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

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.

Benson's cofibrants, Gorenstein projectives and a related conjecture

In this short article, we will be principally investigating two classes of modules over any given group ring – the class of Gorenstein projectives and the class of Benson's cofibrants. We begin by studying various properties of these two classes and studying some of these properties comparatively against each other. There is a conjecture made by Fotini Dembegioti and Olympia Talelli that these two classes should coincide over the integral group ring for any group. We make this conjecture over group rings over commutative rings of finite global dimension and prove it for some classes of groups while also proving other related results involving the two classes of modules mentioned.

Comparative analysis of the accuracy requirements of the equipment for determining the mean integral refractive index of air using different realizations of the gradient method

The speed of propagation of electromagnetic waves in the Earth’s atmosphere differs from the speed of their propagation in a vacuum, which is one of the main factors that have a significant impact on the accuracy of long distance measurement. This influence is taken into account in long distance measurement with the correction for the mean integral group refractive index of air, which depends on such meteorological parameters as temperature, atmospheric pressure and relative air humidity. The purpose of this work is to compare the accuracy requirements for equipment designed to measure temperature, pressure, and relative humidity required to determine the above correction by the gradient method using the Euler-Maclaurin quadrature formula (hereafter, the Euler-Maclaurin method) and the formula based on Hermite interpolation polynomials (hereafter, the Hermite method). The requirements for the uncertainty of measurements carried out with the sensors of meteorological parameters, allowing to find the mean integral group refractive index of air, providing length measurements of the baselines of up to 5 km with an expanded uncertainty of not more than 1 mm, are established. Keywords: atmosphere; mean integral group refractive index of air; laser long distance measurement

Global and local properties of finite groups with only finitely many central units in their integral group ring

The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group, we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same; in particular, the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.