15
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • $\begingroup$ Right now I can't even find an example residually-$p$ for infinitely many $p$ and not RTFN. $\endgroup$
    – YCor
    Commented Feb 27, 2013 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). $\endgroup$
    – Ashot Minasyan
    Commented Feb 27, 2013 at 17:37

1 Answer 1

Reset to default
14
$\begingroup$

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 $p \in \Pi$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .