Search results for "group theory"

Some applications of a fundamental theorem by Gluck and Wolf in the character theory of finite groups

1986

Some problems in number theory that arise from group theory

2021

In this expository paper, we present several open problems in number theory that have arisen while doing research in group theory. These problems are on arithmetical functions or partitions. Solving some of these problems would allow to solve some open problem in group theory.

A spectral mapping theorem for perturbed strongly continuous semigroups

1991

Continuity of solutions of linear, degenerate elliptic equations

2009

We consider the simplest form of a second order, linear, degenerate, divergence structure equation in the plane. Under an integrability condition on the degenerate function, we prove that the solutions are continuous.

On a theorem of Berkovich

2002

In a recent paper, Berkovich studied how to describe the nilpotent residual of a group in terms of the nilpotent residuals of some of its subgroups. That study required the knowledge of the structure of the minimal nonnilpotent groups, also called Schmidt groups. The major aim of this paper is to show that this description could be obtained as a consequence of a more complete property, giving birth to some interesting generalizations. This purpose naturally led us to the study of a family of subgroup-closed saturated formations of nilpotent type. An innovative approach to these classes is provided.

Characters and Blocks of Finite Groups

1998

This is a clear, accessible and up-to-date exposition of modular representation theory of finite groups from a character-theoretic viewpoint. After a short review of the necessary background material, the early chapters introduce Brauer characters and blocks and develop their basic properties. The next three chapters study and prove Brauer's first, second and third main theorems in turn. These results are then applied to prove a major application of finite groups, the Glauberman Z*-theorem. Later chapters examine Brauer characters in more detail. The relationship between blocks and normal subgroups is also explored and the modular characters and blocks in p-solvable groups are discussed. Fi…

Blocks with 𝑝-power character degrees

2005

Let B B be a p p -block of a finite group G G . If χ ( 1 ) \chi (1) is a p p -power for all χ ∈ Irr ⁡ ( B ) \chi \in \operatorname {Irr}(B) , then B B is nilpotent.

Segre, Klein, and the Theory of Quadratic Line Complexes

2016

Two of C. Segre’s earliest papers, (Segre 1883a) and (Segre 1884), dealt with the classification of quadratic line complexes, a central topic in line geometry. These papers, the first written together with Gino Loria, were submitted to Felix Klein in 1883 for publication in Mathematische Annalen. Together with the two lengthier works that comprise Segre’s dissertation, (Segre 1883b) and (Segre 1883c), they took up and completed a topic that Klein had worked on a decade earlier (when he was known primarily as an expert on line geometry). Using similar ideas, but a new and freer approach to higher-dimensional geometry, Segre not only refined and widened this earlier work but also gave it a ne…

New Families of Symplectic Runge-Kutta-Nyström Integration Methods

2001

We present new 6-th and 8-th order explicit symplectic Runge-Kutta-Nystrom methods for Hamiltonian systems which are more efficient than other previously known algorithms. The methods use the processing technique and non-trivial flows associated with different elements of the Lie algebra involved in the problem. Both the processor and the kernel are compositions of explicitly computable maps.

Nondivisibility among character degrees II: Nonsolvable groups

2007

We say that a finite group G is an NDAD-group (no divisibility among degrees) if for any 1 < a < b in the set of degrees of the complex irreducible characters of G, a does not divide b. In this article, we determine the nonsolvable NDAD-groups. Together with the work of Lewis, Moreto and Wolf (J. Group Theory 8 (2005)), this settles a problem raised by Berkovich and Zhmud’, which asks for a classification of the NDAD-groups.