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
94 Bertalan Király 2. Notations and some known facts If H is a normal subgroup of G, then I(RH ) (or 1(H) for short) denotes the ideal of RG generated by all elements of the form h — 1 (h £ H). It is well known that I (RH ) is the kernel of the natural epimorphism (f> : RG RG / H induced by the group homomorphism <f) of G onto G/H. It is clear that I(RG) = A(RG). Let F be a free group on the free generators Xi(i £ /), and ZF be its integral group ring (Z denotes the ring of rational integers). Then every homomorphism 4> : F —• G induces a ring homomorphism ^ : ZF —+ RG by letting <J>(Y2n yy) = ^2n y(fi(y), where y £ F and the sum runs over the finite set of n yy £ ZF. If / £ ZF , we denote by Af(RG) the two-sided ideal of RG generated by the elements </>(/), 4> £ Hom(F, G), the set of homomorphism from F to G. In other words Aj(RG) is the ideal generated by the values of / in RG as the elements of G are substituted for the free generators X{-s. An ideal J of RG is called a polynomial ideal if J = Aj(RG) for some / £ ZF, F a free group. It is easy to see that the augmentation ideal A(RG) is a polynomial ideal. Really, A(RG) is generated as an R—module by the elements g — 1 (g £ G), i.e. by the values of the polynomial x — 1. Lemma 2.1. ([2], Corollary 1.9, page 6.) The Lie powers A^(RG)(n > 1) are polynomial ideals in RG. We use the following lemma, too. Lemma 2.2. ([2] Proposition 1.4, page 2.) If f £ ZF, then f defines a polynomial ideal Af(RG) in every group ring RG. Further, if 9 : RG — KH is a ring homomorphism induced by a group homomorphism <p '. G II and a ring homomorphism ip : R —- K , then 0(Aj(RG )) C Aj(KH). (It is assumed that IJJ(1R) = 1 K, where 1/? and 1/^- are identity of the rings R and K , respectively.) Let 0 : RG —RjLG be an epimorphism induced by the ring homomorphism 6 of R onto R/L. By Lemma 2.1 A^(RG)(n > 1) are polynomial ideal and from Lemma 2.2 it follows that (1) 6(A [n ](RG)) = A [n ](R/LG). Let p be a prime and n a natural number. In this case let's denote by G p " the subgroup generated by all elements of the form g p (g £ G).
