Abstract:
Let $\mathfrak{P}$ be a non-empty set of primes. We prove that any $\mathfrak{P}$-bounded nilpotent group is $\mathfrak{P}$-potent, and the tree product $T$ of a finite number of $\mathfrak{P}$-bounded nilpotent groups with proper locally cyclic edge subgroups is residually a finite $\mathfrak{P}$-group if and only if any vertex group of $T$ has no $\mathfrak{P}^{\prime}$-torsion and any edge subgroup of $T$ is $\mathfrak{P}^{\prime}$-isolated in the vertex group containing it. We prove also that the tree product of a finite number of groups with locally cyclic edge subgroups is residually a finite $p$-group if all its vertex groups have this property and any edge subgroup is separable in the corresponding vertex group by the class of finite $p$-groups.
Keywords:potent group, nilpotent group, residual finiteness, residual $p$-finiteness, residual solvability, generalized free product, tree product, fundamental group of a graph of groups.
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