect to some involution and other results on which integral group rings have large subgroups which can be constructed with free subgroups and natural group operations.

Coherent groups of units of integral group rings and direct products of free groups

2016 ◽  
Vol 162 (2) ◽  
pp. 191-209
Author(s):  
ÁNGEL DEL RÍO ◽  
PAVEL ZALESSKII
Keyword(s):  
Direct Product ◽  
Division Algebra ◽  
Finite Groups ◽  
Free Groups ◽  
Group Rings ◽  
Integral Group ◽  
Direct Products ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Finite Dimensional

AbstractWe classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$, the group of units of the integral group ring of G, does not contain a direct product of two non-abelian free groups. This list of groups contains all the groups for which $\mathcal{U}({\mathbb Z} G)$ is coherent. This reduces the problem to classify the finite groups G for which $\mathcal{U}({\mathbb Z} G)$ is coherent to decide about the coherency of a finite list of groups of the form SLn(R), with R an order in a finite dimensional rational division algebra.

Free Subgroups in the Unit Groups of Integral Group Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1342-1352 ◽  
Author(s):  
B. Hartley ◽  
P. F. Pickel
Keyword(s):  
Sufficient Conditions ◽  
Finite Subgroup ◽  
Free Subgroup ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
Necessary And Sufficient ◽  
Free Subgroups

Let G be a group, ZG the group ring of G over the ring Z of integers, and U(ZG) the group of units of ZG. One method of investigating U(ZG) is to choose some property of groups and try to determine the groups G such that U(ZG) enjoys that property. For example Sehgal and Zassenhaus [9] have given necessary and sufficient conditions for U(ZG) to be nilpotent (see also [7]), and the same authors have investigated when U(ZG) is an FC (finite-conjugate) group [10]. For a survey of related questions, see [3]. In this paper we consider when U(ZG) contains a free subgroup of rank 2. We conjecture that if this does not happen, then every finite subgroup of G is normal, from which various other conclusions then follow (see Lemma 4).

Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups

2021 ◽  
Vol 28 (04) ◽  
pp. 561-568
Author(s):  
Jinke Hai ◽  
Lele Zhao
Keyword(s):  
Abelian Group ◽  
Simple Group ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
Coleman Automorphism

Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.

scholarly journals Integral Group Rings with Nilpotent Unit Groups

1976 ◽  
Vol 28 (5) ◽  
pp. 954-960 ◽  
Author(s):  
César Polcino Milies
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Unit Element ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
The Subject ◽  
A Finite Group

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.The study of the nilpotency of U(RG) has been the subject of several papers.

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