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 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.

On Coleman Automorphisms of Extensions of Finite Quasinilpotent Groups by Some Groups

2018 ◽  
Vol 25 (02) ◽  
pp. 181-188 ◽  
Author(s):  
Jinke Hai ◽  
Yixin Zhu
Keyword(s):  
Finite Group ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
Coleman Automorphism ◽  
A Finite Group

Let G be an extension of a finite quasinilpotent group by a finite group. It is shown that under some conditions every Coleman automorphism of G is an inner automorphism. The interest in such automorphisms arose from the study of the normalizer problem for integral group rings. Our theorems generalize some well-known results.

scholarly journals TORSION UNITS IN INTEGRAL GROUP RINGS OF CERTAIN METABELIAN GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 363-385 ◽  
Author(s):  
Martin Hertweck
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Rings ◽  
Integral Group ◽  
Metabelian Groups ◽  
Metacyclic Groups ◽  
Integral Group Rings ◽  
Torsion Unit ◽  
A Finite Group ◽  
Torsion Units

AbstractIt is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

On the Torsion Units of Some Integral Group Rings

2006 ◽  
Vol 13 (02) ◽  
pp. 329-348 ◽  
Author(s):  
Martin Hertweck
Keyword(s):  
Finite Group ◽  
Finite Simple Group ◽  
Fundamental Result ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Torsion Unit ◽  
Zassenhaus Conjecture ◽  
A Finite Group ◽  
Torsion Units

It is shown that any torsion unit of the integral group ring ℤG of a finite group G is rationally conjugate to a trivial unit if G = P ⋊ A with P a normal Sylow p-subgroup of G and A an abelian p′-group (thus confirming a conjecture of Zassenhaus for this particular class of groups). The proof is an application of a fundamental result of Weiss. It is also shown that the Zassenhaus conjecture holds for PSL(2,7), the finite simple group of order 168.

COLEMAN AUTOMORPHISMS OF STANDARD WREATH PRODUCTS OF NILPOTENT GROUPS BY GROUPS WITH PRESCRIBED SYLOW 2-SUBGROUPS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350156 ◽  
Author(s):  
ZHENGXING LI ◽  
JINKE HAI
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Consequence ◽  
Nilpotent Groups ◽  
Wreath Products ◽  
Coleman Automorphism ◽  
A Finite Group ◽  
Generalized Quaternion ◽  
Standard Wreath Product ◽  
The Normalizer Property

Let G = N wr H be the standard wreath product of N by H, where N is a finite nilpotent group and H is a finite group whose Sylow 2-subgroups are either cyclic, dihedral or generalized quaternion. It is shown that every Coleman automorphism of G is inner. As a direct consequence of this result, it is obtained that the normalizer property holds for G.

Determining Units in Some Integral Group Rings

1990 ◽  
Vol 33 (2) ◽  
pp. 242-246 ◽  
Author(s):  
E. G. Goodaire ◽  
E. Jespers ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Dihedral Group ◽  
Group Rings ◽  
Integral Group ◽  
Basic Information ◽  
Integral Group Rings ◽  
Special Cases ◽  
A Finite Group ◽  
Made In

In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].

scholarly journals On the Structure of Integral Group Rings of Sporadic Groups

2000 ◽  
Vol 3 ◽  
pp. 274-306 ◽  
Author(s):  
Frauke M. Bleher ◽  
Wolfgang Kimmerle
Keyword(s):  
Automorphism Groups ◽  
Symmetric Groups ◽  
Computational Procedure ◽  
Isomorphism Problem ◽  
Simple Groups ◽  
Group Rings ◽  
Integral Group ◽  
Sporadic Groups ◽  
Integral Group Rings ◽  
Sporadic Simple Groups

AbstractThe object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the ℚ-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automorphism groups of sporadic simple groups G which are different from G. The proofs are given with the aid of the GAP computer algebra program by applying a computational procedure to the ordinary and modular character tables of the groups. It is also shown that the isomorphism problem of integral group rings has a positive answer for certain almost simple groups, in particular for the double covers of the symmetric groups.

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