Frobenius groups of automorphisms and their fixed points

Suppose that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that the fixed-point subgroup of $F$ is trivial: $C_G(F)=1$. In this situation various properties of $G$ are shown to be close to the corresponding properties of $C_G(H)$. By using Clifford's theorem it is proved that the order $|G|$ is bounded in terms of $|H|$ and $|C_G(H)|$, the rank of $G$ is bounded in terms of $|H|$ and the rank of $C_G(H)$, and that $G$ is nilpotent if $C_G(H)$ is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of $G$ in the case of metacyclic $FH$. The exponent of $G$ is bounded in terms of $|FH|$ and the exponent of $C_G(H)$ by using Lazard's Lie algebra associated with the Jennings--Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of $G$ is bounded in terms of $|H|$ and the nilpotency class of $C_G(H)$ by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms.Comment: 31 page

Reconstitution paléogéographique des dynamiques paysagères durant l'Holocène autour de Xanthos et Létôon dans l'ancienne Lycie (Turquie) : premiers résultats

Selon Strabon, Xanthos et Létôon étaient une cité et un sanctuaire majeurs de Lycie à la période hellénistique (ive-iie siècles av. J.-C.). L’archéologue qui étudie ces deux sites ne peut faire l’hypothèse que le paysage dans lequel ils se situent est resté immobile depuis l’arrivée des premiers habitants au viie siècle av. J.-C. Comprendre les sociétés qui ont occupé cet espace, ainsi que leurs relations avec l’environnement, implique de reconstituer les paysages qu’elles ont habités et aménagés, en s’appuyant sur la compréhension des conditions environnementales et des dynamiques géomorphologiques qui s’y sont exercées. Dans cet article, les premiers résultats issus de l’analyse de séquences carottées et de profils géophysiques sont interprétés et confrontés aux sources historiques, archéologiques et littéraires, et une première reconstitution de l’évolution des paysages est proposée. Les enregistrements sédimentaires indiquent qu’une baie marine a été progressivement fermée par le développement d’une flèche littorale à l’arrière de laquelle s’établissait un système lagunaire. C’est dans ce contexte paysager que se sont développés les deux sites. Face au Létôon un ancien chenal a été identifié, qui permettait peut-être une relation directe, par le fleuve, du sanctuaire avec la mer en aval, et avec la cité de Xanthos en amont. L’étude de la morphologie de la vallée dans son ensemble révèle une forte mobilité des paysages, qui a contraint les hommes à choisir des sites protégés dans des lieux stratégiques pour y installer leurs cités, leurs sanctuaires et leurs voies de communication. La mobilité des paysages a aussi été exploitée de manière symbolique : installé dans un environnement hostile de marécages en plaine alluviale, le sanctuaire de Létôon se pose comme un défi aux lois de la nature

Guide pour les ingénieurs : environnement; comment et où obtenir des informations pour la protection de l'environnement dans un processus industriel

Version anglaise dans la bibliothèque sous le titre: Guide for engineers: environment ..

Different times? Archaeological and environmental data from intra-site and off-sites sequences.: An introduction

International audienc

Different times ? Archaeological and environmental data from intra-site and off-sites sequences. An introduction

International audienceThe aim of this session is to bring together scholars from the fields of archaeology (from Prehistory to the Middle Ages) and natural sciences (geography-geomorphology mostly), in order to discuss the ”thorny” question of reading time in sedimentary sequences inside and beyond settlements, and its use for a joint reconstruction of past events. It is not always easy indeed to interpret correctly the different lines of evidence and establish solid links between the two records, in order to formulate well-built proposals about their relation — in terms of time (contemporaneous, prior, posterior) or causality (related, simply coeval, irrelevant). We wish to put forward the following elements:a)The necessity to take into account for our interpretations not only the density or the precision of time measures (diagnostic artefacts, absolute dates – especially radiocarbon) but also the nature of the dated facts and the adequacy of the elements used for dating with respect to the dated contexts and to the questions underlying the dating. A charcoal found in a house destruction layer and another found in a colluvium that reworked this same layer do not provide the same information in terms of temporal framing — and this, independent of the short- or long-lived character of the charred wood plant-species. The discrepancies generated by such contextual differences are essential for our understanding of the succession or amplitude of past events. Speaking of “time reading” instead of ”time measurement” is a way to draw attention on this important aspect.b)The necessity of a closer dialogue between specialists from the two disciplines that would overcome the separation between intra-site and off-site records, the former being considered as the “ground” of archaeologists, the latter being that of geomorphologists. Although convenient in practical terms and justified to a certain point, this separation minimizes indeed the interaction between the two spaces over the short, middle and long terms (people impacting on their environment but also living with it or “bringing it” at home), and neglects the similarities in the approaches, or the methodological tools, used here and there (core-drills applied inside settlements, study of artefacts’ distribution in areas off, etc.). Ultimately, comparisons between intra-site (i.e. basically anthropogenic) sequences with neighbouring off-site (i.e. basically environmental) sequences need to take into account, in addition to distance distortion, the effects of time delay observed, or estimated, in the recording of mutual impacts.This session is proposed as part of the activities of the Working Group ‘Environmental and Social changes in the Past’, animated in the frame of the Cluster of Excellence ”Dynamite” (Territorial and Spatial Dynamics) of the University Paris 1-Sorbonne. Its topic illustrates the kind of questions asked by the Group’s members and anticipates the kind of answers that could be given in return. Papers or posters focusing on this topic (time measurement/time reading, dialogue archaeological stratigraphy/environmental sequence, building of historical scenarios) are welcome in our session

Post-LGM evolution of environments in the Eregli area: first results from geological indicators

International audienc

The Öküzini marshes : a new upper Pleistocene record on the Anatolian mediterranean coast

International audienc