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
82 Bertalan Király let G be a group with no generalized torsion elements. Then A L J{RG) = 0 if and only if G is a residually-J\f p group for all p £ II. Theorem B. Let T(R +) / 0. If G' is with no generalized torsion elements with respect to the lower central series of G and T(R +) is with no non-trivial elements of infinite p-height then A^{RG) = 0 if and only if G is a residually-Vp group for all p £ II. Proof. Let p an arbitrary prime of II, A^(RG) = 0, and let p s (s > 1) be the order of element a £ T(R +). Since the equation oo oo oo Gw = n =n n( G') p"7*(G)) =« k= 1 n—1 k=l implies that G £ RP p, it is enough to show, that C[ p] = (1). Suppose that g £ G[ p]. Then g £ (G") p" ")k{G) for every n and k and by Lemma 3.1. we have that g- 1 = p sX{k,g) (mod A [k ]{RG)) for every k. From p sa = 0 it follows that a(g — 1) £ A^(RG) for every k. Hence a(g - 1) £ A^(RG) and a{g - 1) = 0. This imphes that g = I: Consequently G[ p] = (1). This means that G is a residually-P p group for all p £ n. Conversely. Let G £ RX> p for p £ n and let 1 ^ g be an arbitrary element of G' . Then there exists a normal subgroup H of G such that G/H £ V v and g £ H. Since G/H £ V p then (G/H)' £ Af p. By the isomorphism G'H/H 2 G' /H D G' we have that g = g(H n G') ± 1. This means that if G £ ~RV p then G' £ RvVp. Using Proposition 3.3. we have that A^(RG') = 0 and from (2) it follows that A^(RG) = 0. Lemma 3.4. Let oo oo yt n p€Tj = ln= 1 Then for a prime p £ P and arbitrary natural numbers k and s y = p sY(p, k, s, y) (mod A^(RG)), where Y{p , k, s, y) £ RG and P is a subset of the set of prime numbers. Proof. Let p £ P. For every natural n we can express y as l y = ^ QiiZi(hi - l), i-i
