A COMMUTATOR IDENTITY PROVED BY MEANS OF THE MAGNUS ALGEBRA

ABSTRACT. Let F be the free group generated by a and b; F = (a,b). Let F n be the n th subgroup f the lower central series of F. Let p be a prime. Let c 1 < c 2 < c 3 <.-- < Cq(p2) be the basic commutators in F of dimension •< p2. Let P1 = (a,b), Pn = (Pn-1,b) for n> 1. Let c z = Pp. Then ot,_.q(p2) ei (a'bP) = l•i=3 cimoa Fp2+l, where ot •> 1. It is shown that the exponents, el are such that (i) e i are divisible bypot for 3 •< i •< q(p+l) and i 4: z, (ii) e iare divisible y pot-1 for q(p+l) + 1 •<i •< q(p2), (iii) e z is divisible y pOt-l, and (iv) e 3 = pa. I. Introduction. This paper is the result of research on the lower central series of a free product of cyclic groups, at least one of which is of finite order. In [1] a procedure for finding presentations of the factor groups of the lower central series of such free products was established. The author has recently succeeded in using this procedure to find many such presentations. The present proofs are very complicated