EXTENSION OF PSEUDOCHARACTERS FROM NORMAL SUBGROUPS

Abstract

We prove general theorems describing the conditions ensuring that a given pseudocharacter on a normal subgroup of a given group can be extended to a pseudocharacter on the group itself and the conditions under which this extension is unique. Namely, let G be a group, let N be a normal subgroup of G, let σ : G/N → G be a section (σ(gN) ∈ gN for every g ∈ G; we also assume that σ(N) = eN = eG), and let ω = ωσ : G → N be defined by the rule ω(g) = n for g = σ(gN)n, g ∈ G, n ∈ N. Let σ(giN) = si ∈ giN, i = 1, 2, and ω(gi) = ni, i = 1, 2, and thus gi = sini, i = 1, 2. This defines a mapping G/N × G/N → N such that ω(s1s2) ∈ N. We prove that an extension of a given pseudocharacter f on N to a pseudocharacter on |G exists if and only if there is a section σ for which the set {f(ω(s1s2)), si ∈ G/N; i = 1, 2} is bounded, and this extension is unique if and only if the group G/N admits no nontrivial pseudocharacters.