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 67 m k-l , nv where Fk(h ) = — and — ( . )• Since is zero in i=l j=i ^ ' i?, we have that h — 1 E ^^(ÄG) which imphes the inclusion I(G P }k) C and completes the proof of the lemma. Lemma 3.2. Let R be a commutative i ing of characteristic p s . Then A [ u\RG) = I(G P). Proof. From (3) and from Lemma 3.1 the inclusion I(G V) C A^(RG) follows. We can readily verify that G/G p is the residually-£> p group and by Theorem 2.2 A [u ]{RG/G p) = 0. (5) By (1) 4>{A^ N\RG)) = AW(RG/G P) for all n > 1, where f : RG RGJG V the natural epimorphism induced by the group homomorphism 4> of G onto G/G P. Consequently f{A^{RG)) C A^{RG/G P) for all n and therefore <F>(AW(RG)) C Then from the isomorphism RG/G P RG/I(G P) and from (5) we conclude that AM (ä G ) C /(G p). Therefore AM(RG) = I(G P). This completes the proof of the lemma. If G is a nilpotent group with a finite p-group as the commutator subgroup and R a commutative ring of characteristic p s then the ideal A(RG) is Lie nilpotent (see Theorem 2.1). Denote R°[A(RG)] the Lie nilpotency index of A(RG) i.e. the natural number n for which A^ N~^(RG) ± A^(RG) = 0 holds. If G = (1) we put T°[A(RG)] = 1. Let r p[G] denote the smallest natural number k (if it exists) such that Gp,k-i / G P ik — • •' — G p. Theorem 3.1. Let R be a commutative ring of characteristic p s. Then: 1) T r[G] = 1 if and only if G = G P, 2) T r[G] = 2 if and only if G £ G' = G P, 3) Tr[G] > 2 if and only if GJG P is a nilpotent group whose derived group is a finite p-group. Proof. The statement 1) follows from Lemma 3.2. 2) Let T r[G] = 2, i.e. A(RG) Í AW{RG) = A^(RG) = ••• = A^(RG). By statement 1) of our theorem G p / G and consequently G / G' . Because G/G' is an Abelian group, A^(RG/G') = 0. From the isomorphism AW{RG/G') = (A^(RG) + I{G'))/I(G'l
