Professor Nikolai F. Kuzennyi was born in the village Kam'yanyi Brid in Cherkasskaya Region on November, 28, 1947. In his childhood, since a tragical accident he became blind and lost eight out of ten fingers. In spite of this, in 1970 he graduated with the highest honors from Herson Pedagogical Institute, and in 1975 was awarded his Ph.D at Mathematics Institute of the National Academy of Science of Ukraine for a thesis in group theory. In 1995 he was awarded his Doctor of Sciences Degree at Kiev University. In 2001 he became a Full Professor. Since 1996 he has been working as a Leading Researcher for Mathematics Institute of the National Academy of Science of Ukraine. He supervised six Ph.D dissertations.

Nikolai F. Kuzennyi belongs to the world famous Ukrainian Algebraic School and he is a student of Professor Sergey N. Chernikov. Being significantly influenced and stimulated by this distinguished mathematician and great man, Nikolai Kuzennyi has begun his work in algebra in the beginning of 1970th. That time, S.N. Chernikov has recently moved to Kiev from Sverdlovsk. He was invited to chair the Department of Algebra of the Mathematics Institute of the National Academy of Sciences in Ukraine. Following his traditions, this outstanding scientist and great teacher formed the core of his famous Kiev Group Theoretical Collective. Young Nikolai Kuzennyi was one of the first members of this Collective where he became a close friend with many other young mathematicians. This warm and fruitful friendship has been continuing for many many years and led to many collaborative works. The influence, the taste and the vision of S.N. Chernikov became one of the most influenced factors that determined research interests of Nikolai Kuzennyi.

N.F. Kuzennyi has achieved his main results in group theory. Employing a very productive approach of studying groups by given properties of their subgroups he focuses on study of finite and infinite groups, in particularly, on generalizing on infinite groups some important results of finite group theory. The efficiency of this approach was well justified by a variety of established great results and newly described classes of groups. For instance, we can mention infinite groups with distinct finiteness conditions, generalized soluble groups, generalized nilpotent groups, and many other classes of groups that have been introduced and described in this passing.

Mathematical talent of N. F. Kuzennyi, his consistent hard work, and his enthusiasm allowed him to make significant contribution to infinite group theory by obtaining many interesting results and describing new classes of groups. Let us briefly mention some of them.

Chronologically, the first research of N.F. Kuzennyi focused on the finite groups with some systems of dispersive subgroups. N.F. Kuzennyi studied these groups in collaboration with S.S. Levishenko. Remind that a finite group $G$ is called \emph{dispersive} if it has a normal series $\{G_i | 1\leqslant i\leqslant n\}$, whose every factor $G_i/G_{i-1}$ is a primary $p_i$-group which is isomorphic to the Sylow $p_i$-subgroup of $G$. In other words, these groups can be constructed from normal primary blocks. The dispersive groups are natural generalizations of supersoluble groups. In that time, the minimal non-supersolublae groups and their formation generalizations have been studied intensively. In a series of their works, N.F. Kyzennyi and S.S. Levishenko have been studied finite soluble minimal non - dispersive groups. These results have been highly praised by experts, in particular, by L.A. Shemetkov. Note that all these investigations are very rich on details and require sophisticated and intricate computations.

In collaboration with L.A. Kurdachenko, V.V Pylaev and N.N. Semko, N.F. Kuzennyi made a significant contribution in study of groups with the dense system of some normal and generalized normal subgroups.

Other central subjects of interests of N.F. Kuzennyi are pronormal and abnormal subgroups. These subgroups appeared in the process of investigation of some important subgroups of finite (soluble) groups such as Sylow subgroups, Hall subgroups, system normalizers, and Carter subgroups. Let $H$ be a subgroup of a group $G$. We recall that a subgroup H is pronormal in $G$ if for each element $g\in G$, $H$ and $H^g$ are conjugate in $\langle H,H^g\rangle$. If $g\in\langle H,H^g\rangle$ we call $H$ \emph{to be abnormal} $G$. Pronormal subgroups have been introduced by P. Hall in his lectures in Cambridge, while the term an abnormal subgroup belongs to R. Carter. These subgroups and their generalizations have shown to be very useful in the finite group theory. It appears to be logical to employ such fruitful concepts to infinite groups. It is noteworthy to admit that the transferring results from finite groups to infinite groups is a very complicated process that is usually driving at significant extension of the volume of the classes of groups under research. Thus in some classes of infinite groups, pronormal subgroups gain such properties that they cannot posses in the finite case. For example, it is well-known that every finite $p$-group has no proper abnormal subgroups. Nevertheless, A.Yu. Olshanskii has constructed a series of impressive examples of infinite finitely generated $p$-groups saturated with abnormal subgroups. In this class, every proper non-identity subgroup of $G$ is maximal, and being non-normal, is abnormal. In general, we quite frequently observe that the situation in infinite groups is significantly different from the situation in the corresponding finite case. It is important to admit that the first results on infinite groups saturated with pronormal subgroups have been obtained by N.F. Kuzennyi in the collaboration with I.Ya.~Subbotin in 1986. For instance, they completely described locally soluble (and in the periodic case locally graded) groups in which all subgroups pronormal, locally soluble groups in which all infinite, all primary, all abelian, all cyclic subgroups pronormal. In this passing, they obtained many interesting results and constructed some sophisticated examples. These investigations established a base for the study of infinite groups with systems of pronormal and abnormal subgroups that successfully has been continuing nowadays by many well known mathematicians around the glob.

Groups saturated with pronormal subgroups are very tightly connected to well-described groups with the transitivity of normality (the $T$-groups). Finite $T$-groups have been studied by many authors. D.J.S. Robinson has obtained the main results on infinite $T$-groups. Generalizing it, N.F. Kuzennyi obtained some interesting results on groups with transitivity condition on abelian normal subgroups.

Together with N.N. Semko, N.F. Kuzennyi has written a cycle of articles dedicated to metahamiltonian groups, that is the groups whose nonabelian subgroups are normal. These groups as a natural generalization of hamiltonian groups have been introduced by N.F. Sesekin and G.M. Romalis. Some their properties have been established by G.M, Romalis, N.F. Sesekin, N.S. Chernikov, A.A. Makhnev and others. N.F. Kuzennyi and N.N. Semko were able to obtain very detailed, constructive and intricate descriptions of all metahamiltonian groups.

Of course, this is a very brief and general description of N.F Kuzennyi research achievements. His results have been published in his two books and numerous journal articles. Unfortunately, we were not able to mention many of his important results in the current brief article.

Nikolai F. Kuzennyi is a very energetic and enthusiastic mathematician who has a lot to come. We would like to congratulate him with his sixties birthday and to wish him great health and many new achievements.

 

V. V. Kirichenko, L. A. Kurdachenko, N. N. Semko, I. Ya. Subbotin