Kourovka Notebook, 19th edition

The latest edition (the 19th) of the Kourovka Notebook has just been released. It now has its own website, https://kourovka-notebook.org/.

The Kourovka Notebook has been going for more than 50 years, longer than my life as a mathematician. It is a problem book for group theory and related areas, and now contains somewhere around a thousand problems, 111 of them new in this edition. The first edition appeared in 1965, and the KN did a remarkable job of encouraging communication across the Iron Curtain in those tense times. There are about 20 problems from that first edition still unsolved, and the first two to be attributed (rather than just labelled “Well known problem”) are by Mal’cev and Magnus. The last two questions ask whether the identical relations of a polycyclic group, or of a matrix group in characteristic zero, have a finite basis. This was a time of great activity in the area of varieties and identical relations of groups, coming shortly after the Oates–Powell Theorem demonstrated that the identical relations of a finite group have a finite basis. This topic is still of interest; the last problem in this year’s crop asks whether there is an infinite group, all of whose proper subgroups are cyclic of prime order p (a Tarski monster), which satisfies no identical relations except for x^p = 1 and its consequences.

I was fortunate enough to become a contributor fairly early in my career: two of my problems from the ninth edition survive, including one which I think it would be timely to return to. Suppose that G is a finite primitive permutation group of rank r, and suppose that the maximum rank of any transitive constituent of a point stabiliser is s. Is it true that either r is bounded by a function of s, or the stabiliser of a point has rank s (and so acts regularly on one of its orbits)? (This formulation is slightly updated.)

The first part of the book (up to page 152) is the list of unsolved problems (or problems which have been solved since the last update a few years ago). Then there is an archive of solved problems, giving references to the solutions (in English if possible). Finally there is an author index of contributors.

The Kourovka Notebook, which as I said was such an important communication tool between group theorists right from the start, is clearly going from strength to strength. Any group theorist, and many people in related parts of algebra, logic and discrete mathematics, will find plenty of challenging problems here.