The Augmentation Terminals of Certain Locally Finite Groups

Abstract

Let G be a group and ZG be the integral group ring of G. We shall write 𝔤 for the augmentation ideal of G; that is to say, the kernel of the homomorphism of ZG onto Z which sends each group element to 1. The powers gλ of 𝔤 are defined inductively for ordinals λ by 𝔤λ = 𝔤μ𝔤, if λ = μ + 1, and otherwise. The first ordinal λ for which gλ = 𝔤λ+1 is called the augmentation terminal or simply the terminal of G. For example, if G is either a cyclic group of prime order or else isomorphic with the additive group of rational numbers then gn > 𝔤ω = 0 for all finite n, so that these groups have terminal ω.The groups with finite terminal are well-known and easily described. If G is one such, then every homomorphic image of G must also have finite terminal.