Symmetric units in integral group rings

In this study, we construct a new public key cryptosystem both like El-Gamal and RSA cryptosystem based on Bass cyclic units in integral group rings. Naturally, the underlying hard problems for this system are both decisional Diffie-Hellman assumption and factorization of integers. Since the system is related to RSA scheme indirectly, key generation process is very efficient. This cryptosystem actually supports and strengthens the current public key cryptosystems in [6], [8] and [20]. Encryption algorithm in the system is constructed via non-commutative operations in integral group rings of dihedral groups. Finally, we briefly discuss the security of our cryptosystem against some known attacks.

In this note we investigate the hypercentral units in integral group rings ℤG,where G is not necessarily torsion. One of the main results obtained is the following (Theorem 3.5): if the set of torsion elements of G is a subgroup T of G and if Z 2(𝒰) is not contained in C 𝒰(T),then T is either an Abelian group of exponent 4 or a Q* group. This extends our earlier result on torsion group rings.

For an arbitrary group G , and a G -adapted ring R (for example, R = ℤ), let 𝒰 be the group of units of the group ring RG , and let Z ∞ (𝒰) denote the union of the terms of the upper central series of 𝒰, the elements of which are called hypercentral units. It is shown that Z ∞ (𝒰) ⩽ ( G ). As a consequence, hypercentral units commute with all unipotent elements, and if G has non-normal finite subgroups with R( G ) denoting their intersection, then [𝒰,Z ∞ (𝒰)] ⩽ R( G ). Further consequences are given as well as concrete examples.

In this note, we show that when G G is a torsion group the second center of the group of units U ( Z G ) U({\mathbb Z}G) of the integral group ring Z G {\mathbb Z}G is generated by its torsion subgroup and by the center of U ( Z G ) U({\mathbb Z}G) . This extends a result of Arora and Passi (1993) from finite groups to torsion groups, and completes the characterization of hypercentral units in Z G {\mathbb Z}G when G G is a torsion group.

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring Z G \mathbb {Z}G for a subgroup closed monomial group G G with the property that every cyclic subgroup of order not a divisor of 4 4 or 6 6 is subnormal in G G . If G G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) . Furthermore, for a generalized strongly monomial group G G , the rank of Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G G .

El objetivo de esta memoria es el grupo de unidades U(ZG) del anillo del grupo con coeficientes enteros ZG de un grupo finito G. Podemos enmarcar el problema de estudio U(ZG)en el problema más general de estudiar el grupo de las unidades de un Z-orden de una álgebra racional semisimple de dimensión finita. Este tipo de órdenes son conocidos con el nombre de órdenes clásicos. En un reciente "survey" sobre los órdenes clásicos, Ernest Kleinert escribía lo siguiente: La aritmética de un orden clásico tiene dos partes naturales, la teoría de módulos y de unidades. La teoría de módulos,entendida como Teoría de Representaciones Enteras, ha tenido un desarrollo sistémico y ha generado poderosas técnicas... La teoría de unidades no está en tal estado y todavía tenemos que suscribir la afirmaciónd de Eichler en la introducción de su artículo de 1935. Todavía hay pocos resultados generales que añadan información sustancial a la información básica de que los grupos de unidades de órdenes son finítamente presentadas

The aim of this thesis is to investigate the circumstances under which group rings over fields have non-zero socle, i.e. contain minimal one-sided ideals. After an introductory chapter, we consider the special case o₊ a periodic abelian group and a non-modular field (that is, a field of characteristic prime to the orders of the elements of the group). This special case, and the background material contained in Chapter III, serve as preparation for our principal results, which concern, locally finite groups. We establish necessary and sufficient conditions on an arbitrary field K and a locally finite group 0 for the group ring KG to contain minimal one-sided ideals: the most important condition is that G should be a Cernikov group. We then examine the structure of KG when these conditions are satisfied. Vie show that KG has a finite series of ideal3 each factor of which i3 a direct sum of quasi-Frobenius rings, and characterize the socle of KG. Me also classify indecomposable KG-modules, and determine (for countable but not necessarily locally finite groups G) necessary and sufficient conditions for all indecomposable KG-modules to be irreducible. In the final chapter we consider non-locally-finite groups, conjecturing that group rings of such groups never contain minimal one-sided ideals. We establish the truth of this conjecture for several classes of groups, and also consider semiartinian group rings.

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring ℤG of a finite group G conjugates to a group element within the rational group algebra ℚG. We investigate the Zassenhaus Conjecture (ZC) and a conjecture by W. Kimmerle about prime graph in the normalized unit group of ℤA6.

We fully describe all normal complements of the trivial units in the integral group rings of the dihedral groups of order 6 and 8.

Glucose serves as a key biomarker of metabolic health, requiring continuous and reliable monitoring. Nonenzymatic impedimetric sensors have emerged as a promising alternative to enzymatic glucose sensors, which dominate current clinical practice. However, their sensitivity is limited by weak molecular recognition and inefficient charge transport at the sensing interface. In this study, we tuned the core flexibility and curvature of azine-linked covalent organic frameworks (COFs) by substituting the central triazine ring in the C3 symmetric building unit with a nitrogen atom. This modification maximized glucose-framework interactions, thereby enhancing signal transduction and detection sensitivity. The results confirmed the sensitivity and selectivity of the COFs toward glucose, with a linear response over a wide concentration range [1 aM-5 mM], surpassing previously reported glucose sensors. The sensors achieved a subattomolar limit of detection (LOD), with excellent stability and reproducibility compared to traditional enzymatic sensors, offering a simpler and more practical alternative for integration into wearable and implantable devices. Experimental findings are supported by molecular simulations, which reveals the mechanism of sensing and charge transfer dynamics. This work establishes a versatile design strategy for enzyme-free designed-to-purpose transducers, opening new pathways for ultrasensitive and selective molecular detection.

Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

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.

Describing the group of units of a group ring is a classical problem. Let [Formula: see text] be a rational prime number. We set [Formula: see text] a primitive root of unity of order [Formula: see text], [Formula: see text] the ring of [Formula: see text]-cyclotomic integers, [Formula: see text] a finite abelian [Formula: see text]-group and [Formula: see text] the group of the units [Formula: see text] of [Formula: see text] such that [Formula: see text], where [Formula: see text] is the augmentation map. We will prove that all the elements of the group [Formula: see text] arise from the units of the group ring [Formula: see text], where [Formula: see text] is the cyclic group of order [Formula: see text]. As an application, we describe explicitly the group of units of the group ring [Formula: see text] when [Formula: see text] is an elementary abelian [Formula: see text]-group and [Formula: see text] is a regular prime number.

We calculate, only in terms of a commutative unital ring R of prime characteristic p and an abelian p-mixed group G, the classical Warfield q-invariants Wα,q(VR(G)) of the group VR(G) of all normalized units in the group ring R(G). This continues our results in (Extr. Math., 2005), (Collect. Math., 2008) and (J. Alg. Appl., 2008).

It is proved that from the isomorphism of the integral group rings of two groups follows the isomorphism of the quotient-groups of these groups by the second commutator subgroup. Groups with certain conditions T1 and T2 are considered. From condition T1 it follows that the group algebra over an integral domain has no zero-divisors, and from T2 that the group of invertible elements of this algebra is trivial. Sufficient conditions are given, under which there are no zero-divisors in the group algebra of a pro-p-group over a field of p elements.