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# Is the free product of arbitrarily many abeliangroupscopiesof${\mathbb{Z}}$and${\mathbb{Z}}/2$ residually nilpotent?

A group is residually nilpotent if the intersection of the terms in its lower central series is the trivial group.

Is the free product of arbitarily (possibly infinite) many abelian groups residually nilpotent? In particular, this would imply that the free product of abelian groups with any free group is residually nilpotent.

Later: Thank you for all the comments and answers. It is good to know that the free product of abelian groups is residually solvable.

The group I am interested in is the free product $F*G$ where $F$ is a free group and $G$ is a free product of arbitrarily many ${\mathbb{Z}}/2$. In the special case ${\mathbb{Z}}*{\mathbb{Z}}/2$, I know that this group is residually nilpotent. This follows from a result of Mal'cev on adjoint groups of residually nilpotent algebras.

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# Is the free product of arbitrarily many abelian groups residually nilpotent?

A group is residually nilpotent if the intersection of the terms in its lower central series is the trivial group.

Is the free product of arbitarily (possibly infinite) many abelian groups residually nilpotent? In particular, this would imply that the free product of abelian groups with any free group is residually nilpotent.