2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65704Let $G$ be a locally compact group. We show that its Fourier algebra $A(G)$ is amenable if and only if $G$ has an abelian subgroup of finite index, and that its Fourier-Stieltjes algebra $B(G)$ is amenable if and only if $G$ has a compact, abelian subgroup of finite index. We then show that $A(G)$ is weakly amenable if the component of the identity of $G$ is abelian, and we prove some partial results towards the converse.16 pages; some, hopefully clarifying revisionsFunctional AnalysisOperator Algebras22D25, 22E99, 43A30, 46H20 (primary), 46H25, 47L50text