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* BERTALAN KIRÁLY Abstract. In this paper we give necessery and sufficient conditions for groups which have finite Lie terminals with respect to commutative ring of characteristic p a where p is a prime and s is a natural number. 1. Introduction. Let R be a commutative ring with identity, G a group and RG its group ring and let A(RG) denote the augmentation ideal of RG, that is the kernel of the ring homomorphism </> : RG —> R which maps the group elements to 1. It is easy to see that as Ä-module A(RG) is a free module with the elements g — 1 (g £ G) as a basis. It is clear that A{RG) is the ideal generated by all elements of the form g — l,g £ G. The Lie powers A^(RG) of A{RG) are defined inductively: A(RG) = A^(RG), Al x +V(RG) = [AW(RG), A(RG)]-RG,if Ais notalimit ordinal, and A^(RG) = n A^(RG) otherwise, where [Ii, M] denote the V< A i?-submodule of RG generated by [k, m] = km — mk, k £ K,m £ M , and for K C RG, K • RG denotes the right ideal generated by K in RG (similary RG • K will denote the left ideal generated by K). It is easy to see that the right ideal A^(RG) is a two-sided ideal of RG for all ordinals A > 1. Evidently there exists a least ordinal T — T r [G] such that A^(RG) = A^ T+ l\RG) which is called the Lie augmentation terminal (or Lie terminal for simple when it is obvious from the context what ring R we are working with) of G with respect to R. If G = (1) we put T R[G] = 1. In general, the question of the classification of groups in regarding to values of the Lie terminals and also of the computation of these terminals, is far from being simple. We are primarily concerned with finding all groups whose the Lie terminals with respect to commutative ring of characteristic p s are finite. In this paper we give necessery and sufficient conditions for groups which have finite Lie terminals with respect to commutative ring of characteristic p s where p is a prime and 5 is a natural number (Theorem 3.1). Research supported by the Hungarian National Foundation for Scientific Research Grant, N-T 4265 and N°T 16432.
