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
66 Bertalan Király The normal subgroups is defined by oo <?p.* = f| (.GYlÁG), n=l where 1k(G) is the kth term of the lower central series of G and G' is the commutator subgroup of G. It is clear, that the factor-group G/G P ik is a re si du ally-V p group for every k. We have the following sequence G — G P i 1 3 G P )2 5 • • • 3 G p (3) oo of normal subgroups G P ik of a group G , where G p = D G P ik. k= 1 Lemma 3.1. Let R be a commutative ring of characteristic p s . Then I(G P tk) Q AW(RG) for all k> 1. Proof. Let the element h — 1 be in I(G P ik)- It will be sufficient to show that h — 1 E A^(RG). For an arbitrary n written the element h as h = h\ h\ • • -h^yk (hi E G',y k E ^k(G)) and using the identity ab-l = (a- l)(b - 1) + (a - 1) + (b - 1) (4) we have that h- 1 = (hf hf •••h^y k- 1 )(y k - 1) + (hf hf • • • h£ - 1) + (y k - 1). By Lemma 2.3, I(~fk(G)) C A^(RG) and hence y k - 1 E A^(RG). Therefore h- 1 EE (h{ h{ 1) (mod ^(ÄG)). Applying (4) repeatedly to (h\ K p 2 • • • h^ -1) from the previous expression it follows that m 77 1 P n / n\ h- 1 EE - 1 )bi - EE - ^ 6* ( mo d i=l 2 — 1 j=l ^ ' where 6; E ÄC. From Lemma 2.3 (cases 1 and 3) we obtain, that the element (h{ — l)7 He in A^ J+ l^(RG) for every i and j. If n > s + k, then p s divides P 7 . for j = 1,2, ..., k — 1. Therefore m m k — 1 h~ 1 EE-i)^- 1) J' 6< = ( m° d i=i i=i j=i
