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.     

Nonsolvable Groups All of Whose Character Degrees are Odd-Square-Free

A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S ≤ G ≤ Aut(S) for a finite simple group S. More generally, we show that if G is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups. If the alternating group A ₇ is involved in G, the structure of G can be further restricted.     

An algorithm and self-reproducing systems

Let G be a group and let N be a normal subgroup of G. We set cd(G|N) to be the degrees of the irreducible characters of G whose kernels do not contain N. We associate a graph with this set. The vertices of this graph are the primes dividing degrees in cd(G|N), and there is an edge between p and q if pq divides some degree in cd(G|N). In this paper, we study this graph when it is disconnected, and we study its diameter when it is connected.     

An infinite family of cubic nonnormal Cayley graphs on nonabelian simple groups

We construct a connected cubic nonnormal Cayley graph on ${\mathrm{A}}_{2^m?1}$ for each integer $m?4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on nonabelian simple groups.     

具有两个p'维非线性不可约特征标的非可解群

McKay猜想是有限群表示理论中的一个重要问题.本文考虑了具有两个p'维不可约特征标的非可解群群G,并证明了McKay猜想对此类群成立.更进一步,当p为奇素数时,下面结果成立:p=3,G≌PSL_2(7)或存在一个正规2-群N,满足G/N≌PSL_2(5).     

Characterization of Groups L2(q) by NSE Where q∈{17, 27, 29}

The authors show that linear simple groups L_2(q) with q ∈ {17, 27, 29} can be uniquely determined by nse(L_2(q)), which is the set of numbers of elements with the same order.     

On Homotopy of connected graphs having the same degree function

Let denote the set of simple graphs having the degree function . It is known that any two graphs from are homotopic in the sense that one can be obtained from the other by a series of simple two-edge switches so that every intermediate graph is in . Let denote the set of simple graphs having at most k components. A natural question arises whether it is possible to extend the above homotopy result on the class , and in particular, whether it is possible to obtain a connected simple graph from another connected simple graph of the same degree function by a series of two edge switches preserving connectivity. In this paper we give an answer to this question. We also consider two-edge switches of a special type and single out some classes of graphs for which these special operations provide the corresponding homotopy between its members.     

特征标次数图是一种连通图的单群

对有限单群G,假设其不可约特征标次数图Δ(G)连通,且图顶点集ρ(G)=π_1∪π_2∪{p},其中|π_1|,|π_2|≥1,π_1∩π_2=θ,且π_1与π_2中顶点不相邻.证明了Δ(G)满足上面的假设的有限单群G只有4种:M_(11),J_1,PSL_3(4)或²B_2(q2B_2(q²),其中q2),其中q²一1是Mersenne素数.     

Hamiltonian paths in Cayley graphs

The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log₂|G|, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction of 3-regular Hamiltonian expanders.     

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.     

Homology Stability for Unitary Groups II

In this paper, the homology stability problem for hyperbolic unitary groups over a local ring with an infinite residue field is studied. We obtain a much better range of homology stability compared to the results existing in the literature. An application of our results is given.     

Algebraic Cayley graphs over finite fields

A new algebraic Cayley graph is constructed using finite fields. It provides a more flexible source of expander graphs. Its connectedness, the number of connected components, and diameter bound are studied via Weil's estimate for character sums. Furthermore, we study the algorithmic problem of computing the number of connected components and establish a link to the integer factorization problem.     

Degree sequence and supereulerian graphs

A sequence d=(d₁,d₂,…,d_n) is graphic if there is a simple graph G with degree sequence d, and such a graph G is called a realization of d. A graphic sequence d is line-hamiltonian if d has a realization G such that L(G) is hamiltonian, and is supereulerian if d has a realization G with a spanning eulerian subgraph. In this paper, it is proved that a nonincreasing graphic sequence d=(d₁,d₂,…,d_n) has a supereulerian realization if and only if d_n≥2 and that d is line-hamiltonian if and only if either d₁=n−1, or ∑_di=1d_i≤∑_dj≥2(d_j−2).