Sergei Adian

About the Burnside problem:

Very much like Fermat's Last Theorem in number theory, Burnside’s problem has acted as a catalyst for research in group theory. The fascination exerted by a problem with an extremely simple formulation which then turns out to be extremely difficult has something irresistible about it to the mind of the mathematician.

Before the work of Novikov and Adian an affirmative answer to the problem was known only for ${\displaystyle n\in \{2,3,4,6\}}$ and the matrix groups. However, this did not hinder the belief in an affirmative answer for any period ${\displaystyle n}$. The only question was to find the right methods for proving it. As later developments showed, this belief was too naive. This just demonstrates that before their work nobody even came close to imagining the nature of the free Burnside group, or the extent to which subtle structures inevitably arose in any serious attempt to investigate it. In fact, there were no methods for proving inequalities in groups given by identities of the form ${\displaystyle X^{n}=1}$.

An approach to solving the problem in the negative was first outlined by P. S. Novikov in his note, which appeared in 1959. However, the concrete realization of his ideas encountered serious difficulties, and in 1960, at the insistence of Novikov and his wife Lyudmila Keldysh, Adian settled down to work on the Burnside problem. Completing the project took intensive efforts from both collaborators in the course of eight years, and in 1968 their famous paper appeared, containing a negative solution of the problem for all odd periods ${\displaystyle n>4381}$, and hence for all multiples of those odd integers as well.

The solution of the Burnside problem was certainly one of the most outstanding and deep mathematical results of the past century. At the same time, this result is one of the hardest theorems: just the inductive step of a complicated induction used in the proof took up a whole issue of volume 32 of Izvestiya, even lengthened by 30 pages. In many respects the work was literally carried to its conclusion by the exceptional persistence of Adian. In that regard it is worth recalling the words of Novikov, who said that he had never met a mathematician more ‘penetrating’ than Adian.

In contrast to the Adian–Rabin theorem, the paper of Adian and Novikov in no way ‘closed’ the Burnside problem. Moreover, over a long period of more than ten years Adian continued to improve and simplify the method they had created and also to adapt the method for solving some other fundamental problems in group theory.

By the beginning of the 1980s, when other contributors appeared who mastered the Novikov–Adian method, the theory already represented a powerful method for constructing and investigating new groups (both periodic and non-periodic) with interesting properties prescribed.