Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
KlRALY, B., Residual Lie nilpotence of the augmentation ideal
84 Bertalan Király Using identity (3) repeatedly to (g — 1) we conclude that g-l = v + w + (y k-l) = v + w (mod A [k ](RG)), l s where v = ^ ( g{ — l)x{, w = (g x — l)x{ and X{ E RG. In the case when 1=1 i=l+ 1 Afl{pi,P2, ... ,Ps] ='0 we assume that v = 0, and if A Pl{pi ,p 2, .. -,p s} = {pi,p 2, • • • we put w = 0. Because 9?'™' e l k(G)CD [k ](G) and gi G G' for every i = 1, 2,..., s, we conclude from (4) that there exists a natural number r t (i = 1, 2,..., 5) such that (5) Pi r'(9i~l )eA^(RG). Also, since i « e n ^ n já r) p£A 1=1 we can express a as a = ppa» (a; E -ß) for each i < I. Then by (5) i av EE Y^ aiP? (9i - 1)«» = 0 (mod A^ (RG)). i= 1 Therefore (6) a(g - 1) = av + aw = aw (mod A^(RG)). If A = Q g then w — 0 and case 2) is proved. By Lemma 3.4. x = p\*Y(pi,k,ri,x) (mod A^(RG)), and so, s wx= ]T pffa -l)x iY(pi,k 1r i ix) (mod A [k ](RG)). i=l+ 1 Hence by (5) (7) wx = 0 (mod A [k ](RG)).
