不可约特征标次数为等差数的有限可解群

设G为有限群,cd(G)表示G的所有复不可约特征标次数的集合.本文研究了不可约特征标次数为等差数的有限可解群,得到两个结果:如果cd(G)={1,1+d,1+2d,…,1+kd},则k≤2或cd(G)={1,2,3,4};如果cd(G)={1,a,a+d,a+2d,…,a+kd},|cd(G)|≥4,(a,d)=1,则cd(G)={1,2,2^e+1,2e+1,2^(e+1)},并给出了d>1时群的结构.     

特征标次数的商与有限群结构

本文考察特征标次数的商如何影响有限群的结构.设G是非线性不可约特征标次数的商都是n次方自由的有限群,首先,对可解群G证明了它的导长能被一个仅依赖于n的函数所界定;其次,给出了当n≤3时有限群G的结构.     

特征标次数的商与有限群结构

本文考察特征标次数的商如何影响有限群的结构.设G是非线性不可约特征标次数的商都是n次方自由的有限群,首先,对可解群G证明了它的导长能被一个仅依赖于n的函数所界定;其次,给出了当n≤3时有限群G的结构.     

有限群的特征标余次数

对于有限群G的一个不可约特征标χ,我们定义它的余次数为cod(χ)=|G:kerχ|/χ(1).本文介绍有限群特征标余次数方面的工作进展和未解决的问题.     

有限群特征标次数商的几点注记

对自然数n ,W(n)表示n中的素因子个数 (计重数 ) .对于有限群G的不可约复特征标集Irr(G) ,令W0 (G) =max{W(|G :kerχ|χ(1) ) | χ∈Irr(G) ,χ(1) >1} ,本文将考察W0 (G)≤ 3时有限群G的群论结构 .     

特征标表中零点个数很少的有限可解群

（１）与（２）均曾研究过有某特征标恰在两个共轭类上取非零值的有限群。本文讨论上述情形的另一极端,即每个不可约特征标至多在ｐ个共轭类上取零值的有限可解群,这里ｐ为群阶的最小素因子,通过对极小非交换商群的分析,我们将刻划这类群的结构。     

关于有限群的特征标次数的商

设Ｇ是有限非Ａｂｅｌ群，Ｉｒｒ（Ｇ）是Ｇ的一切不可约特征标组成的集合。在这篇文章中，我们考虑商集，考察对这些数作的某些假设如何影响Ｇ的结构。例如，设是ｎ的素数分解式。置及．我们证明：如果Ｇ是非可解的且Ｗ（Ｇ）＝４，则Ｇ恰是下述群之一：Ｚ＿ｐ×Ａ＿５，ＳＬ（２，５），Ｓ＿５，ＰＳＬ（２，７），ＰＳＬ（２，１１）和ＰＳＬ（２，１３）。     

不同不可约特征标在某一元素上取不同值的可解群

本文给出了满足下面条件的有限可解群的分类存在某个元素,使得不同的不可约特征标在该元素上取不同的值.     

不同不可约特征标在某一元素上取不同值的可解群

本文给出了满足下面条件的有限可解群的分类:存在某个元素,使得不同的不可约特征标在该元素 上取不同的值.     

特征与可解性

本文研究了具有不可约特征x使得x(1)²=|G:Z(G)|成立的群G的可解性,并证明了:对于这样的群G,当|Irr(G)|≤4时,G是可解群;当4<|Irr(G)|≤6时,在某些条件下,G是有解群.     

Prime Divisibility Among Degrees of Solvable Groups

Abstract

Let G be a finite, nonabelian, solvable group. Following work by D. Benjamin, we conjecture that some prime must divide at least a third of the irreducible character degrees of G. Benjamin was able to show the conjecture is true if all primes divide at most 3 degrees. We extend this result by showing if primes divide at most 4 degrees, then G has at most 12 degrees. We also present an example showing our result is best possible.     

Common Divisor Character Degree Graphs of Solvable Groups with Four Vertices

Let G be a finite solvable group. The common divisor graph Γ(G) attached to G is a character degree graph. Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor (m, n) > 1. In this article, we classify all graphs with four vertices that may occur as Γ(G) for solvable group G.     

The Kernel of the Average Sylow Multiplicity Character and the Solvable Radical

Let G be a finite group, and let p ₁,…, p _m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P ₁,…, P _m , where P _i is a Sylow p _i -subgroup of G, and g ∈ G, denote by m _𝒫(g) the number of factorizations g = g ₁…g _m such that g _i  ∈ P _i . Previously, it was shown that the properly normalized average of m _𝒫 over all 𝒫 is a complex character over G termed the Average Sylow Multiplicity Character. The present article identifies the kernel of this character as the subgroup of G consisting of all g ∈ G such that m _𝒫(gx) = m _𝒫(x) for all 𝒫 and all x ∈ G. This result implies a close connection between the kernel and the solvable radical of G.     

Finite Solvable Groups with an Irreducible Character Vanishing on Just One Class of Elements

The aim of this note is to classify the finite solvable groups with an irreducible character that vanishes on just one class of elements.     

On the Character Degree Graph of Solvable Groups, I Three Primes

For any finite solvable group G we show that if three primes dividing the degrees of certain irreducible characters of G are given, then there exists an irreducible character of G with degree divisible by at least two of the given primes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solitary Solvable Groups

A subgroup H of a group G is a solitary subgroup of G if G does not contain another isomorphic copy of H. Combining together the concepts of solitary subgroups and solvable groups, we define (normal) solitary solvable groups and (normal) strongly solitary solvable groups. We derive several results that hold for these groups and we discuss classes of groups that, under certain hypotheses, are (normal) solitary solvable and (normal) strongly solitary solvable. We also derive several results about p-groups that are solitary solvable.     

The Average Sylow Multiplicity Character and the Solvable Residual

Let G be a finite group, and let p₁,…, p_m be the distinct prime divisors of |G|. Given a sequence 𝒫 =P₁,…, P_m, of Sylow p_i-subgroups of G, and g ∈ G, denote by m_𝒫(g) the number of factorizations g = g₁…g_m such that g_i ∈ P_i. The properly normalized average of m_𝒫 over all 𝒫 is a complex character over G whose kernel contains the solvable radical of G [7 Levy , D. ( 2010 ). The average Sylow multiplicity character and solvability of finite groups . Communications in Algebra. 38 : 632 – 644 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The present paper characterizes the solvable residual of G in terms of this character.     

Three-Dimensional Topological Loops with Solvable Multiplication Groups

We prove that each 3-dimensional connected topological loop L having a solvable Lie group of dimension ≤5 as the multiplication group of L is centrally nilpotent of class 2. Moreover, we classify the solvable non-nilpotent Lie groups G which are multiplication groups for 3-dimensional simply connected topological loops L and dim G ≤ 5. These groups are direct products of proper connected Lie groups and have dimension 5. We find also the inner mapping groups of L.