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Let be a partition of the set of all primes, and let be a finite group. The group is said to be -primary if is a -group for some and -complete if has a Hall -subgroup for each . A subgroup of is (i) -subnormal in if it has a subgroup series such that either or is -primary for each ; (ii) modular in if (1) for all such that and (2) for all such that ; (iii) -quasinormal in if is -subnormal and modular in . Finite solvable groups in which the -quasinormality of subgroups is a transitive relation are described. Some known results are generalized. [ABSTRACT FROM AUTHOR]

Let be a finite group and let be a partition of the set of all primes . The group is -primary if is a -group for some ; while is -nilpotent if is the direct product of -primary subgroups; and is a Schmidt group if is nonnilpotent but each proper subgroup in is nilpotent. A subgroup of is -abnormal in if for all subgroups in , where , the quotient group is not -primary. We describe the structure of finite groups whose every non- -nilpotent Schmidt subgroup is -abnormal. [ABSTRACT FROM AUTHOR]

Let 퐺 be a finite group, and let 픉 be a nonempty formation. Then the intersection of the normalizers of the 픉 -residuals of all subgroups of 퐺 is called the 픉 -norm of 퐺 and is denoted by subscript 푁 픉 퐺 . A group 퐺 is called 픉 -critical if 퐺 픉 , but 푈 픉 for any proper subgroup 푈 of 퐺 . We say that a finite group 퐺 is generalized 픉 -critical if 퐺 contains a normal subgroup 푁 such that 푁 Φ 퐺 and the quotient group 퐺 푁 is 픉 -critical. In this publication, we prove the following result: If 퐺 does not belong to the nonempty hereditary formation 픉 then the 픉 -norm subscript 푁 픉 퐺 of 퐺 coincides with the intersection of the normalizers of the 픉 -residuals of all generalized 픉 -critical subgroups of 퐺 . In particular absent the norm 푁 퐺 of 퐺 coincides with the intersection of the normalizers of all cyclic subgroups of 퐺 of prime power order. [ABSTRACT FROM AUTHOR]

All the groups considered in this paper are finite, and always denotes a finite group; is a partition of the set of all primes, i.e., , where and for all . A group is said to be -primary if is a -group for some , and -solvable if every chief factor of is -primary. A set of subgroups of a group is called a complete Hall -set of if every element of the set is a Hall -subgroup for some , and contains exactly one Hall -subgroup of the group for all such that . A subgroup of a group is said to be - -subnormal in if contains a series of subgroups such that either or the group is -solvable for all . We say that a subgroup of a group is weakly - -subnormal in if contains - -subnormal subgroups and such that and . In the present paper, we study conditions under which a group is -solvable. In particular, we prove that a group is -solvable if and only if at least one of the following two conditions is satisfied: (i) has a complete Hall -set all of whose elements are weakly - -subnormal in ; (ii) in every maximal chain of subgroups of the groups , at least one of the subgroups , , or is weakly - -subnormal in . [ABSTRACT FROM AUTHOR]

We study some conditions for a group to be -solvable or meta- -nilpotent. [ABSTRACT FROM AUTHOR]

The paper presents an analysis of echocardiographic data of patients with PDA having different hemodynamic significance. Presently, the concept of “degree of hemodynamic significance” remains controversial. Criteria for determining the hemodynamic significance of PDA in term infants are not described. In our work, we used a classification of three degrees of hemodynamic significance: non-hemodynamically significant PDA – insignificant, moderately hemodynamically significant – moderately significant and large PDA – hemodynamically significant PDA. Aim – to work out an algorithm for determining the degree of PDA hemodynamic significance for prediction of its further development. The study involved children under 18 years old (80% of them ‒ children under 3 years old) who were examined in the cardiac surgery department of the SI “IGUS n.a.V.T. Zaytsev NAMS of Ukraine” during 2013- 2017 yy. Gestational age (from 37 weeks), the presence of left to right PDA shunt, absence of a ASD shunt and other congenital heart defects served as enrollment criteria for the trial. Two prognostic tables were formed with the main criteria for determining hemodynamic significance of PDA by the sum of the signs according to gradations (insignificant, moderately significant and hemodynamically significant) as an outcome of the trial. Based on the determination of the main and additional criteria of hemodynamic significance and the strength of their influence, a diagnostic algorithm is formed for a patient with PDA with the possibility of predicting a further clinical scenario. For determination of hemodynamic significance, an increase in pulmonary gradient with an information index of 1.135 and a prognostic factor of +5.90/-1.75 were dominant, while the left ventricular dilatation located from the A4C with an informational index of 1.020 and a prognostic coefficient of +3.69/-2.50 dominated in determining the degree of hemodynamic significance in the presence of secondary changes. The proposed mechanism for determining the level of hemodynamic significance of PDA for children with borderline changes will predict more precisely the development of the disease, determine the patient’s observation tactics and regulate the decision-making process of surgical intervention. [ABSTRACT FROM AUTHOR]