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

86 Bertalan Király Theorem C. Let the derived group G' contain a generalized torsion element of G with respect to the lower central series of G. Then A{RG) is residually Lie nilpotent if and only if there exists a non-empty subset Í2 of the set of primes such that fl

p

£QJ

P{R) = 0, G is discriminated by the class of groups Vn and every proper subset A of the set il at least one of the conditions (1) n

p<E

AJ

p(R) = 0 (2) G is discriminated by the class of groups 2)q\a holds. Proof. Let A^(RG) = 0. Let us first consider the case when G' contains a non-trival torsion element. Then there exists a p-element g in G' with p G Cl. Then by (4) for every k there exists a natural number m such that (8) P

m{g - 1) G A^(RG). If a G J

P{R), then for each m we can write element a as a = p

ma

m (a

m G R). Therefore a(^-l) G A^(RG) for every k, that is a(g-l) G A^(RG). Hence a(g — 1) = 0 and so, a = 0. Consequently J

P(R) = 0. Now we show, that G is discriminated by V{

py. Let oo oo he n n(GT'7*(G). = l 1 = 1 Then oo oo n r\n(GYMG)) k=11=1 and by Lemma 3.4. for every k and m (9) h- 1 EE p

mY(p, k,m,h- 1) (mod A

[k

] {RG)). By (8) and (9) we have that (g - 1 ){h - 1) = p

m{g - 1 )(h - 1)Y{p, m,k,h- 1) (mod A™ {RG)) for every k. This implies that (g - 1 ){h - 1) G AM{RG) and so, (g - 1 ){h - 1) = 0. Prom this equation we have that the characteristic of R is p (= 2) and from (9) it follows that h- 1 G A^(RG). Therefore h = 1 and so oo oo n n(G'fMG)

=a). k= 1 t=

1