# Subgroups of free products

Let $G$ be a free product of 2 groups with $G \neq Z_2*Z_2$ . i would like to know if the following assertion is correct :

Every almost nilpotent subgroup of G is contained in a unique maximal almost nilpotent subgroup .

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How could it be true? Take $G=C_2\ast C_3$, and consider the two subgroups $C_2$ and $C_3$. The only subgroup that contains both of them is all of $G$, which is certainly not virtually nilpotent (being virtually free and all). –  user641 May 17 '12 at 17:59

As the modular group $\,C_2*C_3\,$ is not torsion free, we cannot be sure of the existence of max. virt. nilpotent subgroups, as remarked in the above paper, yet I still don't fully understand Steve's point as why would we care about what happens to a subgroup containing two subgroups?