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

Residual Lie nilpotence of the augmentation ideal 83 where hi £ (G')

p™ ^k(G), a.i E R and every z

l is from a set of coset representatives of (G'Y 7k(G) in G. For a large enough n by Lemma 3.1. hi- 1 EE p

sX(k,hi) (mod A

[k

]{RG)) for every i (z = 1,2,..., /) and the proof follow. If g E G' is a generalized torsion element of a group G then Q,

g denotes the set of the prime divisors of the order of the elements gjk{G) E G/^k{G) for every k = 2, 3,.... Lemma 3.5. Let g E G' be a generalized torsion element of a group G, A an arbitrary subset ofü

g, a E n

pgAJ

P(R) and let oo oo n nnvr'^c)). p(Efig\A k1 Then one of the following statements (1) if A is a proper subset of £l

g, then a(g — l)a: E A^(RG) (2) if A = n

5, then a(g - 1) E AH(#G) (3) if A - 0, then (.g - l)x E AM(i?G) holds. Proof. It is enough to show that for an arbitrary natural number k the elements a(g — 1), (g — l)x

, a(g - l)x are in the ideal A^(RG). If 9 e Jk{G) then by Lemma 2.5. (g1) E AW(RG), and the statements follow. Now let g £ "fk{G) and let be the prime factorization of the order of the elements gjk{G

) of the nilpotent group G/^k{G). It is clear that pi E £l

g for every i = 1,2,..., s. Let A a subset of Cl

g. With loss of generality we may assume that pi,p

2,

... ,pi E A and pi ^ A for i > I. Let g = gig

2 • • •

g

slk(G

) be the decomposition of the element g~fk{G) of the nilpotent group Gj^kiG) in the product of p

%-elements g

z^k{G) (

i = 1,2,. . .,5). Then 9 - 9x92 • • • g

sVk, 9i E G\ i = 1, 2,..., 5 for a suitable yk E 7k{G). Then there exists m; (i = 1,2,..., s) such that 9Í

r' G 7*(<?)•