Special issue
dedicated to Professor I.Ya. Subbotin on the occasion of his 60-th birthday

Professor Igor Ya. Subbotin was born in Kiev on March 25, 1950. In 1967, Igor Subbotin became a student of an algebra course lectured by Sergey N. Chernikov who is known not only as a great mathematician and one of the founders of infinite group theory, but also as a very influential and caring teacher. Among his numerous students, we can list such prominent mathematicians as V.M. Glushkov, M.I. Kargapolov, V.S. Charin, V.P. Shunkov, Yu.M. Gorchakov, D.I. Zaitsev, L.A. Kurdachenko, and others. S.N. Chernikov's brilliant lectures, his great personality and his caring teaching style deeply impressed young Igor and determined his choice of the future research area. S.N. Chernikov then had just moved to Kiev from Sverdlovsk to chair the Department of Algebra at the Mathematics Institute of the National Academy of Sciences in Ukraine. He also taught algebra courses at Kiev Pedagogical Institute. Soon, I. Subbotin became one of the first members of the newly established Kiev Group Theory Seminar at Institute of Mathematics of National Academy of Sciences of Ukraine. At this seminar, Igor Subbotin made close friends with many other young mathematicians. This warm and fruitful friendship has been continuing for decades and led to many collaborative works.

In 1972, Igor Subbotin graduated with highest honors from Kiev Pedagogical Institute (now Kiev National Pedagogical University), and in 1978 he was awarded his PhD at Mathematics Institute of the National Academy of Science of Ukraine for a thesis in group theory. In 1980, he became an Associate Professor of the Department of Higher Mathematics at Kiev Polytechnic Institute (now Kiev National Technology University). In 1993, he began his teaching and research career at National University, California, USA, where he has been working since that time. He is now a Full Professor and Lead Faculty for mathematics programs at that school. I.Ya. Subbotin has always highly valued the great support of his research and teaching activities provided by the National University.

Igor Subbotin is very active in research. His list of publications includes more than 80 articles in algebra published in major mathematics journals in many countries including Ukraine, USA, Great Britain, Italy, Spain, Russia, China, Brazil, Hungary, Czech Republic and Turkey. He also authored more than 30 articles in mathematics education dedicated mostly to theoretical base of some topics of high school and college mathematics.

In early seventies, I.Ya. Subbotin began to investigate some aspects of the normal structure of finite and infinite groups. This area of research takes its roots in works of R. Dedekind, O.Yu. Schmidt, S.N. Chernikov, R. Baer and O. Taussky. I.Ya. Subbotin focused on studying groups based on given properties of their normal and related to normal subgroups. The efficiency of this approach was well justified by achieving a variety of important results and newly described classes of groups that have been introduced and described in this passing. I.Ya. Subbotin made the first steps in research by studying some generalizations of well known $T$-groups, i.e. groups with transitivity of normality. Generalizing it, I.Ya. Subbotin studied finite and infinite groups G in which every subgroups of $[G, G]$ is $G$-invariant; quasicentral extensions and quasicentral products of groups. He also fully described in detail all finite and infinite groups with quaisecentralizer condition on normal subgroups. Together with N.F. Kuzennyui he obtained a description of some subclasses of groups with quaisicentralizer condition on normal non-abelian subgroups.

This research naturally led I.Ya. Subbotin to investigation of infinite groups satisfying some conditions related to normality, such as pronormality, abnormality and their generalization. Pronormal and abnormal subgroups have been introduced by P. Hall. These subgroups play a key role in investigation of finite (soluble) groups. It appears to be logical to employ such fruitful concepts to infinite groups. However, the transferring results from finite groups to infinite groups is a very complicated process. Thus, in some classes of infinite groups, pronormal subgroups gain some properties they cannot posses in the finite case. In contrast to the finite case, where a finite $p$-group has no proper abnormal subgroups, A.Yu. Olshanskii has constructed a series of impressive examples of infinite finitely generated $p$-groups whose proper subgroups are abnormal. In general, we observe quite frequently 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 I.Ya. Subbotin in collaboration with N.F. Kuzennyi in 1980--1990. 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, including examples of non-splitting extensions of groups based on the well-known techniques developed by Yu.M. Gorchakov and L.G. Kovach, B.H. Neumann and H. de Vries for their famous constructions.

Abnormal, pronormal, and Carter subgroups play a key role in investigation of arrangement of subgroups in groups. Conditions related to the subgroup arrangement allowed algebraists to introduce and describe many important classes of groups. The roots of such investigations lie in the works of P. Hall, R. Carter, J. Rose, and Z. Borevich. Numerous interesting results in this area have been obtained lately by many authors. In collaboration with L.A. Kurdachenko, N.F. Kuzennyi, J. Otal, G. Vincenzi, A. Russo and others, I.Ya. Subbotin investigated some important properties of pronormal, Carter, abnormal, and contranormal subgroups of infinite groups and their influence on the groups structure. Some new criteria of local nilpotency and nilpotency in infinite groups related to these subgroups and natural generalization of the notion of a Carter subgroup for the case of infinite groups have been established on this way. Also, some important classes of groups saturated with above-mentioned subgroups and groups with transitivity of these subgroups properties have been described by these authors.

One of the intensively developing areas of investigation in group theory is the study of influence of the factor-groups on the structure of a group. The book of L.A. Kurdachenko, J. Otal, I.Ya. Subbotin, \textit{Groups with Prescribed Quotient Groups and Associated module theory}, WORLD SCIENTIFIC: New Jersey, London, Singapore, Hong Kong -- 2002 collected and presented the main results of these studies from a single general point of view. In this text, the authors demonstrated quite clearly the capability of the module theory technique in solving group theory problems. In general, modules over group rings were efficiently used in many group theory studies. Together with L.A. Kurdachenko and J. Otal, I.Ya. Subbotin investigated some interesting types of modules close to artinian and noetherian modules. These results, as well as many more, have been presented in the book L.A. Kurdachenko, J. Otal and I.Ya. Subbotin, \textit{Artinian Modules Over Group Rings}, Frontiers in Mathematics. BIRKH\"{A}USER: Basel -- 2007.

Theory of finite dimensional linear groups is one of the most developed algebraic theories. However, the theory of infinite dimensional linear groups is only at its initial level. The concept of groups of finite central dimension, introduced by L.A. Kurdachenko, opens new opportunities for observing and describing some naturally defined classes of infinite dimensional linear groups that are close to ordinary finite-dimensional groups. Based on this approach, quite interesting results on infinite-dimensional linear groups have been obtained lately by L.A. Kurdachenko, M. Evans, M. Dixon, I.Ya. Subbotin, J. Otal, J.M. Munos-Escolano, O.Yu. Dashkova, and N.N. Semko. I.Ya. Subbotin also takes part in the current investigation of infinite dimensional linear groups related to the concept of $G$-invariancy.

Of course, all that is a very short and general description of I. Subbotin's research achievements, and we were not able to mention many of his important results in this brief article.

I.Ya. Subbotin is known as a great teacher. His lectures are very popular among his students. He received the National University Presidents Professoriate Award three times.

I.Ya. Subbotin is a very energetic and enthusiastic mathematician with more achievements to come.

We warmly congratulate him on his 60th birthday and wish him strong health and many successful years of research and teaching.

A. Ballester-Bolinches, N.N. Bilotskii,
M.R. Dixon, R.I. Grigorchuk,
V.V. Kirichenko, L.A. Kurdachenko,
N.F. Kuzennyj, J. Otal, N.N. Semko,
P. Shumyatsky, V.I. Sushchansky,
E. Zelmanov