EXTENSION OF PSEUDOCHARACTERS FROM NORMAL SUBGROUPS, II

Abstract

In the paper “Extension of pseudocharacters from normal sub-groups,” Proceedings of the Jangjeon Mathematical Society, Vol. 18, no. 4, pp. 427–433 (2015), we have proven 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 w = wσ : G → N be defined by the rule w(g) = n for g = σ(gN)n, g ∈ G, n ∈ N. Let σ(giN) = si ∈ giN, i = 1, 2, and w(gi) = ni, i = 1, 2, and thus gi = sini, i = 1, 2. This defines a mapping G/N × G/N → N such that w(s1s2) ∈ N, and we have proven (op. cit.) 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(w(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. Here we give an explicit formula for the extension (if it exists).