27 - Addendum to: “Groups with exponent four”

Summary

In the quarter of a century which has elapsed since the St Andrews meeting in 1981 a lot of information has been gathered about Burnside groups, whether free or restricted. We will use RBP generally as shorthand for the restricted Burnside problem and in this case the work of Kostrikin and Zelmanov and has established the local finiteness property. Their success in obtaining an affirmative answer to the RBP for all prime-power exponents and so, by the theorem of Hall & Higman, for all finite exponents has naturally directed attention to the structure of such groups — in particular, questions about order, class and derived length.

Much of the work — including further studies of B(r, 4) — has depended on

1. (a) Computer-aided calculations, using Todd–Coxeter coset enumeration at first, and more recently employing nilpotent quotient algorithms which produce presentations of groups through power-commutator relations. Pioneers in this work were John Leech and I. D. Macdonald and; the procedures involved are described in Appendix B of.

2. (b) Linearising problems about group commutators by studying the associated Lie algebras, yielding connections between commutator identities in certain groups and identities in associated Lie rings. Kostrikin in his book refers to classical papers of Magnus, Grün, Zassenhaus and Baer in the early nineteen-forties, followed by contributions of Lazard and Higman in the nineteen-fifties, which drew attention to the connection between the RBP for a prime exponent p and local nilpotency of an associated Lie algebra of characteristic p.