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
The Lie augmentation terminals of groups 95 If K, L are two subgroups of G, then we denote by (K, L) the subgroup generated by all commutators (g, h ) — g~ 1h~ 1gh, g G K , h G L. The n t h term of the lower central series of G is defined inductively: 71(G) = G, 72(G) = G" is the derived group (G,G) of G, and 7 n(G) = (7 n_i(G),G). The normal subgroups G P ik (k = 1,2,...) is defined by 00 »7 = 1 We have the following sequence of normal subgroups G P ii of a group G G = G' P ii 3 G P i 2 ^ • • • ^ Gp, CX) where G» — fl G„ t. In [1] the following theorem was proved. Theorem 2.1. ft be a commutative ring with identity of characteristic p n , where p a prime number. Then 1- rn[G] = 1 if and only if G — G P, 2- T R[G] — 2 if and only if G / G" = G P , 3. T R [G] > 2 if and only ifG/G p is a nilpotent group whose derived group is a finite p-group. 3. The Lie augmentation terminal It is clear, that if G is an Abelian group, then AW(RG) = 0 . Therefore we may assume that the derived group G' = 72(G) of G is non-trivial. We considere the case char R = m = p"'p^ 2 • • -P™'{ s > !)• Let Iff in) = {PI>P2, • • •, Ps } and R P i = R/p" ' R (pi G II(m)). If 9 is the homomorphism of RG onto R P lG , then by (1) (2) 9{A [ N\RG)) = A [N ] (R P L G) (3) AW{R P IG) = {AL N\RG) + PI'RG)/P"' RG. Theorem 3.1. Let G be a. non-Abelian group and R be a commutative ring with identity of non-zero characteristic rn = p" .. .p" " (s > 1) Then the Lie augmentation terminal of G with respect to R is finite if and onli if for every Pi G n(m) one of the following conditions holds:
