A well-known theorem of Gruenberg implies that a finitely generated residually torsion-free nilpotent group is residually $p$-finite for all primes $p$. What about the converse?

Question: Are there any examples of a finitely generated group $G$ with the properties that $G$ is residually $p$-finite for all primes $p$, but $G$ is not residually torsion-free nilpotent?

  • $\begingroup$ Right now I can't even find an example residually-$p$ for infinitely many $p$ and not RTFN. $\endgroup$
    – YCor
    Feb 27 '13 at 17:16
  • $\begingroup$ @Yves: Neither can I. For any finite collection of primes $\Pi$, ${\rm SL}(3,\mathbb{Z})$ has a finite index subgroup $K$ which is residually-$p$ for each $p \in\Pi$. But of course $K$ is not RTFN, as it has property (T). $\endgroup$ Feb 27 '13 at 17:37

Browsing through the archive of solved problems of Kourovka Notebook, I accidentally saw that the same question was asked by Yu.V. Kuz'min in 1999 (see question 14.52). Apparently the required examples were produced a long time ago by B. Hartley (Hartley, B., `On residually finite p-groups.' Symposia Mathematica, Vol. XVII (Convegno sui Gruppi Infiniti, INDAM, Roma, 1973), pp. 225–234. Academic Press, London, 1976. MR0419611 (54 #7629)).

Hartley showed that for any set of primes $\Pi$ there exists a $3$-generated center-by-metabelian group $G$ which is not residually torsion-free nilpotent and which is residually $p$-finite iff $p \in \Pi$.


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