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

Abstract

‎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 $nneq1operatorname{mod}(p-1)$‎, ‎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.