On Subgroups Determined by Ideals of an Integral Group Ring

Abstract

Let G be a group, ZG the integral group ring of G and I(G) its augmentation ideal. Recall that I(G) is the kernel of the ring homomorphism ∈: ZG → Z given by ∈(∑n _i g _i ) = ∑n _i , n _i ∈ Z, g _i ∈ G, and it is generated as a free Z-module by the elements g - 1, g ∈ G, g ≠ e. For n ≥ 1, Iⁿ(G) denote the nth associative power of I(G). For an ideal J of ZG, let G ∩ (1 + J) = {x ∈ G/x - 1 ∈ J}. Observe that for x, y ∈ G ∩ (1+ J), z ∈ G,

$$\begin{array}{*{20}{c}} {xy - 1 = x\left( {y - 1} \right) + x - 1 \in J} \\ {{x^{ - 1}} - 1 = - {x^{ - 1}}\left( {x - 1} \right) \in J,} \\ \end{array}$$

and

$${z^{ - 1}}xz - 1 = {z^{ - 1}}\left( {x - 1} \right)z \in J,$$

which imply that G ∩ (1 + J) is a normal subgroup of G. This subgroup is called the subgroup of G determined by the ideal J of ZG. When J = Iⁿ(G), n ≥ 1, the subgroup G ∩ (1 + Iⁿ(G)) = D _n (G) is called the nth integral dimension subgroup of G and as been well studied during the last forty years—but we donot discuss this problem here (cf. Gupta, 1987; Gupta et al., 1984; Gupta and Kuzmin-unpublished; Passi et al., 1968, 1974, 1979, 1987, 1983; and Sandling 1972a & b; the list of references for dimension subgroups is by no means exhaustive).