Dimension Subgroups

Let ℤ[G] be the integral group ring of a group G and let g be its augmentation ideal. For each natural number n ≥ 1, D _n (G) = G ∩ (1 + gⁿ) is a normal subgroup of G called the nth integral dimension subgroup of G. It is easy to see that the decreasing series

G = D ₁ (G) ⊇ D ₂ (G) ⊇ . . . ⊇ D _n (G) ⊇ . . .

is a central series in G, i.e., [G, D _n (G)] ⊑ D_n+1(G) for all n ≥ 1. Therefore, γ _n (G) ⊑ D _n (G) for all n ≥ 1, where γ _n (G) is the nth term in the lower central series of G. The identification of dimension subgroups, and, in particular, whether γ _n (G) = D _n (G), has been a subject of intensive investigation for the last over fifty years. It is now known that, whereas D _n (G) = γ _n (G) for n = 1, 2, 3 for every group G (see [Pas79]), there exist groups G whose series {D _n (G)}_n>-1 of dimension subgroups differs from the lower central series {γ _n (G)}_n≥1 ([Rip72], [Tah77b], [Tah78b], [Gup90]). The various developments in this area have been reported in [Pas79] and [Gup87c]. In the present exposition, we will primarily concentrate on the results that have appeared since the publication of [Gup87c]. We particularly focus attention on the fourth and the fifth dimension subgroups. We recall the description of the fifth dimension subgroup due to Tahara (Theorem 2.29) and give a proof of one of his theorems which states that, for every group G, D₅(G)⁶ ⊑ γ₅(G) (see Theorem 2.27). The proof here is, hopefully, shorter than the original one.