Recognizing the Automorphism Groups of Mathieu Groups Through Their Orders and Large Degrees of Their Irreducible Characters

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.     

A characterization of almost simple K 3-groups

It is a well-known fact that characters of a finite group can give important information about the group's structure. Also it was proved by the third author of this article 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 almost simple K₃-groups by their orders and at most three irreducible character degrees of their character tables.     

A New Characterization of Simple K 3-Groups by Their Orders and Large Degrees of Their Irreducible Characters

It is a well-known fact that characters of a finite group can give important information of the structure of the group. Also it was proved by the second author that a finite simple group can uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite group by using less information of its character table, and successfully characterize K ₃-groups by their orders and one or two irreducible character degrees of their character tables.     

有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了[4]中的问题1，并说明[6，定理]的条件可以极大地减弱．     

Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

It is known that a number of algebraic properties of the braidgroups extend to arbitrary finite Coxeter-type Artin groups.Here we show how to extend the results to more general groupsthat we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellativemonoid where the expressions of a given element have boundedlengths, and where left and right lowest common multiples exist.A Garside monoid is a Gaussian monoid in which the left andright lowest common multiples satisfy an additional symmetrycondition. A Gaussian group is the group of fractions of a Gaussianmonoid, and a Garside group is the group of fractions of a Garsidemonoid. Braid groups and, more generally, finite Coxeter-typeArtin groups are Garside groups. We determine algorithmic criteriain terms of presentations for recognizing Gaussian and Garsidemonoids and groups, and exhibit infinite families of such groups.We describe simple algorithms that solve the word problem ina Gaussian group, show that these algorithms have a quadraticcomplexity if the group is a Garside group, and prove that Garsidegroups have quadratic isoperimetric inequalities. We constructnormal forms for Gaussian groups, and prove that, in the caseof a Garside group, the language of normal forms is regular,symmetric, and geodesic, has the 5-fellow traveller property,and has the uniqueness property. This shows in particular thatGarside groups are geodesically fully biautomatic. Finally,we consider an automorphism of a finite Coxeter-type Artin groupderived from an automorphism of its defining Coxeter graph,and prove that the subgroup of elements fixed by this automorphismis also a finite Coxeter-type Artin group that can be explicitlydetermined. 1991 Mathematics Subject Classification: primary20F05, 20F36; secondary 20B40, 20M05.     

A New Characterization of the Mathieu Groups and Alternating Groups with Degrees Primes

In the past thirty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians. Such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacy classes, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups and so on. Here the authors continue this topic in a new area tending to characterize finite simple groups with given orders by some special conjugacy class sizes, such as largest conjugacy class sizes, smallest conjugacy class sizes greater than 1 and so on.     

Finite Groups in Which Each Irreducible Character has at Most Two Zeros

Let G be a finite group, Irr(G) denotes the set of irreducible complex characters of G and gG the conjugacy class of G containing element g. A well-known theorem of Burnside([1,Theorem 3.15]) states that every nonlinear X ∈ Irr(G) has a zero on G, that is, an element x (or a conjugacy class xG) of G with X(x) = 0. So, if the number of zeros of character table is very small, we may expect, the structure of group is heavily restricted. For example, [2, Proposition 2.7] claimes that G is a Frobenius group with a complement of order 2 if each row in charcter table has at most one zero (its proof uses the classification of simple groups). In this note, we characterize the finite group G satisfying the following hypothesis:     

关于有限群某些子群的X-半置换性

Let A be a subgroup of a group G and X a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we try to use the X-semipermutability of some subgroups to characterize the structure of finite groups.     

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.     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

Double Coset Decompositions of Groups

We show that residually finite or word hyperbolic groups which can be decomposed as a finite union of double cosets of a cyclic subgroup are necessarily virtually cyclic, and we apply this result to the study of Frobenius permutation groups. We show that in general, finite double coset decompositions of a group can be interpreted as an obstruction to splitting a group as a free product with amalgamation or an HNN extension.     

Fusions of Character Tables and Schur Rings of Abelian Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. The theory is developed in terms of the S-rings of Schur and Wielandt. We discuss certain classes of p-groups which fuse from abelian groups and give examples of such groups which do not. We also show that a large class of simple groups do not fuse from abelian groups. The methods to show fusion include the use of extensions which are Camina pairs, but other techniques on S-rings are also developed.     

Some Properties of Bell Groups

For any integer n ≠ 0,1, a group G is said to be “n-Bell” if it satisfies the identity [x ⁿ ,y] = [x,y ⁿ ]. It is known that if G is an n-Bell group, then the factor group G/Z ₂(G) has finite exponent dividing 12n ⁵(n ? 1)⁵. In this article we show that this bound can be improved. Moreover, we prove that every n-Bell group is n-nilpotent; consequently, using results of Baer on finite n-nilpotent groups, we give the structure of locally finite n-Bell groups. Finally, we are concerned with locally graded n-Bell groups for special values of n.     

论有限群元素共轭类的平均长度

本文研究有限群元素共轭类的平均长度问题.利用初等群论方法和有限群特征标理论,在共轭类平均长度为某一定数时,获得了对有限群结构的刻划,且对有限群数量性质的研究是有意义的.     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.     

Groups whose cyclic subgroups have bounded permutability properties

It follows from classical results of Neumann and Macdonald that a group G has finite commuator subgroup if and only if either the normalizers of cyclic subgroups of G have boundedly finite indices or cyclic subgroups of G have bounded indices in their normal closures. In this paper, groups with a similar condition are considered, when normality is replaced by permutability.      

Groups Saturated by a Finite Set of Groups

We study the periodic groups that are saturated by a finite set of finite (nonabelian simple) groups, obtaining some information about the elements of the saturating set.     

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.     

Generalized Covering Groups

Let v and be two varieties of groups defined by the sets of laws V and W, respectively. In this paper we construct a -v-covering group of a finite v-perfect group G, and show that every automorphism of G may be lifted to its -v-covering group. These generalize the previous work of the first two authors (1999). We also characterize the -v-irreducible extensions in some sense.AMS Mathematics Subject Classification (2000) Primary 20E34 20E36 Secondary 20E10     

Simply Connected, Adjoint and Universal Groups of Lie Type

The special linear group is the simply connected group and theprojective linear group is the adjoint group of Lie type A_n.They are distinguished sections of the (reductive) general lineargroup which certainly is of this type as well (root system).We shall characterize the general linear group as the universalgroup of type A_n. Indeed we shall introduce corresponding algebraicgroups and finite groups for each Lie type (to indecomposableroot systems). Knowledge of the universal group implies knowledgeof the related simply connected and adjoint groups; in certainrespects the universal group even appears to be better behaved(automorphisms, Schur multiplier, character table).