Finite p-groups with a Frobenius group of automorphisms whose kernel is a cyclic p-group

Suppose that a finite p-group G admits a Frobenius group of automorphisms FH with kernel F that is a cyclic p-group and with complement H. It is proved that if the fixed-point subgroup CG(H) of the complement is nilpotent of class c, then G has a characteristic subgroup of index bounded in terms of c, jCG(F)j, and jFj whose nilpotency class is bounded in terms of c and jHj only. Examples show that the condition of F being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms FH. It is also proved that G has a characteristic subgroup of (jCG(F)j; jFj)-bounded index whose order and rank are bounded in terms of jHj and the order and rank of CG(H), respectively, and whose exponent is bounded in terms of the exponent of CG(H)

Finite groups and Lie rings with an automorphism of order 2n2^n

Suppose that a finite group $G$ admits an automorphism $\varphi$ of order $2^n$ such that the fixed-point subgroup $C_G(\varphi ^{2^{n-1}})$ of the involution $\varphi ^{2^{n-1}}$ is nilpotent of class $c$. Let $m=|C_G(\varphi)|$ be the number of fixed points of $\varphi$. It is proved that $G$ has a characteristic soluble subgroup of derived length bounded in terms of $n,c$ whose index is bounded in terms of $m,n,c$. A similar result is also proved for Lie rings.Comment: minor corrections and addition

Metabolomics application in maternal-fetal medicine

Metabolomics in maternal-fetal medicine is still an "embryonic" science. However, there is already an increasing interest in metabolome of normal and complicated pregnancies, and neonatal outcomes. Tissues used for metabolomics interrogations of pregnant women, fetuses and newborns are amniotic fluid, blood, plasma, cord blood, placenta, urine, and vaginal secretions. All published papers highlight the strong correlation between biomarkers found in these tissues and fetal malformations, preterm delivery, premature rupture of membranes, gestational diabetes mellitus, preeclampsia, neonatal asphyxia, and hypoxic-ischemic encephalopathy. The aim of this review is to summarize and comment on original data available in relevant published works in order to emphasize the clinical potential of metabolomics in obstetrics in the immediate future

Locally finite groups containing a 22-element with Chernikov centralizer

Suppose that a locally finite group $G$ has a $2$-element $g$ with Chernikov centralizer. It is proved that if the involution in $\langle g\rangle$ has nilpotent centralizer, then $G$ has a soluble subgroup of finite index

Finite groups and Lie rings with an automorphism of order 2n

Abstract. Suppose that a finite group G admits an automorphism ϕ of order 2n such that the fixed-point subgroup CG (ϕ2n−1) of the involution ϕ2n−1 is nilpotent of class c. Let m = |CG (ϕ)| be the number of fixed points of ϕ. It is proved that G has a characteristic soluble subgroup of derived length bounded in terms of n, c whose index is bounded in terms of m, n, c. A similar result is also proved for Lie rings

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

USE OF INFORMATION TECHNOLOGIES FOR THE PREVENTION OF PROFESSIONAL BURNOUT OF TEACHERS

Professional burnout of the high school tutors and its negative impact on the educational process and the state of health of teachers, have been revealed in the article, signs, indicating the systemic nature of the problem, have been described. Strategies of overcoming such professional burnout have been considered, the preference of the problem-oriented strategy has been noted. The results of the analysis of the time spent by the teacher in the formation of assignments have been presented. The expediency of using information technologies as a tool to reduce the load on the teacher by reducing the time required to prepare for practical and laboratory classes has been substantiated. The stages of preparation for classes, that should be automated, have been highlighted