Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)
KIRÁLY, B., The Lie augmentation terminals of groups
96 Bertalan Király 1. G = G P i 2. G £ G' = G' P l 3. G/G P i is a nilpotent group whose derived group is a finite p;-group. Proof. Let 'pi £ II(m) and let one of the conditions hold: G — G P i or (j ^ G 1 — G Pt or G/G P i is a nilpotent group whose derived group is a finite pj-group. From (2),(3) and Theorem 2.1 it follows, that for every pi G II(m) there exists ki > 1 such that A^ k'\R P iG) = Al k' + 1\R P tG) where R P i = R/p^R. If k = max-_j {k{} , then A [k ](R P tG) = A [k+1 ]{R P iG) for all pi G n(m). Since AW(R P iG) = (A^( R.G) + p"' RG)/p™' RG for all n and every P i G n(m), then from the previous isomorphism it follows, that an arbitrary element x G can be written as x = Xi + p^cii, where Xi G A^ k+ 1\RG), a t- G RG. If mi = m/pthen mix = m,^,; since m^p"' is zero in /?. We have T : mi I x = ^ rriiXi. p,en(mj / piGn(m) Obviously m z and p\ l ' are coprime numbers and for all pi G II(m) p "' divides rrij for j / ?'. Therefore mi 91 1 the characteristic rn of the ring R are coprime numbers. Consequently Yl P ieu(m) ^ invertible in R. So x = a ^^ iri-iXi , P.snfrn) where a^ p, 6n(m) = L Hence x G A^+^ÄG) and x G A^(RG) - A^ + 1^(RG). Conversely. Let T R(G ) = n > 1, i.e. ^ = j4 n+ 1(ÄG) Then for every prime pi G II (rn) ^ AW(R P jG) = A^ k+ 1\R P iG) holds for a suitable k < n and Theorem 2.1 completes the proof.
