Suitable sets for subgroups of the σ-product of discrete groups

Determine whether every subgroup H of the σ-product G of the product ∏_{α<τ} G_α (each G_α a non-trivial discrete group), endowed with the subspace group topology 𝔽_G from the Tychonoff product topology, has a generating suitable set.

Background

The σ-product G of a family of discrete groups {G_α} is considered as a subgroup of the full product ∏{α<τ} Gα, carrying the subspace group topology 𝔽_G. The question asks whether every subgroup of this σ-product possesses a generating suitable set.

The authors recall this as an open problem from Sanchis and Tkachenko (2002) and later provide results addressing it (e.g., Theorem t4).