The major part of this paper was presented to the University of Manchester for the degree of M.Sc. It is a great pleasure to record my gratitude to Dr B H Neumann, both for suggesting this topic to me and for his generous advice and guidance throughout: I also thank the University College of North Staffordshire for the Research Studentship which enabled me to carry out this work.

Most of the well-known theorems of Sylow for finite groups and of P Hall for finite soluble groups have been extended to certain restricted classes of infinite groups. To show the limitations of such generalizations, examples are here constructed of infinite groups subject to stringent but natural restrictions, groups in which certain Sylow or Hall theorems fail.

This is a well-motivated, entertaining, deceptively leisurely, and elegant survey of the state of knowledge of torsion-free nilpotent varieties of groups. The discussion centres around the curious fact (discovered by the author) that there are exactly 39 such varieties of class ≤ 5 (while there are infinitely many of class 6), each of which he identifies. However the article is more wide-ranging than this might suggest; the author starts from scratch (with the definition of a variety of groups), suggests reasons for the decline of activity in variety theory in the seventies, describes how the basic problems of variety theory link up with the study of torsion-free nilpotent varieties, and throughout illustrates the techniques used in that study. For those familiar with variety theory this article is illuminating; to those ignorant of that theory, who might wish to discover what it is about, it can also be strongly recommended.

Professor László Kovács is internationally rated for the very high level of research he has carried out in group theory and representation theory. During the 1960s and 1970s especially, he achieved outstanding results on varieties of groups. In recent years, his key achievements have been in the modular theory of free Lie algebras. He has published over a hundred scientific papers. He has been an invited key speaker in numerous international conferences. He has been a visiting professor at numerous universities throughout the world. He kept in touch with Debrecen colleagues even after their departure abroad. From the 1980s onwards he has been a visiting professor at our university. He has conducted joint research with Debrecen algebraists and arranged Australian invitations for them. He has served on the editorial boards of the University of Debrecen's international mathematical journals.

My own personal interaction with Laci started in my honours year. I was lucky enough to have solved a problem in combinatorial group theory and I sent a draft of a paper to a dozen or so group theorists. Laci and I exchanged emails and I looked into the possibility of pursuing a PhD with him, but what discouraged me from going to the Australian National University was his plans to take long service leave, and possible retirement. Plus, I was also in contact with Cheryl Praeger who was effectively luring me to the west, and that's what I eventually ended up doing. Laci ended up being one of my PhD examiners and took great interest in my work. He came to Perth at least a couple of times during my PhD and first postdoctoral position, and we had long enjoyable discussions in my office, on politics (one of Laci's pet subjects) and mathematics. Laci was one of the integral members of the golden era of group theory at the Australian National University, and he will be sorely missed by the group theory community.

Laci was a splendid mathematician, without any personal ambition, the prototype of an old-world learned person also beyond mathematics. I liked him very much.

Laci was very helpful. He wrote letters of recommendation for many people. Once I asked him a question. He immediately knew somebody who was dealing with that question, and also sent me a copy of a relevant paper. A lot of knowledge has passed away with him.