Products of powers in groups

In this note we show that a product of Nth powers in a group cannot in general be expressed as a product of fewer Nth powers. This extends a result of Lyndon and Newman [1]. THEOREM. Let F be a free group of rank n with basis x1, ... , xn, let U1, ... , Ur be elements of F, and let N be an integer greater than 1. If (*) N1 N= UN U N then m?n. For the proof it will suffice to exhibit a group G and elements x1, * , xn in G such that, if ul, ... *, ur are any elements of G satisfying (*), then m!n. Choose a prime p dividing N and write N=qM, where q=pe for some e> 1 and p does not divide M. Let P be the ring of polynomials over GF(p) in noncommuting indeterminates X1, ... , X,. Let f be the ideal in P generated by X1, * , Xn, and let R=P/lf+l; we shall write Xi also for the image of Xi in R. Let G be the group of units in R. (Thus G is a finite group of exponent pq.) The elements xi= 1 +Xi belong to G, since they have inverses x?1=1-X?+X2-*+(-1)qXi. Now x1= (+Xj)=1+Xjq, whence xN =xqM (1+Xiq)M= 1+Mxi. it follows that n (I)~ ~~~~x ... * * N = I + M xi. Let u1, ... *, ur be in G. We may write Uj= 1 + ,i ocjixi+ Dj where Dj is in /2. Then uq =(1 + jtXi + D )J = I + (:j ocjiXi + DJ) = I + (X vlx + Q _ 1 i . .. Yci xi ... YiQ Received by the editors January 15, 1973. AMS (MOS) subject class/ifcations (1970). Primary 20F10, 20E05. 1 The author wishes to acknowledge the support of the National Science Foundation, and also the hospitality of the University of Birmingham. ? American Mathematical Society 1973 419 This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 05:54:43 UTC All use subject to http://about.jstor.org/terms 420 R. LYNDON, T. McDONOUGH AND M. NEWMAN summed over all il, *, q such that 1 i1, , i_ ,a q = I t1 i <n), ji (4') oc (1? i h < n;i s h). Let A=(oca-1) and B= (ocji), m-by-n matrices over GF(p). Then (3') and (4') assert that (5) ATB = In where AT is the transpose of A and In is the n-by-n identity matrix. It follows that n=rank(Jn)< rank(B)_ m. REFERENCE 1. Roger C. Lyndon and Morris Newman, Commutators as products of squares, Proc. Amer. Math. Soc. 39 (1973), 267-272. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MICHIGAN, ANN ARBOR, MICHIGAN 48104 (Current address of Roger Lyndon) DEPARTMENT DE MATHEMATIQUES, UNIVERSITE DE MONTPELLIER, 34 MONTPELLIER, FRANCE DEPARTMENT OF MATHEMATICS, UNIVERSITY COLLEGE OF WALES, ABERYSTWYTH, CARDIGANSHIRE, WALES (Current address of Thomas McDonough) NATIONAL BUREAU OF STANDARDS, WASHINGTON, D.C. 20234 (Current address of Morris Newman) This content downloaded from 157.55.39.215 on Tue, 30 Aug 2016 05:54:43 UTC All use subject to http://about.jstor.org/terms