On the structure of some non-periodic groups whose subgroups of infinite special rank are transitively normal

A group G has a finite special rank r, if every finitely generated subgroup of G can be generated by at most r elements, and there exists a finitely generated subgroup H which has exactly r generators. This paper is devoted to genera lized radical non-Abelian groups of infinite special rank whose subgroups of infinite special rank are transitively normal.

On the structure of some non-periodic groups whose subgroups of infinite special rank are transitively normal

A group $G$ has a finite special rank $r$ if every finitely generated subgroup of $G$ is generated by at most $r$ elements and there is a finitely generated subgroup of $G$ which has exactly $r$ generators. If there is not such $r$, then we say that $G$ has infinite special rank. In this paper, we study generalized radical non-abelian groups of infinite special rank whose subgroups of infinite special rank are transitively normal.

The special rank of virtual knot groups

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications 24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The [Formula: see text]-orbifold group of a link, Math. Z. 200 (1989) 187–208], we prove that classical knots with special rank [Formula: see text] are [Formula: see text]-bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered.

TYPES OF DISCIPLINARY PENALTIES APPLICABLE TO POLICE OFFICERS

The relevance of the article is that the special content and educational goals of legal liability in general and its individual types are expressed in the variability of legal penalties. For example, the most severe criminal liability involves the application to violators, criminals, a wide range of special sanctions that correspond to the severity of the crime committed by each of them. At the same time, diversity is also inherent in disciplinary action against police officers. It is determined that reprimand and severe reprimand are similar to reprimand disciplinary sanctions and mostly embody a psychological and evaluative form of influence on a police officer. That is, their use shows that the management of a police body or unit condemns the relevant actions of the police officer. However, in contrast to the remarks, reprimands and severe reprimands have a greater disciplinary intensity, as they are applied for a longer period and, as a result, can be a punishment for more serious disciplinary offenses. At the same time, among many other penalties, reprimands and severe reprimands can be described as insignificant, because they do not cause any other additional legal consequences. It was found that the reduction of a special rank by one degree provides not only a formalized condemnation of illegal actions or inaction of a police officer, which constitute a disciplinary offense, but also limit his employment status, by: first, assigning the latter a rank that does not actually correspond the term of service of a police officer; secondly, the restriction of his financial support, because the lower special rank provides a lower salary for such a rank, which is part of the pay structure of police officers. It was emphasized that the dismissal in general and as a disciplinary sanction is expressed in the complete termination of employment between the relevant police officer and the National Police of Ukraine. In contrast to dismissal, which involves the loss of a part of a police officer's function, the termination of employment leads to the loss of a person's status as a police officer as a whole. In addition, a police officer dismissed from the police for a disciplinary offense may not be recruited by the police for three years from the date of execution of the dismissal order.