1958/07/01 by K. T. Chen, Ralph H. Fox, Roger C. Lyndon · 363 citations
Mathematics · Medicine · Dentistry · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Mathematics #Quotient #Differential calculus #Series (stratigraphy) #Calculus (dental) #Differential (mechanical device) #Time-scale calculus #Algebra over a field #Pure mathematics #Multivariable calculus #Medicine #Dentistry
paper · doi:10.2307/1970044
published in Annals of Mathematics 68(1), 81 (Princeton University)
openalex publication_date 1958/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/12
The quotient groups Qn(G) =GnGn+i of the lower central series G = G1 D G, D G, D * * of a finitely generated group G are finitely generated abelian groups. Our object is to develop an algorithm for the calculation of Qn from any given finite presentation of G. As a preliminary step, the special case of a free group X is considered. It is known [2], [7] that, for a free group X of rank q, the group Qn(X) is a free abelian group whose rank is the Witt number 0b,(q), and a basis for QJ(X) has been exhibited by M. Hall [42]. Our approach is somewhat different in that we construct, by means of the free differential calculus, a basis for the dual group Qn* = Hom [QnJ]. The corresponding dual basis of Q. is not the same as the Hall basis, although it bears a superficial resemblance to it. In the course of this construction we re-prove Witt's result [7] that the elements of Xn are just those for which the non-constant terms of the Magnus expansion are all of degree at least n, in short, that the lower central groups coincide with the dimension groups of Magnus [2]. Further, we derive a complete set of finite identities for the coefficients in the Magnus expansion of an element of X. The algorithm for Qn(G) is to be found in the last section. The authors wish to thank Julian Brody for his help in simplifying the arguments, and for selection of the example in ?4.