Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)
KIRÁLY, B., On the powers of the augmentation ideal of a group ring
78 Bertalan Király The n-th term of the lower central sériés of G is defined inductively: 7I(G) = G, l 2{G) = G' is the derived subgroup [G, G] of G, and 7 n(G) = hn-l(G),G]. We shall also use the following well known fact: I{ln(G)) Ç A n(RG) for all n > 1. 3. The augmentation terminais Throughout this paper R will dénoté a commutative ring with unity of non-zero characteristic and also p will dénoté a prime number. Let p be a prime and n a natural number. Then we shall dénoté by G pn the subgroup generated by all elements of the form g v ,g £ G. The normal subgroup G Pi k are defined by oo = n Gp n^( G)> n= 1 where 7k(G) is the fc-th term of the lower central sériés of G. It is clear, that the factor-group G/G vy k is residually-A/'p group for every k. We have the following sequence (1) G = G P ji D G V 2 5 • • • 2 G p 00 of normal subgroup s G Pj k of a group G, where G p = Q G Pi k. k= 1 Lemma 3.1. Let R be a commutative ring of characteristic p s. Then I(G P} k) Ç A k(RG ) for ail k > 1. PROOF. Let the element h— 1 be in I(G Pi k). IT will be sufficient to show n n n that h - le A k(RG). Writing the element h as h = h[ h\ • • • h^ y k (hi E G , Vk £ lk(G)) and using the identity (2) ab - 1 = (a - 1 )(b - 1) + (a - 1) + (b - 1) we have that h- 1 = (hf hf y k-l)(y k-l) + (hf h? • • • h£ - 1) + (y k - 1).
