Understanding Wall's theorem on dependence of Lie relators in Burnside groups

Document Type: Original Article


Christ Church, University of Oxford, Oxford, OX1 1DP, England.


‎G.E‎. ‎Wall [J‎. ‎Algebra 104 (1986)‎, ‎no‎. ‎1‎, ‎1--22; Lecture‎ ‎Notes in Mathematics‎, ‎pp. 191--197‎, ‎1456‎, ‎Springer-Verlag‎, ‎Berlin‎, ‎1990] gave two different proofs of a remarkable result about the‎ multilinear Lie relators satisfied by groups of prime power exponent $q$‎. ‎He‎ ‎showed that if $q$ is a power of the prime $p$‎, ‎and if $f$ is a multilinear‎ ‎Lie relator in $n$ variables where $n\neq1\operatorname{mod}(p-1)$‎, ‎then $f=0$‎ ‎is a consequence of multilinear Lie relators in fewer than $n$ variables‎. ‎For‎ ‎years I have struggled to understand his proofs‎, ‎and while I still have not‎ ‎the slightest clue about his proof in [J‎. ‎Algebra 104 (1986)‎, ‎no‎. ‎1‎, ‎1--22]‎, ‎I finally have some understanding‎ ‎of his proof in [Lecture‎ ‎Notes in Mathematics‎, ‎pp. 91--197‎, ‎1456‎, ‎Springer-Verlag‎, ‎Berlin‎, ‎1990]‎. ‎In this note I offer my insights into Wall's second proof‎ ‎of this theorem‎.


M. Hall, The Theory of Groups, The Macmillan Co., New York, 1959.
G. Havas and M. Vaughan-Lee, On counterexamples to the Hughes conjecture, J. Algebra 322 (2009), no. 3, 791–801.
W. Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann. of Math. (2) 52 (1950) 111–126.
I.N. Sanov, Establishment of a connection between periodic groups with period a prime number and Lie rings, Izv. Akad. Nauk SSSR, Ser. Mat. 16 (1952) 23–58.
M.R. Vaughan-Lee, The restricted Burnside problem. Second edition, London Mathematical Society  Monographs.
New Series, 8, The Clarendon Press, Oxford University Press, New York, 1993.
G.E. Wall, On Hughes’ Hp-problem, Proc. Internat. Conf. Theory of Groups (Canberra, 1965), pp. 357–362, Gordon and Breach, New York, 1967.
G.E. Wall, On the multilinear identities which hold in the Lie ring of a group of prime-power exponent, J. Algebra 104 (1986), no. 1, 1–22.
G.E. Wall, Dependence of Lie relators for Burnside varieties, pp. 191–197, Lecture Notes in Mathematics, 1456, Springer-Verlag, Berlin, 1990.