Metrizability and the Fréchet-Urysohn property in topological groups

A question of Arhangel’skii, whether weakly first countable topological groups are metrizable, is answered in two ways: if the Hausdorff axiom is assumed, the answer is yes, but in general a weakly first countable topological group need not be pseudometrizable. The former result is obtained as a corollary of a more general sufficient condition for a sequential group to be Fréchet-Urysohn. A general necessary and sufficient condition for a sequential group to be Fréchet-Urysohn is given, and a number of questions are raised. Examples are given to show in what respect the theorems of the paper are the "best possible".