Sjogren's theorem on dimension subgroups

Abstract

Let G be a group, ZG its integral group ring, and g its augmentation ideal. The nth dimension subgroup Dn(G) of G (over Z) is defined by Dn(G) = G N (1 + gn). It is well known and not difficult to prove that Z)n(G) contains Gn, the nth term of the lower central series of G. The group Dn(G)/Gn was long known to be periodic; this follows from work of Jennings on dimension subgroups over the rationals (see [5, 11, 14, 15]). It was at one time conjectured that D n ( G ) = G n for all G and n, but this conjecture was refuted in a striking paper of Rips [17], who constructed a finite 2-group G such that [D4(G)/G4[ -2. Tahara [19,20] has constructed other groups for which certain dimension subgroups differ from the appropriate terms in the lower central series, but to the authors' knowledge, no group G has yet been constructed such that a quotient D~(G)/Gn contains a non-trival element of odd order. In a remarkable paper [18], Sjogren has shown that the periodicity of Dn(G)/Gn can be strengthened as follows: