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The aim of this paper is to describe the structure of the finitely generatedsubgroups of a family of branch groups, which includes the first Grigorchukgroup and the Gupta-Sidki 3-group. This description is made via the notion ofblock subgroup. We then use this to show that all groups in the above familyare subgroup separable (LERF). These results are obtained as a corollary of amore general structural statement on subdirect products of just infinitegroups.

Finitely generated subgroups of branch groups and subdirect products of just infinite groups