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

Quadratic equations over free groups are NP-complete

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete

A lattice in more than two Kac--Moody groups is arithmetic

Let $\Gamma$ be an irreducible lattice in a product of n infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic group over a local field and $\Gamma$ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n is at least 2: either $\Gamma$ is an S-arithmetic (hence linear) group, or it is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.Comment: Subsection 2.B was modified and an example was added ther

Клиническая эффективность и безопасность применения иммунной плазмы реконвалесцентов для лечения COVID-19

Aim: to evaluate the efficacy and safety of convalescent plasma therapy for patients with severe SARS-CoV-2 infection.Materials and methods: the study included 64 patients with laboratory-confirmed severe new coronavirus infection. The control group consisted of 58 patients who, in addition to standard therapy, received a transfusion of plasma from donors who had recovered from COVID-19. The effectiveness of immune plasma was assessed by the duration of fever, the level of oxygen (SpO2 %) in dynamics, the detection of SARSCoV-2 RNA in nasopharyngeal and oropharyngeal swabs using PCR method in dynamics, as well as by the dynamics of blood tests results. Adverse events (any medically adverse events that occurred after immune plasma transfusion) were recorded as safety criteria.Results: patients who received convalescent plasma, showed a significantly shorter period of SARS-CoV-2 replication compared with the control group. The use of immune plasma did not have a statistically significant effect on the duration of the fever, as well as the dynamics of blood oxygenation. Also, there were no significant differences compared with the control group when assessing blood tests parameters.Conclusion: The use of COVID-19 convalescent plasma to treat severe COVID-19 did not show significant clinical effect but reduced the period of viral replication. It also showed no unexpected or serious adverse events. Цель. Оценка эффективности и безопасности применения иммунной плазмы реконвалесцентов для лечения COVID-19 у пациентов с тяжелой степенью тяжести.Материалы и методы. В исследование были включены 64 пациента с подтвержденной новой коронавирусной инфекцией тяжёлой степени тяжести, которым, помимо стандартной терапии, проводили трансфузию иммунной плазмы реконвалесцентов новой коронавирусной инфекции. Контрольную группу составили 58 пациентов, получавших базисную терапию. Эффективность терапии иммунной плазмы оценивалась по длительности лихорадки, уровню сатурации (SpO2 %), выявлению РНК SARS-CoV-2 методом ПЦР в отделяемом со слизистой рото-/носоглотки в динамике, а также по изменениям общеклинических и биохимических показателей крови. В качестве критериев безопасности регистрировались нежелательные явления (любые неблагоприятные с медицинской точки зрения события, возникшие после трансфузии иммунной плазмы).Результаты. У пациентов, получавших иммунную плазму реконвалесцентов, отмечался достоверно более короткий период вирусной репликации SARS-CoV-2 по сравнению с контрольной группой. На продолжительность лихорадки, а также динамику насыщения кислородом крови применение иммунной плазмы статистически значимого влияния не оказывало. Также не отмечалось достоверных различий по сравнению с контрольной группой при оценке общеклинических, биохимических показателей крови.Заключение. Терапия пациентов с тяжёлым течением COVID-19 иммунной плазмой реконвалесцентов, перенёсших инфекцию, вызванную SARS-CoV-2, достоверно не оказывала значимого клинического эффекта, однако сокращала период репликации вируса. Трансфузия иммунной плазмы также не вызывала серьёзных нежелательных явлений