Solvable Groups Satisfying the Two-prime Hypothesis,Part I

In this paper, finite solvable groups satisfying the “-prime hypothesis” are considered. Specifically, a bound on the number of irreducible character degrees of such a group is obtained when . The general situation is also considered, and generalizations of the -prime hypothesis are analyzed. Presented by A. Verschoren.     

Nonsolvable Normal Subgroups with Few Induced Character Degrees

In this note, we classify the finite groups admitting a nonsolvable normal subgroup with three induced irreducible character degrees.     

Recognizing the Automorphism Groups of Mathieu Groups

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

The Number of Irreducible Character Degrees of Solvable Groups Satisfying the One-Prime Hypothesis

Let G be a finite group, and write cd(G) for the set of degrees of irreducible characters of G. We say G satisfies the one-prime hypothesis if whenever a and b are distinct degrees in cd(G), then the greatest common divisor of a and b is either 1 or a prime. We show that if G is a solvable group satisfying the one-prime hypothesis, then |cd(G)|≤9. We also construct a solvable group G satisfying the one-prime hypothesis with |cd(G)|=9 which shows that the bound found in this paper is the best possible bound. Presented by D. Passman Mathematics Subject Classification (2000) 20C15.     

Solvable Groups Satisfying the Two-Prime Hypothesis II

In this paper, we consider solvable groups that satisfy the two-prime hypothesis. We prove that if G is such a group and G has no nonabelian nilpotent quotients, then |cd G|?≤?462,515. Combining this result with the result from part I, we deduce that if G is any such group, then the same bound holds.     

On the Character Degrees of Suzuki p-Groups of Type A and C

In this note, we construct the irreducible characters of Suzuki p-groups of types A _p (m, θ) and C _p (m, θ, ?).     

Some Skew Linear Groups Satisfying Generalized Group Identities

In this article, we consider a type of generalized group identity and extend some earlier results. For example, we show that, if D is a division ring with infinite center, then every subnormal subgroup of GL_n(D) satisfying a generalized group identity over GL_n(D) is central.     

辫子群的全部二维不可约表示

通过直接解矩阵方程给出了Bn群的全部二维不可约表示．     

自同构群阶为prq~2的有限群

本文给出满足|Aut(G)|=prq~2,q>2的有限群 G的完全分类。     

The Commuting Graph of Minimal Nonsolvable Groups

The purpose of this paper is to prove that if G is a finite minimal nonsolvable group (i.e. G is not solvable but every proper quotient of G is solvable), then the commuting graph of G has diameter 3. We give an example showing that this result is the best possible. This result is related to the structure of finite quotients of the multiplicative group of a finite-dimensional division algebra.     

On Complemented Subgroups of Finite Groups

51. IntroductionIt is quite clear that the ekistence of complements for some families of subgroups of agroup gives a lot ofinfor~ion about its structure. FOr instance, Hall[6] proved that a groupG is supersoluble with elementary abelian Sylow subgroups if and only if G is complemellted,that is, every subgroup of G is comPlemeded in G. The same anchor also proved that agroup is soluble if and only if every Sylow subgroup is complemellted (see [3;I,3.5]). Morerecelltly, Arad and Wardll] pro…     

Nonsolvable D 2-groups

Let G be a finite group. Let Irr₁(G) be the set of nonlinear irreducible characters of G and cd₁(G) the set of degrees of the characters in Irr₁(G). A group G is said to be a D₂-group if |cd₁(G)| = |Irr₁(G)| - 2. The main purpose of this paper is to classify nonsolvable D₂-groups.     

On the Genus of the Nilpotent Graphs of Finite Groups

The nilpotent graph of a group G is a simple graph whose vertex set is G?nil(G), where nil(G) = {y ∈ G | ? x, y ? is nilpotent ? x ∈ G}, and two distinct vertices x and y are adjacent if ? x, y ? is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.     

一类不能作为自同构群的奇阶群

本文考虑如下问题：怎样的有限群可以作为另一个有限群的全自同构群？我们首先证明，若有限群Ｋ有一个正规Ｓｙｌｏｗｐ－子群使得｜Ｋ：Ｚ（Ｋ）｜ｐ＝ｐ２，那么Ｋ有２阶自同构．利用这个结果，我们证明了，若奇阶群Ｇ具有阶Ｐｓｍ（１≤ｓ≤３），ｐ为｜Ｇ｜的最小素因子，ｐｍ，ｍ无立方因子，则Ｇ不可能作为全自同构群．     

Regular Functions Satisfying Irregular Ordinary Differential Equations

This paper deals with entire solutions to linear ordinary differential equations in the complex domain. We show that certain entire solutions to singular equations, cannot satisfy any normalized equation without singularities. We provide two proofs of this result, one based on the indicial equation and the other using the Frobenius notion of irreducibility. Our examples include the entire Bessel function.     

Irreducible Subgroups of GL 1(D) Satisfying Group Identities

ABSTRACT

Let D be a finite dimensional F -central division algebra and G an irreducible subgroup of D*: = GL ₁(D). Here we investigate the structure of D under various group identities on G. In particular, it is shown that when [D:F] = p ², p a prime, then D is cyclic if and only if D* contains a nonabelian subgroup satisfying a group identity.     

A Characteristic Property of Finite p -Nilpotent Groups

Abstract

An element x of a group G is called Q-central if there exists a central chief factor H/K of G such that x ∈ H\K. It is proved that a finite group G is p-nilpotent if and only if every element in G _p  \Φ(G _p ) is Q-central. We will adapt Doerk and Hawkes [Doerk, K., Hawkes, T. (1992). Finite Soluble Groups. Berlin–New York: Walter de Gruyter, pp. 892] for notations and basic results.     

Finite Groups with Given Quantitative Non-Nilpotent Subgroups

The aim of this article is to give a new characterization of the solvability of finite groups by some quantitative properties of their non-nilpotent proper subgroups.     

Direct Products with Abelian Automorphism Groups

The article considers when the direct product of two finite groups has an Abelian automorphism group.