Convergence preserving mappings on topological groups

It is well known that a mapping φ : R → R is convergence preserving, that is, whenever an infinite series ∑ a n converges, the series ∑ φ ( a n ) converges, if and only if there exists m ∈ R such that φ ( x ) = m x in some neighborhood of 0. We explore convergence preserving mappings on Hausdorff topological groups, showing in particular, that if G × G is a Fréchet group, and H has no small subgroups, then a mapping φ : G → H is convergence preserving if and only if there is a neighborhood of the identity in G on which φ is a sequentially continuous homomorphism.