Degree Graphs of Simple Linear and Unitary Groups

Let G be a finite group and let cd (G) be the set of irreducible character degrees of G. The degree graph Δ(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). We determine the graph Δ(G) for the finite simple groups of types A _?(q) and ² A _? (q ²), that is, for the simple linear and unitary groups.     

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.     

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.     

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.     

Non-expander Cayley Graphs of Simple Groups

For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.     

Efficient Simple Groups

Abstract

We prove that the simple group L ₃(5) which has order 372000 is efficient by providing an efficient presentation for it. This leaves one simple group with order less than one million, S ₄(4) which has order 979200, whose efficiency or otherwise remains to be determined.     

Spectra of Automorphic Extensions of Finite Simple Exceptional Groups of Lie Type

The spectrum ω (G) of a finite group G is the set of orders of elements of G. Let S be a simple exceptional group of type E ₆ or E ₇ . We describe all finite groups G such that S ≤ G ≤ Aut S and ω (G) = ω (S) and completes the study of the recognition-by-spectrum problem for all simple exceptional groups of Lie type.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

Verifying Huppert's Conjecture for PSL3(q) and PSU3(q 2)

Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ? H × A, where A is an abelian group. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.     

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).     

Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form × D,where h has infinite order and D is finite.     

Dynamics of Simple Lie Groups on Lorentz Manifolds

We give an alternate proof of N. Kowalsky's theorem describing the collection of connected simple Lie groups with finite center which admit a nontrivial, nonproper action by isometries of a connected Lorentz manifold.     

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.     

OD-Characterization of Alternating and Symmetric Groups of Degree p+5

In this paper,it is proved that all the alternating groups A_(p+5) are ODcharacterizable and the symmetric groups S_(p+5) are 3-fold OD-characterizable,where p + 4 is a composite number and p + 6 is a prime and 5≠p∈π(1000!).     

On the Vertices of Simple Modules for Symmetric Groups

In this article, we are concerned with the computation of vertices of some series of simple modules for the symmetric group of n letters in odd characteristic. In the first, the second, and the third section of this work we recall some more or less general results that are needed in our proofs. The fourth section contains new ingredients (in terms of dimension) that play an important role in our proofs of the main results in the last section.     

一类仿射型本原非同步置换群的构造

同步置换群的研究是置换群理论中的一个前沿课题.通过在有限域上定义广义Paley图,来讨论这类图的团数(clique number)和色数(chromatic number)相等的条件.然后证明构造的广义Paley图和一类仿射群的轨道图同构,进而讨论这类仿射群的同步性.在此基础上给出一类本原非同步群的构造.     

Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type

Let G = G(q) be a finite almost simple exceptional group ofLie type over the field of q elements, where q = p^a and p isprime. The main result of the paper determines all maximal subgroupsM of G(q) such that M is an almost simple group which is alsoof Lie type in characteristic p, under the condition that rank(M)> rank(G). The conclusion is that either M is a subgroupof maximal rank, or it is of the same type as G over a subfieldof F_q, or (G, M) is one of (, F₄(q)), (, C₄(q)), (E₇(q),³D₄(q)). This completes work of the first author with Saxl andTesterman, in which the same conclusion was obtained under someextra assumptions.