Group-norms are vector-space norms but with the scalars restricted to units (invertibles), ±1. The Birkhoff–Kakutani theorem (a first-countable Hausdorff topological group has a right-invariant metric) we view as a normability theorem rather than a metrization theorem, a relative of Kolmogorov’s normability theorem for topological vector spaces (the condition for whose normability is that the origin have a convex bounded neighbourhood). The groups here need not be abelian, so one has left-sided and right-sided versions. Proved here is the Analytic Baire Theorem: if a normed group contains an (either-sided) non-meagre analytic set, it is Baire, separable and (modulo a meagre set) itself analytic. Other results here include the ‘Analytic Shift Theorem’ and the ‘Squared Pettis Theorem’, category relatives of the Steinhaus Difference Theorem.