On $\kappa$-Frechet-Urysohn topological groups

We characterize $\kappa$-Fréchet–Urysohn topological groups. Using this characterization we show that: (1) a hemicompact topological group is $\kappa$-Fréchet–Urysohn iff it is locally compact, and (2) if $F$ is a closed metrizable subspace of a topological vector space (tvs) $E$ such that the quotient $E\mathrm{/}F$ is a $\kappa$-Fréchet–Urysohn space, then also $E$ is a $\kappa$-Fréchet–Urysohn space. Consequently, the product of a $\kappa$-Fréchet–Urysohn tvs and a metrizable tvs is a $\kappa$-Fréchet–Urysohn space. Under Martin’s Axiom, we construct a countable Boolean $\kappa$-Fréchet–Urysohn group which is not a $k_{\mathbb{R}}$-space.