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

Residual Lie nilpotence of the augmentation ideal 85 If A = 0, then v = 0, and so, (g - l)x = vx + wx = wx = 0 (mod ^(ÄG)) and case 3) is proved. Also, since a(g - l)x = avx + awx (mod A^(RG)) from congruences (6) and (7) the proof (of case 1)) follows. We recall that for a prime p J\f

p denotes the class of nilpotent groups whose derived groups are p-groups of bounded exponent, and if Cl a subset of the set of primes, then JVQ = U

p^QAÍ

p and Vn = U

P£nV

p. Let a group G be discriminated by the class of groups Vp (r / 0) and let </i, g

2,.. ., g

n be a finite set of distinct elements of G'

. Then there exists a normal subgroup H of G such that g^H ^ gjH if i ^ j and G/H E Vp. Therefore (GjR

)' E M

p for any prime p E T. By the isomorphism G'H

/H = G'/H n G' we have gJI^G') f g

j(HnG') if i^j {i,j = 1,2 ,...,n). This means, that if G is discriminated by the class Vp

, then G' is discriminated by the class of groups Afp. Lemma 3.6. Let O be a non-empty subset of the set of primes such that r\peuJp[R) = 0 and a group G is discriminated by the class of groups VQ. If for every proper subset A of the set fi at least one of the conditions (1) n

pe

Aj

p(R) = o (2) G is discriminated by the class of groups holds, then A^(RG) = 0. Proof. Let n x = ^a

i9

ieAM{RG). i=

1 By Lemma 2.3. it is enough to show that A^(RG) = 0 for all groups G E VQ. So let G E VQ. Then G is a nilpotent group and by (2) A

[üj

](RG) C A

U(RG')RG. Clearly, G' E J\Í

Q. If G is discriminated by the class of groups 'Vp, where r is an arbitrary non-empty subset of H, then G' is discriminated by the clas Afp, which was showed above. Then G' satisfies Theorem 2.6. and so, A

U(RG') = 0. Consequently A^(RG) = 0.