A note on morphisms to wreath products

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].

A note on the support of units in group rings

We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isomorphic to a subgroup of the group base. Moreover we study the orders of torsion units in integral group rings of finite almost quasisimple groups and the existence of torsion-free normal subgroups of finite index in the unit group.

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.

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.

Preface. 1. Groups. 2. Rings, Modules and Algebras. 3. Group Rings. 4. A Glance at Group Representations. 5. Group Characters. 6. Ideals in Group Rings. 7. Algebraic Elements. 8. Units of Group Rings. 9. The Isomorphism Problem. 10. Free Group of Units. 11. Properties of the Unit Group. Bibliography. Index.

We obtain restrictions on units of even order in the integral group ring of a finite group by studying their actions on the reductions modulo 4 of lattices over the 2‐adic group ring . This improves the “lattice method” which considers reductions modulo primes , but is of limited use for essentially due to the fact that . Our methods yield results in cases where has blocks, whose defect groups are Klein four groups or dihedral groups of order 8. This allows us to disprove the existence of units of order for almost simple groups with socle where and to answer the prime graph question affirmatively for many such groups.

Many of the best-known cryptosystems, particularly public key systems, are numbertheory based and work within the setting of abelian groups. With increased computing power the security of these systems is constantly threatened. Consequently, non-commutative algebraic structures have been investigated as possible sources of cryptographic platforms. We discuss some of the group-theoretic problems exploited in the security of these new cryptosystems. We also describe Hurley's method for encryption and decryption using specially chosen units in group rings.

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

The basis of a significant amount of cryptographic systems for information protection are different computationally hard problems. One of these problems is finding the discrete logarithm value in a certain finite group. The problem is to obtain for any two given elements of this group such natural number that the first element to the power of the number equals the second element. In order to implement the cryptosystem, they have to choose an appropriate finite group and an element of high multiplicative order in it, so that computing the discrete logarithm is a hard problem. Powerful quantum computers will solve in polynomial time the discrete logarithm problem in the most common finite groups (multiplicative group of prime or extended finite field, group of points of elliptic curve over a finite field). That is why, as one of directions, they study groups consisting of invertible elements of group rings specified by various rings and groups. In the paper, the issue of finding high order units for special group rings, defined by finite field and dihedral group, is explore

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.

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.

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).

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

For any finite group G the group U(Z[G]) of units in the integral group ring Z[G]is an arithmetic group in a reductive algebraic group, namely the Zariski closure of SL1 (Q[G]). In particular, the isomorphism type of the Q-algebra Q[G] determines the commensurability class of U(Z[G]); we show that, to a large extent, the converse is true. In fact, subject to a certain restriction on the Q-representations of G the converse is exactly true.