Special issue


This special issue of the journal is dedicated to the 60th birthday of an outstanding Ukrainian mathematician, Professor Leonid A. Kurdachenko. Leonid Kurdachenko was born on October 22, 1949 in Dnipropetrovsk. In 1971 he graduated with the high honors from the Mechanics and Mathematics School of Dnipropetrovsk University. In 1969 Leonid Kurdachenko met Sergey N. Chernikov and in 1971 he became his postgraduate student. S. N. Chernikov 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. It is worthy to mention that among his numerous students we can list such prominent mathematicians as V. M. Glushkov, M. I. Kargapolov, V. S. Charin, V. P. Shunkov, Yu. I. Gorchakov, D. I. Zaitsev, and others. In 1974 Leonid Kurdachenko was awarded a Ph. D. degree and began to work as an assistant professor at the Department of Algebra and Geometry of Dnipropetrovsk University. In 1976 he became an associate professor and in 1977 the Department Chair. Leonid Kurdachenko defended his Doctor of Sciences thesis in 1992 at St Petersburg University. He has been also awarded a Doctor of Sciences Degree by Kiev University. Since 1996 Leonid Kurdachenko is a Full Professor and Chair of the Department of Algebra and Geometry of Dnipropetrovsk University. In 1998 the president of Ukraine awarded him with an honorary title "Distinguished Scientist of Ukraine". Leonid Kurdachenko is very active in research. His list of publications includes more than 150 articles published in major mathematics journals in many countries including Ukraine, USA, Germany, Great Britain, Italy, Spain, China, Greece, Brazil, Hungary, Czech Republic, Turkey, and Russia. Leonid Kurdachenko is a world class expert and lead researcher in infinite groups and associated module theory. In early seventies, he began his research on the FCgroups and their generalizations. The theory of $FC$groups (i.e., the groups with finite classes of conjugate elements) takes its roots in works of R. Baer, S. N. Chernikov, B. Neumann, and P. Hall. They have developed this theory mostly in the frame of periodic groups. Continuing the study of periodic $FC$groups in collaboration with M.~Tomkinson and J.~Otal, Leonid Kurdachenko initiated serious studies of nonperiodic $FC$groups. He showed that nonperiodic $FC$groups have their own nature, which is different from that of periodic $FC$groups. Thus, the new conditions of embedding periodic $FC$groups into direct products of finite groups have been obtained, the study of groups with minimax classes of conjugate elements has been initiated, investigation of groups with Chernikov classes of conjugate elements and groups with almost polycyclic classes of conjugate elements have been continued. In this passing, some broad generalizations of classical theorems of I.Schur and B.Neumann have been proved by L. A. Kurdachenko in collaboration with S. Franciosi and F. de Giovanni. One of the mainstreams in group theory (both infinite and finite) is the study of the influence of systems of subgroups on the structure of a group. L. Kurdachenko's significant input in this area is wellknown. Inspired by D. I. Zaitsev, he successfully continued the study of groups with weak minimal and maximal conditions, initiated by R. Baer and D. I. Zaitsev. L. Kurdachenko wrote a series of articles on groups with weak minimal and maximal conditions on normal subgroups. Together with his collaborators V. E. Goretskii, G. Cutolo, H. Smith, I. Ya. Subbotin, L. S. Kazarin, P. Shumyatskii and others, he investigates groups with weak maximal and minimal conditions on different important families of subgroups. Investigation of groups satisfying some 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 due to P. Hall, R. Carter, J. Rose, and Z. Borevich. Lately, numerous interesting results in this area have been obtained by many authors. L. Kurdachenko's input in this area is hard to overestimate. In collaboration with I. Ya. Subbotin, J. Otal, G. Vincenzi, A. Russo and others, he was able to classify some important properties of pronormal, Carter, abnormal, and contrarnormal subgroups of infinite groups and investigate their influence on the groups structure. They established new criteria of local nilpotency and nilpotency in infinite groups related to these subgroups, generalized the notion of a Carter subgroup for the case of infinite groups, and described some important classes of groups saturated with above mentioned subgroups and groups with transitivity of these subgroups properties. Lately together with A. BallesterBolinches and T. Pedraza, L. Kurdachenko published a cycle of interesting articles dedicated to properties of permutable and Sylowpermutable ($S$permutable) subgroups in infinite periodic groups. One of the intensively developing areas of investigation in group theory is the study of influence of the factorgroups on the structure of a group. The papers of L. Kurdachenko and his coauthors (I. Ya. Subbotin, J. Otal, S. Franciosi and F. de Giovanni, V. V. Pylaev) play an important role here. The book of L. A. Kurdachenko, J. Otal, I. Ya. Subbotin, ``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 monograph, the authors demonstrated quite clearly the capability of the module theory technique in solving group theory problems. Modules over group rings were efficiently used by L.~A.~Kurdachenko in many group theory studies. He focuses stepbystep on certain problems of module theory. One of his important achievements here was a description of injective envelope of artinian modules over abelian groups of finite rank. Together with I. Ya. Subbotin, L. A. Kurdachenko investigated some interesting types of modules close to artinian and noetherian modules. This as well as many other results have been presented in the book L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, ``Artinian modules over group ring'', Frontiers in Mathematics. (BIRKH\"{A}USER: Basel  2007). Of course, all that is a very short and general description of L. A. Kurdachenko's research achievements, and we were not able to mention many of his important results in this brief article. Leonid Kurdachenko is known as a great teacher. His lectures are very popular among his students. He successfully supervised 5 Ph.D. students. He has many friends and collaborators around the globe. L. A. Kurdachenko 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. BallesterBolinches, R. I. Grigorchuk, M. R. Dixon, 

Table of contents ADM  Number 4 (2009) 

A. BallesterBolinches, J. C. Beidleman, H. Heineken, M. C. PedrazaAguilera 
A survey on pairwise mutually permutable products of finite groups 

A. BallesterBolinches, J. Cossey, R. EstebanRomero 
A characterization via graphs of the soluble groups in which permutability is transitive 

A. BallesterBolinches, L. M. Ezquerro, A. N. Skiba 
On subgroups which cover or avoid chief factors of a finite group 

M. R. Dixon, I. Ya. Subbotin 
Groups with finiteness conditions on some subgroup systems: a contemporary stage 
full text in  pdf  ps   
M. De Falco, F. de Giovanni, C. Musella 
Groups with many selfnormalizing subgroups  
A. M. Gaglione, S. Lipschutz, D. Spellman 
Some properties of nilpotent groups  
R. Grigorchuk  On a question of Wiegold and torsion images of Coxeter groups  
V. V. Kirichenko, O. Yu. Dashkova 
On some aspects of the theory of modules over group rings  
J. M. MunozEscolano, J. Otal, N. N. Semko 
The structure of infinite dimensional linear groups satisfying certain finiteness conditions 

J. Otal, N. N. Semko 
Groups with small cocentralizers  
A. Russo, G. Vincenzi 
Groups with many generalized FCsubgroup  
A. J. Slupik, V. I. Sushchansky 
Minimal generating sets and Cayley graphs of Sylow psubgroups of finite symmetric groups 

I. Ya. Subbotin  On arrangement of subgroups in groups and related topics: some recent developments 

all abstracts in  pdf  ps   