A Generalization of <Emphasis Type="Italic">T</Emphasis>-Groups

This paper identifies a certain class of locally supersoluble groups (called soluble hall-T groups) which contains the soluble T-groups as well as the nilpotent groups. The main result states that the product of a normal soluble hall-T subgroup and a subnormal locally supersoluble subgroup is always locally supersoluble.AMS Subject Classification (1991): 20E25, 20F16, 20F19     

关于SQ-补子群

假定Fitting子群F(G)或广义Fitting子群F^*(G)的某些子群在G中SQ-补来研究包含超可解群的饱和群系s,这里G∈s.一些已知结果被推广.     

两个正规p-超可解子群的积

本文研究了两个正规的p-超可解子群的积构成的极小非P-超可解群的结构的问题.利用有限群论和群类论的一些基本方法,获得了两个正规的p-超可解子群的积仍为P-超可解群的一些充分条件,并推广了文[1]中关于超可解群的情况.     

New characterizations of finite supersoluble groups

Let A be a subgroup of a group G and X a nonempty subset of G.A is called an X- semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T_1 of T there exists an clement x∈X such that AT_i~x=T_i~xA.On the basis of this concept we obtain some new characterizations of finite supersoluble groups.     

The influence of <Emphasis Type="Italic">X</Emphasis>-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be 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 investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771172, 10771180)     

The influence of X-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be 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 investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

On coherence of graph products of groups and Coxeter groups

Graph products of groups and Coxeter groups are defined via vertex-edge-labeled graphs. We show that if the graph has a special shape, then the corresponding group is coherent, i.e. every finitely generated subgroup is finitely presented.     

Subgroup separability of graphs of abelian groups

In the present paper we give necessary and sufficient conditions for the subgroup separability of the fundamental group of a finite graph of groups with finitely generated abelian vertex groups.

     

Subgroup Separability and Linearity of Certain Graphs of Groups

We show that if the fundamental group of a finite graph of groups with finitely generated Abelian vertex groups is subgroup separable, then it is linear.     

Fibre products, non-positive curvature, and decision problems

We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of . Received: October 7, 1999.     

Free Subgroups of Groups with Nontrivial Floyd Boundary

Abstract

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205–218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group.     

Palindromic Width of Finitely Generated Solvable Groups

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step ≥3, we prove that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of G is finite. We also prove that the palindromic width of ??? with respect to the set of standard generators is 3.     

Average-case complexity and decision problems in group theory

We investigate the average-case complexity of decision problems for finitely generated groups, in particular, the word and membership problems. Using our recent results on “generic-case complexity”, we show that if a finitely generated group G has word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then the average-case complexity of the word problem of G is linear time, uniformly with respect to the collection of all length-invariant measures on G. This results applies to many of the groups usually studied in geometric group theory: for example, all braid groups B_n, all groups of hyperbolic knots, many Coxeter groups and all Artin groups of extra-large type.     

Locally Finite-Indicable Groups

A group is locally ?-indicable if every finitely generated subgroup has a nontrivial homomorphism onto a nontrivial ?-group. If ? is a quasi-variety, then the class L(?) of locally ?-indicable groups coincides with the class N(?) of groups which have normal systems with factors in ?. It is not known if ? must be a quasi-variety in order for the equality L(?) = N(?) to hold. We show here that if ? is the class of all finite groups, which is the union of an ascending sequence of quasi-varieties, then L(?) ≠ N(?). Examples of finitely generated groups in L(?)\ N(?) are also constructed.     

Finite groups with some given systems of Xm‐semipermutable subgroups

Let X be a nonempty subset of a group G. We call a subgroup A of G an X_m‐semipermutable subgroup of G if A has a minimal supplement T in G such that for every maximal subgroup M of any Hall subgroup T₁ of T there exists an element x ∈ X such that AM^x = M^xA. In this paper, we study the structure of finite groups with some given systems of X_m‐semipermutable subgroups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

G-covering systems of subgroups for the class of supersoluble groups

Let ? be a class of groups. Given a group G, assign to G some set of its subgroups Σ = Σ(G). We say that Σ is a G-covering system of subgroups for ? (or, in other words, an ?-covering system of subgroups in G) if G ∈ ? wherever either Σ = ? or Σ ≠ ? and every subgroup in Σ belongs to ?. In this paper, we provide some nontrivial sets of subgroups of a finite group G which are G-covering subgroup systems for the class of supersoluble groups. These are the generalizations of some recent results, such as in [1–3].     

On the Prime Radical of PI-Representable Groups

The notion of PI-representable groups is introduced; these are subgroups of invertible elements of a PI-algebra over a field. It is shown that a PI-representable group has a largest locally solvable normal subgroup, and this subgroup coincides with the prime radical of the group. The prime radical of a finitely generated PI-representable group is solvable. The class of PI-representable groups is a generalization of the class of linear groups because in the groups of the former class the largest locally solvable normal subgroup can be not solvable.     

Hall’s Theorem for Limit Groups

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela. Received: May 2006 Revision: May 2007 Accepted: May 2007     

Coxeter Groups, 2-Completion, Perimeter Reduction and Subgroup Separability

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have a uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index.