The equations satisfied by GGS-groups and the abelian group structure of the Gupta–Sidki group

The group Γ of p -adic automorphisms of the p -adic rooted tree can naively be identified with the p -group F p N , using pointwise addition + in the portraits of the automorphisms. We prove that the operation + is also internal for the Gupta–Sidki group, as well as for some other discrete subgroups of Γ . In order to get this, we introduce the notion of equation, or pattern, for subgroups of Γ , and we describe all equations for these groups.