Isomorphisms between lattices of almost normal subgroups

A subgroupH of a groupG is said to bealmost normal inG if it has only finitely many conjugates inG. The setan(G) of almost normal subgroups ofG is a sublattice of the lattice of all subgroups ofG. Isomorphisms between lattices of almost normal subgroups ofFC-soluble groups are considered in this paper. In particular, properties of images of normal subgroups under such an isomorphism are investigated.     

Amenable subgroups of semi-simple groups and proximal flows

We present a classification of maximal amenable subgroups of a semi-simple groupG. The result is that modulo a technical connectivity condition, there are precisely 2′ conjugacy classes of such subgroups ofG and we shall describe them explicitly. Herel is the split rank of the groupG. These groups are the isotropy groups of the action ofG on the Satake-Furstenberg compactification of the associated symmetric space and our results give necessary and sufficient conditions for a subgroup to have a fixed point in this compactification. We also study the action ofG on the set of all measures on its maximal boundary. One consequence of this is a proof that the algebraic hull of an amenable subgroup of a linear group is amenable. Supported in part by NSF Grant No. MPS-74-19876.     

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp ² possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.     

On a graph associated to invariant conjugacy classes of finite groups

LetA andG be finite groups of coprime orders such thatA acts by automorphisms onG. We define theA-invariant conjugacy class graph ofG to be the graph having as vertices the noncentralA-invariant conjugacy classes ofG, and two vertices are connected by an edge if their cardinalities are not coprime. We prove that when the graph is disconnected thenG is solvable.     

Representing Groups by Colourings of Graphs

The topic of this paper is representing groups by edge-coloured graphs. Every edge-coloured graph determines a group of graph automorphisms which preserve the colours of the edges. An edge colouring of a graph G is called a perfect one iff every colour class is a perfect matching in G. We prove that every group H and all of its subgroups can be represented (up to isomorphism) by a group of colour preserving automorphisms related to some perfect colouring of the same graph.     

Prime ideals in crossed products of finite groups

LetR * G be a crossed product of the finite groupG over the ringR. In this paper we discuss the relationship between the prime ideals ofR*G and theG-prime ideals ofR. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime idealsP ofR*G such thatP∩R is a fixedG-prime ideal ofR. As an application, we prove that ifG is a finite group of automorphisms ofR, then the prime (primitive) ranks ofR and of the fixed ringR ^G are equal provided •G•⁻∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group.     

The groups of the semigroup of compact subsets of a topological group

The compact subsets of a topological groupG form a semigroup,S(G), when multiplication is defined by set product. This semigroup is a topological semigroup when given the Vietoris topology. It would be expected that the subgroups ofS(G) should in some way be related to the groupG. This is the case. It is shown that the subgroups ofS(G) are both algebraically and topologically exactly the groups obtained as quotients of certain subgroups ofG. One consequence of this is that every subgroup ofS(G) is a topological group. Conditions are also given for these subgroups to be open or closed. Green's relations inS(G) have a particularly nice formulation. As a result, the relationsD andJ are equal inS(G). Moreover, the Schützenberger group of aD-class is a topological group that is topologically isomorphic to a quotient of certain subgroups ofG.     

ZS-metacyclic groups and their endomorphism near-rings

ItG is a group written additively, the inner automorphisms and the endomorphisms additively generate near-ringsI(G) andE(G) respectively. IfI(G)=E(G), i.e., if every endomorphism is a sum of inner automorphisms, we callG anI-E group. In this paper we describe a class ofI-E groups which includes two of the four known classes ofI-E groups and which contains infinitely many other examples. The order ofI(G) is obtained and its radical determined.Supported in part by the Natural Sciences and Engineering Research Council of Canada.     

A comparison of length definitions for maps of modules over local rings

LetM be a finitely generated free module over a local ring. An automorphism ofM can be written as a product of automorphisms that are elements of a given generating groupG of the set of all automorphisms ofM. The minimal number of elements required for such a product is called the length of. We study two decompositions using similar generating groups and compare the two resulting lengths.     

On galois descent for Hochschild and cyclic homology

LetG be a finite group acting by automorphisms on an algebraS over some commutative ringk. We show that if the action ofG restricted to the center ofS is Galois in the sense of [C-H-R], thenHH _*(S ^G)≊HH _* (S) ^G. An analogous result holds for cyclic homology, provided the order ofG is invertible ink. The author was supported in part by a grant from the NSF.     

Groups whose subgroups have small automizers

IfH is a subgroup of a groupG, theautomizer ofH inG is the group of all automorphisms ofH induced by elements of its normalizerN _G (H). the subgroupH is said to havesmall automizer ifAut _G (H)=Inn(H), i.e. ifN _G (H)=HC _G (H). This article is devoted to the study of groups for which many subgroups have small automizer. In Memoriam Valeria Fedri R. Brandl wishes to express his sincerest thanks for the warm hospitality offered by the Department of Mathematics of the University of Napoli “Federico II” for the time of writing this paper.     

The Bogomolov multiplier of rigid finite groups

The Bogomolov multiplier of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. This invariant of G plays an important role in birational geometry of quotient spaces V/G. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of G in the sense that it has no outer class-preserving automorphisms.     

A lattice theoretic characterization of prefrattini subgroups

Consider an interval [H,G] in the lattice of subgroups of a finite soluble groupG. We define a certain set of subgroups in the lattice [H,G], and prove that they are conjugate inG. ForH=1 one gets the prefrattini subgroups ofG.     

Subgroups generated by root-elements of groups generated byk-root subgroups

LetG be a Chevalley group over the fieldk and the class of long root subgroups ofG or an abstract group generated by a class of k-root subgroups. Then elements ofA, A are called root-elements. This paper is concerned with the question, whether a subgroupU ofG generated by root-elements is already generated byl-root subgroups,l a subfield ofk. Especially if it is shown that this is the case, whenU is a fixpoint subgroup under some set of automorphisms.     

On density properties of certain subgroups with boundedness conditions

LetG be a locally compact group acting on a topological space. Here we define some boundedness conditions for the action. For a nondiscrete locally compact vector spaceV andgG L (V), layering structures forV and the projective spaceP (V) ofV are obtained. From the layering structures, we derive then density properties of subgroups ofG with boundedness conditions. We generalize the Borel density theorem and Prasad's theorem on automorphisms of algebraic semi-simple groups. Some new results onp-adic groups are added.Partially supported by N. S. F. Grant 7702168.     

Weakly power automorphisms of groups

An automorphism α of a group G is called a weakly power automorphism if it maps every non-periodic subgroup of G onto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic group G of derived length at most 3 are power automorphisms, i.e. they fix all subgroups of G. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism.     

Les beaux automorphismes

Assume that the class of partial automorphisms of the monster model of a complete theory has the amalgamation property. The beautiful automorphisms are the automorphisms of models ofT which: 1. are strong, i.e. leave the algebraic closure (inT ^eq) of the empty set pointwise fixed, 2. are obtained by the Fraïsse construction using the amalgamation property that we have just mentioned. We show that all the beautiful automorphisms have the same theory (in the language ofT plus one unary function symbol for the automorphism), and we study this theory. In particular, we examine the question of the saturation of the beautiful automorphisms. We also prove that in some cases (in particular if the theory is -stable andG-trivial), almost all (in the sense of Baire categoricity) automorphisms of the saturated countable model are beautiful and conjugate.     

On p-nilpotency of finite groups with some subgroups π-quasinormally embedded

Summary A subgroup H of a group G is said to be π-quasinormal in G if it permutes with every Sylow subgroup of G, and H is said to be π-quasinormally embedded in G if for each prime dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some π-quasinormal subgroups of G. We characterize p-nilpotentcy of finite groups with the assumption that some maximal subgroups, 2-maximal subgroups, minimal subgroups and 2-minimal subgroups are π-quasinormally embedded, respectively.     

<Emphasis Type="Italic">G</Emphasis>-covering subgroup systems for the classes of supersoluble and nilpotent groups

LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups. Research of the first author is supported by the NNSF of China (Grant No. 10171086) and QLGCF of Jiangsu Province and a Croucher Fellowship of Hong Kong. Research of the second author is partially supported by a UGC (HK) grant #2060176 (2001/2002).     

Invariant class sizes and solvability of finite groups under coprime action

Suppose that A and G are finite groups of relatively prime orders such that A acts on G via automorphisms, and then, A acts on the set of conjugacy classes of G. We study several arithmetical properties on the sizes of certain classes which are left fixed by A and how they reflect on the A‐invariant structure and may imply the solvability of G.