Affine algebraic groups with periodic components

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.Comment: 20 page

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

Теоретичне обґрунтування застосування великогабаритної техніки під час виконання робіт із ліквідації завалів на дорогах і прилеглих територіях до об’єкта НС

. Problem statement. For dismantling of mass debris, the use of large-sized machinery with special equipment is expedient. The analysis of the performance of works in the elimination of the collapse of buildings and structures suggests that the use of large-sized machinery (bulldozers, scrapers, motor graders, etc.) reduces the time for dismantling debris on roads and supporting areas of site. At the same time, there are no justifications for their effective and safe use in the analysis of debris, there is no possibility to predict the complexity and timing of such work. For these types of work, it is necessary to improve the working equipment of large-sized machinery, in particular bulldozers. To determine the types and number of mechanization allow data concerning the destruction of facilities. The most important element of these machines is the working equipment, the perfection of which determines the efficiency of these units. One of the perspective areas for improving the working equipment of bulldozers is the use of three-section hinged and connected dumps. The aim of research. The purpose of this work is to develop a theoretical model for disassembling debris which accumulates in front of a three-section dumps. Conclusion. The approved calculated scheme and mathematical dependencies to determine the volume of elementary components allowed to propose a viable mathematical model for predicting the size of the displacement prism during the transportation of debris, both with traditional and sectional dumps, and to identify dangerous areas. Developed theoretical model fairly objectively reflects the change of the shape and volume of debris, which accumulate before the sectional dumps and can be used in the engineering forecast of this indicator and the determination of the danger zone when performing work on dismantling debris.Постановка проблемы. Для разборки массовых завалов целесообразно использование крупногабаритной техники со специальным оборудованием. Анализ выполнения работ по ликвидации последствий при обрушении зданий и сооружений свидетельствует о том, что применение крупногабаритных машин (бульдозеров, скреперов, автогрейдеров и т. п.) позволяет сократить время на разборку завалов на дорогах и прилегающих территориях объектов. В то же время отсутствуют обоснования их эффективного и безопасного применения при разборе завалов. Для таких видов работ необходимо совершенствование рабочего оборудования крупногабаритной техники, в частности, бульдозеров. Определить виды и количество средств механизации позволяют данные по характеру разрушений объектов. Наиболее важным элементом этих машин является рабочее оборудование, совершенство которого определяет эффективность агрегатов. Одним из перспективных направлений совершенствования рабочего оборудования бульдозеров является применение трехсекционных шарнирно - соединенных отвалов. Цель исследований ‑ разработка теоретической модели для разборки завалов. Вывод. Принятая расчетная схема и математические зависимости по определению объемов элементарных составляющих частей позволили предложить дееспособную математическую модель для прогноза величины призмы перемещения при транспортировке обломков.Постановка проблеми. Для розбирання масових завалів доцільне використання великогабаритної техніки зі спеціальним обладнанням. Визначити види та кількість засобів механізації дозволяють дані щодо характеру руйнувань об’єктів. Найважливіший елемент цих машин ‑ робоче обладнання, досконалість якого визначає ефективність агрегатів. Один із перспективних напрямів удосконалення робочого обладнання бульдозерів ‑ це застосування трисекційних шарнірно з’єднаних відвалів. Мета досліджень ‑ розроблення теоретичної моделі для розбирання завалів, що накопичуються перед трисекційним відвалом, з урахуванням безпеки проведення робіт. Висновок. Розроблена теоретична модель досить об’єктивно відображає закономірність зміни форми і об’єму уламків, що накопичуються перед секційним відвалом, та може використовуватись для інженерного прогнозу цього показника і визначення небезпечної зони під час  розбирання завалів

Automorphisms of prime order almost regular in the sense of rank

Several finite groups admitting automorphisms of prime order which are almost regular in the sense of rank are presented. Three theorems are presented in this context, in which the first theorem describes that if a finite nilpotent group G admits an automorphism Φ of prime order p with fixed-point sub group CG(Φ) of rank r, then G has a characteristic subgroup C such that its nilpotency class is p-bounded and the quotient group G/C has (p,r)-bounded rank. The second theorem presents that a group G contains a nilpotent periodic normal subgroup H of nilpotency class c for which the quotient group G/H has finite rank r. While, the third theorem provides that if a finite nilpotent group G of derived length d admits an automorphism of prime order p with centralizer of rank r, then the group G has a characteristic subgroup C such that its nilpotency class is p-bounded and the quotient group G/C has (p,r,d)-bounded rank

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism � of order 4 having exactly m < � fixed points, then (a) the subgroup G, �2 contains a subgroup of m-bounded index in G, �2 which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup V, �2 is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism � of order 4 having exactly m < � fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup V, �2f(m), generated by all f(m)th powers of elements in V, �