A CHARACTERIZATION OF M_π-GROUPS

Let π be a set of primes. Isaacs established the π-theory of characters, which generalizes the theory of Brauer modular characters. Motivated by Isaacs＇s work, we introduce the definition of Mπ-groups and provide a characterization of Mπ-groups.

A New Characterization of Simple K_3-groups

In this paper, We show that the simple K\-3-groups can be characterized by the orders of their maximal abelian subgroups. That is, we have Theorem Let G be a finite group and M a simple K \-3-group. Then G is isomorphic to M if and only if the set of the orders of the maximal abelian subgoups of G is the same as that of M .

Periodic Bifurcations in Descendant Trees of Finite <i>p</i>-Groups

Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.

The K_2-groups over Finaite Commutative Rings

The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.

Full-Rank Factoring of Elementary 2-Groups with Equal Size Factors

In order to answer a question motivated by constructing substitution boxes in block ciphers we will exhibit an infinite family of full-rank factorizations of elementary 2-groups into two factors having equal sizes.

On Matrix Гα-group Inverse and Гα-sharp Ordering

The Γα-group inverse and the Γα-sharp ordering in matrix set are introduced. Some characterizations and properties of the Γα-sharp ordering are given. The relations between the Γα-sharp ordering and the Γα-minus ordering are discussed.

The Unitary Equivalence of Operators and K_0-Group of Commutant Algebras

We show that two irreducible operators on H are unitari1y equivalentif and only if W*(A B)’≌M2(C), and give an answer to the open question posedby J. B. Conway (Subnormal Operators, πPitman, Advanced Publishing Program,Boston, London, Melbourne, 1981) for irreducible operator. We also show that ifT, T1 and T2 are irreducible operators with T T1≌T T2, then T1≌T2. Finally,weshow that K0 (A(D))≌Z, giving a new result on the K0-group of Banach algebras.

The Isomorphism Problem of Normalized Unit Groups of Group Algebras of a Class of Finite 2-groups

Let p be a prime and F_p be a finite field of p elements.Let F_(pG)denote the group algebra of the finite p-group G over the field F_(p)and V(F_(pG))denote the group of normalized units in F_(pG).Suppose that G and H are finite p-groups given by a central extension of the form 1→Z_(p)^(m)→G→Z_(p)×···×Z_(p)→1 and G'≌Z_(p),m≥1.Then V(F_(p)G)≌V(F_(p)H)if and only if G≌H.Balogh and Bovdi only solved the isomorphism problem when p is odd.In this paper,the case p=2 is determined.

Finite Groups All of Whose Second Maximal Subgroups Are PSC*-Groups

This paper discusses the influence of minimal subgroups on the structure of finite groups and gives the structures of finite groups all of whose second maximal subgroups are PSC*-groups.

Finite p-Groups all of Whose Subgroups of Index p^(3) are Abelian

Suppose that G is a finite p-group.If all subgroups of index p^(t)of G are abelian and at least one subgroup of index p^(t−1)of G is not abelian,then G is called an A_(t)-group.We useA0-group to denote an abelian group.From the definition,we know every finite non-abelian p-group can be regarded as an A_(t)-group for some positive integer t.A_(1)-groups and A_(2)-groups have been classified.Classifying A_(3)-groups is an old problem.In this paper,some general properties about A_(t)-groups are given.A_(3)-groups are completely classified up to isomorphism.Moreover,we determine the Frattini subgroup,the derived subgroup and the center of every A_(3)-group,and give the number of A_(1)-subgroups and the triple(μ_(0),μ_(1),μ_(2))of every A_(3)-group,whereμi denotes the number of A_(i)-subgroups of index p of A_(3)-groups.

On K_(0)-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X2 is not a sufficient condition of K0(B(X)) = 0.

Finite p-groups all of whose proper subgroups have small derived subgroups

Let G be a finite p-group.If the order of the derived subgroup of each proper subgroup of G divides pi,G is called a Di-group.In this paper,we give a characterization of all D1-groups.This is an answer to a question introduced by Berkovich.

Category X(R) and A-groups

New groups: X-groups are constructed. These groups are easy to calculateand they reflect more general properties of rings than K0-groups. As their applications, some characteristics of group rings with torsion-free K0-groups are obtained.

Finite p-Groups with Exactly One A_1-Subgroup of Given Structure of Order p^3

Abstract In this paper, we classified the finite p-groups with exactly one A1-subgroup of given structure of order p^3.

On a Generalisation of Finite T-Groups

Letσ={σ_(i)|i∈I}be some partition of all primes P and G a finite group.A subgroup H of G is said to beσ-subnormal in G if there exists a subgroup chain H=H_(0)≤H_(1)≤・・・≤Hn=G such that either H_(i−1)is normal in Hi or Hi/(H_(i−1))Hi is a finiteσj-group for some j∈I for i=1,...,n.We call a finite group G a T_(σ)-group if everyσ-subnormal subgroup is normal in G.In this paper,we analyse the structure of the T_(σ)-groups and give some characterisations of the T_(σ)-groups.

The topology of foliations formed by the generic K-orbits of a subclass of the indecomposable MD_5-groups

The paper is a continuation of the authors' works in 2007. We consider the MD_5-foliations associated to connected and simply connected MD_5-groups such that their Lie algebras have 4-dimensional commutative derived ideals. In this study, we give a topological classification of all considered MD_5-foliations. A description of these foliations by certain fibrations or suitable actions of R2 and the Connes' C*-algebras of the foliations which come from fibrations are also given.

Simple groups with orders 2~a3~b5~cp^d, 2~a3~b7~cp^d and 2~a3~b5~c7~d

This work deals with the power exponent 1rand 2r respectively of the maximal and second-maximal prime factors of the order of simple K4-group, and the classification for simple 4{5,7}K--group G (i.e. G can not be divided by 5 nor by 7 or ()Gp= 4 ), simple 5 -4K-group G (i.e. G can not divided by 5 and ()Gp=4) and simple 7-4K-group G (i.e. G can not divided by 7 and ()Gp= 4). It is derived that 1r =1, 2 and 4, and 2r is not greater than 4. All the simple 4K-groups with order 235,237abcdabcdpp and 2357abcd are obtained.

A Classification of Finite Metahamiltonian p-Groups

A finite non-abelian group G is called metahamiltonian if every subgroup of G is either abelian or normal in G.If G is non-nilpotent,then the structure of G has been determined.If G is nilpotent,then the structure of G is determined by the structure of its Sylow subgroups.However,the classification of finite metahamiltonian p-groups is an unsolved problem.In this paper,finite metahamiltonian p-groups are completely classified up to isomorphism.

Classification Theory for Inductive Limit C^＊-algebras

This paper is a survey on the recent work of the authors and their col-laborators on the Classification of Inductive Limit C*-algebras. Some examples are presented to explain several important ideas.

K_0-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an "approximate inverse" of the Riesz decomposition theorem. In the paper, we give a more precise characterization of approximate invariants of strongly irreducible operators. The main result is: For any T ∈ L(H) with connected spectrum and ε > 0, there exists a strongly irreducible operator A, such that A - T < ε, V (A (A)) =～ N, K0(A (A)) ～= Z, and A (A)/rad A (A) is commutative, where A (A) denotes the commutant of A and rad A (A) denotes the Jacobson radical of A (A). The research is inspired by the recent similarity classification technique of Cowen-Douglas operators of Jiang Chunlan.