关于半局部群环

Let R be an associative ring with identity.R is said to be semilocal if R/J(R)is(semisimple)Artinian,where J(R)denotes the Jacobson radical of R.In this paper,we give necessary and sufficient conditions for the group ring RG to be semilocal,where G is a locally finite nilpotent group.     

Semisimple crossed products of groups and rings

Let K?G be a crossed product of a multiplicative group G over an associative ring K with 1 and let C(G) be the center of G. If K has no C(G)-invariant ideals, then the Jacobson radical of the center of K?G is a nil ideal. In addition, if G is a ZA-group, then K?G is semisimple if and only if K?G has no central nilpotent elements.     

INVERSE MONOIDS OF GRAPHS

. IntroductionGraph endomorphism and its regularity property have been investigated in some literatures (of. [1--41 for examples). The invertibility is a stronger algebraic property thanregUlarity in semigroup theory. It is commonly agreed that inverse semigroups are the mostpromising class of semigroups for study. In this paper we first present a combinatorial characterization of an inverse monoid of a graph (Theorem 2.3). Then using this we prove thata bipartite graph with an inverse monoi…     

Structure of Quantum Doubles of Finite Clifford Monoids

Abstract

In this paper, over a field k, we give the structure theorem of the quantum double of a finite Clifford monoid through bicrossed products and quantum doubles of groups. By this result, it is shown that the quantum double of a finite Clifford monoid is semisimple (resp. von Neumann regular) if and only if the semigroup is a finite group and the characteristic p of k does not divide the order of this group.     

Amenability and Linear Cellular Automata Over Semisimple Modules of Finite Length

Let M be a semisimple left module of finite length over a ring R and let G be an amenable group. We show that an R-linear cellular automaton τ:M^G → M^G is surjective if and only if it is pre-injective.     

Linear associative algebras with finitely many idempotents

Let A be a linear (i.e., finite-dimensional) associative algebra with unity defined over K, an algebraically closed field. Then A with respect to its multiplication is an algebraic monoid over k, denoted by A_M, and with respect to the the bracket forms a Lie algebra over K, denoted by A_L. The following theorem is established A_M is nilpotent as an algebraic monoid (equivalentlyA_L is so as a Lie algebra) if and only if the set of idempotents of A is finite if and only if all irreducible closed submonoids of codimension 1 are nilpotent.     

A Note on Prehomomorphisms

We show that if G is a free group with basis X then any map θ from X to an inverse monoid S extends to a monoid prehomomorphism ψ: G\rightarrow S. As an application we give an affirmative answer to a problem of M. Petrich. 1980 Mathematics Subject Classification: Primary 20M10. September 14, 1999     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

子群的θ-偶和群的结构

研究极大子群和2-极大子群的θ-偶对群结构的影响.设G是有限群,本文得到了:如果G的每一个极大子群M都有极大θ-偶(C,D),使MC=G且C/D是2-闭的,那么G可解;如果G的每一个2-极大子群H都有θ-偶(C,D),使C/D幂零且G=HC,那么G是幂零.     

有限幂零群通过单群扩张的整群环的正规化子性质

设G是一个有限幂零群通过单群的扩张,即G有一个幂零正规子群N,使得G/N是单群.本文证明了这样的有限群G具有正规化子性质.特别地,内可解群有正规化子性质.     

Finite groups in which every self-centralizing subgroup is nilpotent or subnormal or a TI-subgroup

Czechoslovak Mathematical Journal - Let G be a finite group. We prove that if every self-centralizing subgroup of G is nilpotent or subnormal or a TI-subgroup, then every subgroup of G is nilpotent...     

Fundamental groups,inverse schützenberger automata,and monoid presentations

This paper gives decidable conditions for when a finitely generated subgroup of a free group is the fundamental group of a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Also, generalizations are given to specific types of inverse monoids as well as to monoids which are "nearly inverse." This result has applications to computing membership for inverse monoids in a Mal'cev product of the pseudovariety of semilattices with a pseudovariety of groups.

This paper also shows that there is a bijection between strongly connected inverse automata and subgroups of a free group, generated by positive words. Hence, we also obtain that it is decidable whether a finite strongly connected inverse automaton is a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Again, we have generalizations to other types of inverse monoids and to "nearly inverse" monoids. We show that it is undecidable whether a finite strongly connected inverse automaton is a Schützenberger automaton of a monoid presentation of anE-unitary inverse monoid.     

Locally nilpotent groups satisfying the weak minimum or maximum condition for subgroups of bounded nilpotency class

We study locally nilpotent groups containing subgroups of classc, c>1, and satisfying the weak maximum condition or the weak minimum condition on c-nilpotent subgroups. It is proved that nilpotent groups of this type are minimax and periodic locally nilpotent groups of this type are Chernikov groups. It is also proved that if a group G is either nilpotent or periodic locally nilpotent and if all of its c-nilpotent subgroups are of finite rank, then G is of finite rank. If G is a non-periodic locally nilpotent group, these results, in general, are not valid.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 384–389, March, 1992.     

有限Abel群的一个特征

本文证明了有限群Ｇ是Ａｂｅｌ群当且仅当Ｇ_ｒ满足下列条件：（Ⅰ） Ｇ有一个幂自同构 ａ使得 ＣＧ（ａ）是一个初等 ＡｂｅｌＺ一群．（Ⅱ）Ｇ没有子群与２－群＜ａ,ｂ｜ａ~２~ｎ＝ｂ~２~ｍ＝１,ａ~ｂ＝ａ~（１＋２）~（ｎ－１）＞同构,其中ｎ≥３,ｎ≥ｍ．利用该结果,作者还证明若有限群Ｇ有一个幂自同构ａ使得Ｃ_Ｇ（ａ）是一个初等Ａｂｅｌ２－群,则Ｇ是幂零群     

On theta pairs for a maximal subgroup

For a maximal eubgroup M of a finite group G, a 8-pair is any pair of subgroups (C,D) of G such that (i) D?G, D≤C, (ii) - G, - M and (iii) C/D has no proper normal subgroup of G/D. A partial order may be defined on the family of 8-pairs. Let △(M) - {(C,D)|(C,D) is a maximal 8-pair and CM - G}. The purpose of this note is to prove: (1) A group G is solvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) such that C/D ie nilpctent. (2) If a group G is S₄-free, then G ia eupersolvable if and only if, for each maximal subgroup M of G, △(M) contains a 8-pair (C,D) auch that C/D is cyclic     

Orbit decompositions of orthogonal monoids and the orders of finite orthogonal monoids

In this paper we determine the \(G\times G\) orbits of both an even orthogonal monoid and an even special orthogonal monoid, where G is the unit group of the even special orthogonal monoid. We then use the orbit decompositions to compute the orders of these monoids over a finite field.     

S-半正规性与P-幂零性

In this paper, we deal mainly with the following problem: if every 2-maximal subgroup of a Sylow p-subgroup of a finite group G is S-seminormal in G, what conditions force G to be p-nilpotent? As an application of main results, some sufficient conditions for finite nilpotent groups and finite supersolvable groups are obtained.     

Inverse semigroups with idempotent-fixing automorphisms

A celebrated result of J. Thompson says that if a finite group \(G\) has a fixed-point-free automorphism of prime order, then \(G\) is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.     

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.     

Nilpotent elements in group rings

The main theorem gives necessary and sufficient conditions for the rational group algebra QG to be without (nonzero) nilpotent elements if G is a nilpotent or F·C group. For finite groups G, a characterisation of group rings RG over a commutative ring with the same property is given. As an application those nilpotent or F·C groups are characterised which have the group of units U(KG) solvable for certain fields K.This work has been supported by N.R.C. Grant No. A-5300.