Polynomial Sequences in Groups

Given a groupGwith lower central seriesG = G1 G2 G3 •••, we say that a sequenceg: → Gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatorDg(n) = g(n) − 1g(n + 1)dtimes takes its values inGk. We introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group. As an application we obtain the following extension of the Hall–Petresco theorem: THEOREM.LetG = G1 G2 G3 •••be the lower central series of a group G.Let x Gk,y Gland letp, qbe polynomials → of degrees k and l,respectively. Then there is a sequencez0 G,zi Gifori ,such thatfor all n .