Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)

KlRALY, B., The Lie augmentation terminals of group

68 Bertalan Király which follows from (2), we conclude that A^(RG) C I{G'). By Lemma 2.3 we obtain the inclusion A^(RG) D I{G'). Consequently A^(RG) = I{G'). Since T

r[G] = 2,AW(RG) = A^(RG). Then from Lemma 3.2 we have the equality A^(RG) = J{G

P). Therefore 1(G

P) = 1(G') and G

P = G'

. Conversely. If G / G' = G

P, then A^(RG/G

P) = 0 because G/G

P is an Abelian group. From this equality it follows that A^(RG) C I(G

P) and by Lemma 3.2 A^(RG) = A^(RG). Since G ± G

P

I A(RG) ± A^(RG). Consequently = 2 which prove 2) of our theorem. 3) Suppose that T

R[G] = n > 2. From the statements 1) and 2) it follows that G / GP and G' G

P. It is very simple to see that G/G

P

ii are residually—V

p groups and consequently, by Theorem 2.2, A^{RG/G

P}

I) = 0 for all i > 1. Because T

r[G] is finite then • • • D A

[

N"

1

] (RG) D A^

1 (RG) = AT

N+

1Í (RG) = ••• = A

["

] (RG) and hence • • O A^

n~

l\RG/G

P

ii) D A^(RG/G

P>

1) = = A^(RG/G

P}

t) = ••• = A^(RG/G

p>

i). It follows that T

R[G/G

p,i] are finite and not greater than T

r[G] for all i > 1. Then there exists a natural number k < n such that AW(RG/G

Pt

i) = 0 (6) for all i. Then from (2) we have that A^ {RG) C I(G

PJ

I) for all i. If i = fc, by Lemma 3.1 we obtain that A^(RG) = I(G

P,

K)- Hence I{G

PF

K) Q I(G

PI

I)) and therefore, G

P

ii D G

P}

k for all i > 1. This implies that • • • 2 G

Pt

k — G

P}

k+1 = • • • = G

p (7) and by (6) we have that AW(RG/G

P) = 0. (8) By Theorem 2.1 it follows that G/G

p is a nilpotent group whose commutator subgroup is a finite p-group.