On the length of chains of proper subgroups covering a topological group

We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-σ-bounded topological group G admits an increasing chain 〈Gα : α < b(L)〉 of its proper subgroups such that: (i) ⋃ α Gα = G; and (ii) For every σ-bounded subgroup H of G there exists α such that H ⊂ Gα. In case of the group Sym(ω) of all permutations of ω with the topology inherited from ω this improves upon earlier results of S. Thomas.