20EIOThis is an informal report on the present status of the displayed problems in Hanna Neumann's book Varieties cf groups [50]. The reader should have the book at hand, not only for notation and terminology but also because we do not re-state the problems nor repeat the comments available there. Our aim is to be up to date, not to present a complete historical survey; superseded references will be mostly ignored regardless of their significance at the time. Details of solutions will not be quoted from papers already published, unless needed to motivate further questions.The discussion and the problems highlighted in it reflect our personal interests rather than any considered value-judgement.The preparation of this report was made much easier by access to the notes Hanna Neumann had kept on these problems. We are indebted to several colleagues who took part in a seminar on this topic and especially to Elizabeth Ormerod for keeping a record of these conversations. The report has also gained a lot from the response of conference participants; in particular, Professor Kostrikin supplied much useful information. Of course, all errors and omissions are our own responsibility: we shall be very grateful for information leading to corrections or additions. PROBLEM (page 6 in [50]). As Hanna Neumann wrote, "this is of no great consequence". The answer is negative; see Kovacs and Vaughan-Lee [47]. PROBLEMS 2, 3, 11 (pages 22, 92). The celebrated "finite basis problem" asked whether every variety can be defined by a finite set of laws. Ol'sanskii proved (in about 1968, unpublished) that this is equivalent to the problem: is the set of varieties countable? (See also Kovacs [42].) A positive answer would have created a simple situation to report on; however, in general, the answer is negative. A comprehensive survey of the positive partial results is beyond the scope of this report, and the listing of open questions provoked by the complexity of the situation is also without any claim to completeness. The negative answer was first obtained by Ol'sanskii [55] in September 1969: he proved that there are continuously many loeally finite varieties of soluble length at most 5 and exponent dividing 8pq whenever p, q are distinet, odd primes. This settled Problems 2 and 3. By Deeember 1969, Vaughan-Lee [62] construeted (by entirely different means) continuously many varieties within ~ A !2)2; and, early in 1970, Adyan ([l], see also [3]) gave an infinite independent set of very simple two-variable laws.Given the negative solution, Zorn's Lemma yields the existenee of at least one just non-finitely-based variety (a variety minimal with respeet to not being definable by a finite set of laws). One may then ask:
