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
80 Bertalan Király Let p be a prime and n a natural number. Then G p H is the subgroup of G generated by all elements of the form g p , g E G. For a prime p and a natural number k the normal subgroup G\ p^\ of G is defined by oo G [P M = n(GT"7*(G). n1 We have the following sequence G = G\ P ii] D G[ P : 2] ^ • •. 2 G[ p ] of normal subgroups G[ P ik] of G, where oo Gip] = n gm • k= 1 It is clear, that G/{G') p n 1k{G) are in V p, and G/G^^] and G/G[ p] are re si du ally-V p groups for every k and n. Lemma 3.1. If n > ks and h E {G') p n ~f k{G), then h - 1 = p sX(k , h) (mod AW (RG )) for a suitable X(k,h ) E A^(RG). Proof. Let h E (G") p" ^k(G). We can write element h as h = h p h\ •••hf Vk where hi E G',YK E 7K{G). Using the identity (3) ab - 1 = (a - 1)(6 - 1) + (a - 1) + (b - 1) to h — 1 we have that h- 1 = (hf hf hf - 1 )(y k - 1) + (hf hf... hf - 1) + (y k - 1). By Lemma 2.5. /(TA^G)) C A^(RG) and hence y k - IE A^(RG). Therefore h- 1 = (hf hf ••• hf - 1) (mod A [k ]( RG)).
