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.     

特征标次数的重数与可解群结构

非线性不可约特征标次数的重数全部为1的有限群的分类是熟知的.对可解群,本文讨论更一般的,即非线性不可约特征标次数的重数都与群阶互素的有限群的纯群论性质.特别地,得到了非线性不可约特征标次数的重数均小于2p的奇阶群G的分类结果.这里p为群阶|G|的最小素因子.     

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.     

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.     

Fitting Height and Character Degree Graphs

Given a group G, Γ(G) is the graph whose vertices are the primes that divide the degree of some irreducible character and two vertices p and q are joined by an edge if pq divides the degree of some irreducible character of G. By a definition of Lewis, a graph Γ has bounded Fitting height if the Fitting height of any solvable group G with Γ(G)=Γ is bounded (in terms of Γ). In this note, we prove that there exists a universal constant C such that if Γ has bounded Fitting height and Γ(G)=Γ then h(G)≤C. This solves a problem raised by Lewis. Research supported by the Spanish Ministerio de Educación y Ciencia, MTM2004-06067-C02-01 and MTM2004-04665, the FEDER and Programa Ramón y Cajal.     

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.     

连通图的度序列及连通平面图的低度点个数

美国数学家Bondy给出了一个非负整数序列为简单图的度序列的充要条件.本文对此进行了发展,证明了一个正整数序列为连通简单图的度序列的充要条件;然后在此基础上又探讨了平面图的低度点个数问题并定义了描述连通平面图的低度点个数的一个概念φ(n,m),并对某些低阶平面图求出了φ(n,m)的值.最后给出了φ(n,m)的上下界.     

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.     

Cut Vertices and Degree One Vertices of Zero-Divisor Graphs

This article continues to examine cut vertices in the zero-divisor graphs of commutative rings with 1. The main result is that, with only seven known exceptions, the zero-divisor graph of a commutative ring has a cut vertex if and only if the graph has a degree one vertex. This naturally leads to an examination of the degree one vertices of zero-divisor graphs.     

The Inducibility of Graphs on Four Vertices

We consider the problem of determining the maximum induced density of a graph H in any graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H. The exact value of this quantity is known only for a handful of small graphs and a specific set of complete multipartite graphs. Answering questions of Brown–Sidorenko and Exoo, we determine the inducibility of K_{1, 1, 2} and the paw graph. The proof is obtained using semidefinite programming techniques based on a modern language of extremal graph theory, which we describe in full detail in an accessible setting.     

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.     

图中的度序列

给出了一个非减的非负整数序列是某个图的度序列的一个新刻划.     

定向可图的度偶序列

n为非负整数序列，若存在以该序列为度序列的图，则称n为可图的，特别的，若此图是一个定向图，该序列则称为是定向可图的，本提出了一个判断序列是否为定向可图的充分必要条件，并且在定理的证明过程中给出了一个在定理条件下构造所求定向图的有效算法。     

门槛图与度极大图

证明了门槛图与度极大图是一类图的两种不同说法,同时用图的对角限制极左矩阵刻画这一类图的结构。     

Groups with Four Character Degrees and Derived Length Four

In this article, we consider solvable groups that have four character degrees and derived length four.     

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.     

维数分别为51和52的对称群的OD-刻画

利用有限群的阶和它的次数型分别对对称群S51和S_（52）进行了刻画,得到：对称群S_（51）和S_（52）均可3-重OD-刻画.     

Bubble-sort图和Modified Bubble-sort图的自同构群

Bubble-Sort图和Modified Bubble-Sort图是两类特殊的Cayley图,由于其在网络构建中的应用而受到广泛关注.本文完全确定了这两类图的自同构群.     

Degree Graphs of Simple Groups of Exceptional Lie Type

Abstract

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type.     

6p阶可解对称图的分类

设ｘ是简单无向图，Ｇ是Ａｕｔ（Ｘ）的一个于群，Ｘ称为Ｇ－对称的，如果Ｇ在ｘ的１－孤（即两相邻顶点构成的有序偶）集合上的作用是传递的；ｘ称为对称图，如果Ｘ是Ａｕｔ（ｘ）－对称的；ｘ称为可解对称的，如果Ａｕｔ（Ｘ）包含可解子群Ｇ，使Ｘ是Ｇ－对称的．本文给出了具有６Ｐ个顶点的可解对称图的一个分类，这里ｐ≥５是素数．