Topologies on groups determined by sets of convergent sequences

A Hausdorff topological group (G,τ) is called an s-group and τ is called an s-topology if there is a set S of sequences in G such that τ is the finest Hausdorff group topology on G in which every sequence of S converges to the unit. The class S of all s-groups contains all sequential Hausdorff groups and it is finitely multiplicative. A quotient group of an s-group is an s-group. For a non-discrete topological group (G,τ) the following three assertions are equivalent: (1) (G,τ) is an s-group, (2) (G,τ) is a quotient group of a Graev free topological group over a metrizable space, (3) (G,τ) is a quotient group of a Graev free topological group over a sequential Tychonoff space. The Abelian version of this characterization of s-groups holds as well.