Units In Group Rings Research Articles

Aritmethic lattices of SO(1,n) and units of group rings

We establish that standard arithmetic subgroups of a special orthogonal group SO(1,n) are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings. We also prove that finite quotients of group of units of any of these group rings determines the original group ring.

On Gardam's and Murray's units in group rings

We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.

On the unit groups of rings with involution

On the unit groups of rings with involution

Units of group rings and a conjecture of H. J. Zassenhaus

Units of group rings and a conjecture of H. J. Zassenhaus

Fuchs' problem for 2-groups

Nearly 60 years ago, László Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question of which 2-groups are realizable as unit groups of finite rings, a necessary step toward determining which nilpotent groups are realizable. We prove that all 2-groups of exponent 4 are realizable in characteristic 2. Moreover, while some groups of exponent greater than 4 are realizable as unit groups of rings, we establish general constraints on the exponent of a realizable 2-group. These constraints are used to describe examples of 2-groups that cannot be the group of units of a finite ring.

An application of blocks to torsion units in group rings

We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p p -block of a finite group G G , the normalized unit group of the integral group ring of G G contains an element of order p q pq if and only if so does G G for q q a prime different from p p . Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.

HeLP: a GAP package for torsion units in integral group rings

We briefly summarize the background of the HeLP-method for torsion units in group rings and present some functionality of a GAP-package implementing it.

Bounded Engel and solvable unitary units in group rings

Let FG be the group ring of a group G over a field F. We consider the group of unitary units of FG with respect to the classical involution. Under suitable restrictions upon F, we show that if the unitary units of FG are both bounded Engel and solvable, then the entire unit group of FG is nilpotent. This extends a result of Fisher, Parmenter and Sehgal.

Units of group rings, the Bogomolov multiplier and the fake degree conjecture

AbstractLet π be a finite p-group and ${\mathbb{F}_{q}}$ a finite field with q = pn elements. Denote by $\I_{\mathbb{F}_{q}}$ the augmentation ideal of the group ring ${\mathbb{F}_{q}}$[π]. We have found a surprising relation between the abelianization of 1 + $\I_{\mathbb{F}_{q}}$, the Bogomolov multiplier B0(π) of π and the number of conjugacy classes k(π) of π: $$ \left | (1+\I_{\Fq})_{\ab} \right |=q^{\kk(\pi)-1}|\!\B_0(\pi)|. In particular, if π is a finite p-group with a non-trivial Bogomolov multiplier, then 1 + $\I_{\mathbb{F}_{q}}$ is a counterexample to the fake degree conjecture proposed by M. Isaacs.

SON CONSTRUCTION OF CRYPTOGRAPHIC SYSTEMS OVER UNITS OF GROUP RINGS

Encryption schemes can be derived from the units which are known as invertible elements in a group ring. Besides there are many studies on units in group rings in the literature, we can also see some studies of units in terms of applicability to cryptography and coding theory. In this work, we shall establish the relations between units and RSA problem. Our motivation is to construct a much stronger public key cryptosystem. It is clear that we must consider a mathematical hard problem to do this. Thus, we investigate some interesting properties of units which are no deterministic algorithm in general. Our notations follow (8).

On the Torsion Units of Integral Adjacency Algebras of Finite Association Schemes

Torsion units of group rings have been studied extensively since the 1960s. As association schemes are generalization of groups, it is natural to ask about torsion units of association scheme rings. In this paper we establish some results about torsion units of association scheme rings analogous to basic results for torsion units of group rings.

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

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.

A Survey on *-Group Identities on Units of Group Rings

Let F be an infinite field of characteristic different from 2 and G a torsion group. Let the group ring FG have an involution induced from an involution on G. We survey recent results establishing the conditions under which the unit group of FG satisfies a *-group identity. The surprising outcome is that the unit group satisfies a *-group identity if and only if the symmetric units of FG satisfy a group identity.

The ranks of central unit groups of integral group rings of alternating groups

Let G be a finite group and U(Z(Z G)) be the group of units of the center Z(Z G) of the integral group ring Z G (the central unit group of the ring Z G). The purpose of the present work is to study the ranks r n of groups U(Z(ZAn)), i.e., of central unit groups of integral group rings of alternating groups A n . We shall find all values n for r n = 1 and propose an approach on how to describe the groups U(Z(ZAn)) in these cases, and we will present some results of calculations of r n for n ≤ 600.

Codes from zero-divisors and units in group rings

A new construction method for codes using encodings from group rings is described and presented. They consist primarily of two types: zero-divisor and unit derived codes. Previous codes from group rings focused on ideals; for example, cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals. The authors use the result that a group ring is isomorphic to a certain well-defined ring of matrices and thus, every group ring element has an associated matrix. This allows matrix algebra to be used as needed in the study and production of codes, enabling the creation of standard generator and check matrices. Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties, such as being LDPC or self-dual, may be expressed as properties within the group ring, thus enabling the construction of codes with these properties. The methods are general enabling the construction of codes with many types of group rings. There is no restriction on the ring and thus codes over the integers, over matrix rings or even over group rings themselves are possible and fruitful.

On Symmetric Elements and Symmetric Units in Group Rings

ABSTRACT Let R be a commutative ring, G a group, and RG its group ring. Let ϕ: RG → RG denote the R-linear extension of an involution ϕ defined on G. An element x in RG is said to be symmetric if ϕ (x) = x. A characterization is given of when the symmetric elements (RG)ϕ of RG form a ring. For many domains R it is also shown that (RG)ϕ is a ring if and only if the symmetric units form a group. The results obtained extend earlier work of Bovdi (2001), Bovdi et al. (1996), Bovdi and Parmenter (1997), Broche Cristo (2003, to appear), Giambruno and Sehgal (1993), and Lee (1999), who dealt with the case that ϕ is the involution * mapping g ∈ G onto g−1.

PYTHAGOREAN TRIPLES AND UNITS IN INTEGRAL GROUP RINGS

In this note, we describe a simple method for finding units of group rings of the form Z[G] = Z[H]#Z[C2] for H an abelian group, and apply this to the case G = D4, the dihedral group of order 8. Here, units may be described as integer points on hyperboloids, and, defining units u, v to be equivalent if they differ by an inner automorphism, we see that this equivalence relation partitions each hyperboloid into finitely many classes.

Nilpotent Symmetric Units in Group Rings

Let F be a field of characteristic different from 2 and G a torsion group. We let ∗ denote the involution on the group ring, FG, which sends each group element to its inverse. Let U +(FG) denote the set of units which are symmetric with respect to ∗. We say that U +(FG) is nilpotent if it satisfies the group identity (x 1 ,…, x n ) = 1, for some n ≥ 2. In this paper, we determine the conditions under which U +(FG) is nilpotent.

Units in group rings of crystallographic groups

In \cite{ms98-1}, the authors initiated a technique of using affine representations to study the groups of units of integral group rings of crystallographic groups. In this paper, we use this approach for some special classes of crystallographic groups. F

Normal subgroups of the group of units in group rings of torsion groups

Normal subgroups of the group of units in group rings of torsion groups