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
The Lie augmentation terminals of groups 65 Consequently A [ n\RG/L) ^ (A^(RG) + I(RL))/I(RL) (2) for all n > 1. If /C denotes a class of groups (by which we understand that /C contains all groups of order 1 and, with each H E /C, all isomorphic copies of H) we define the class R/C of residually-/C groups by letting G E RAT if and only if: whenever 1 / ^ G G, there exists a normal subgroup H g of the group G such that Gj Hg E K and g H g. We use the following notations for standard group classes: V: nilpotent groups whose derived groups are torsion-free nilpotent groups and V p: nilpotent groups whose derived groups are p-groups of finite exponent. Let p be a prime and n a natural number. Then we shall denote by the subgroup generated by all elements of the form g p ,g E G. If Ii, L are two subgroups of G, then we shall denote by ( Ii , L) the subgroup generated by all commutators (g,h) = g~ lh~ lgh, g E K , h E L. The nth term of the lower central series of G is defined inductively: 71(G) = G, 12(G) = G' is the commutator subgroup (G,G) of G, and ln(G) = (ln-l(G),G). In this paper we shall use also the following theorems: Theorem 2.1. ([1]) Let G be a non-Abelian group, R a commutative ring with identity. Then A^ (RG) — 0 for some n > 2 if and only if G is nilpotent, G' is a finite p-group and p is nilpotent in R. 00 The ideal JJR) of a ring R is defined by JJR) = n p nR. n— 1 Theorem 2.2. ([2], Theorem 2.13, page 85) Let G be a residually V p-group and J p(R) = 0, then A^(RG) = 0. We shall use the following lemma, which gives some elementary properties of the Lie powers A^(RG) of A(RG). Lemma 2.3. ([2], Proposition 1.7, page 4) For an arbitrary natural numbers n and m are true: 1) I(ln(G))C AW(RG) 2) [AW(RG),AW(RG)] C A^ m\RG) 3) AW(RG) • AW(RG) C A\ n+ m~ l\RG). 3. The Lie augmentation terminals. Throughout this section R will denote a commutative ring with identity of characteristic p s .
