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

Acta Acad. Paed.. Agriensis, Sectio Mathematicae 28 (2001) 93-113 THE LIE AUGMENTATION TERMINALS OF GROUPS Bertalan Király (Eger, Hungary) Abstract. In this paper we give necessary and sufficient conditions for groups which have finite Lie terminals with respect to commutative ring of non-zero characteristic m, where m is a composite number. AMS Classification Number: 16D25 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 <j> : RG —• R which maps the group elements to 1. It is easy to see that as i?-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 9 - 1 (g e G). The Lie powers AW(RG) of A{RG) are defined inductively: A(RG) = AM(RG), AL

X+

1\RG) = [AW(RG),A(RG)]-RG, if A is not a limit ordinal, and AW(RG) = n AM(RG) otherwise, where [K

, M] denotes the v<\ R—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 (similarly 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. We have the following sequence A{RG) D A

2(RG) D ... of ideals of RG. Evidently there exists the least ordinal r = T

R[G] such that AW(RG) = AL

T+

1L (RG) which is called the Lie augmentation terminal (or Lie terminal for simple) of G with respect to R. In this paper we give necessary and sufficient conditions for groups which have finite Lie terminal with respect to a commutative ring of non-zero characteristic. *Research supported by the Hungarian National Foundation for Scientific Research Grant, No T025029.