6
$\begingroup$

It is well-known fact that every finite $p$-group $G$ is nilpotent. I am asking to have a counter example when $G$ is infinite $p$-group. Thanks.

$\endgroup$
  • 2
    $\begingroup$ There is an example on grouprops. $\endgroup$ – JSchlather Dec 10 '12 at 10:02
  • 2
    $\begingroup$ Tarski monsters! A Tarski monster is a finitely-generated, infinite $p$-group such that every proper, non-trivial subgroup is cyclic of order $p$. It is not too difficult to see that such a group is simple, and as it clearly isn't abelian. Thus, not nilpotent. Tarski monsters exist for $p>>1$, which is a result from the early 80s by Ol'shanskii. He has written a book, but it is translated from the Russian and I was told it is "unreadable" by someone much cleverer than myself. (The book is called "Geometry of Defining Relations in Groups" c1991 if you are interested.) $\endgroup$ – user1729 Dec 10 '12 at 10:17
  • $\begingroup$ To see that these groups are simple, see this old answer of mine (about infinite simple groups not having elements of order $2$). $\endgroup$ – user1729 Dec 10 '12 at 10:22
  • 1
    $\begingroup$ Hmm...certainly, there are infinite abelian $p$-groups (for example, prufer quasi-cyclic groups), so there are infinite nilpotent $p$-groups. However, infinite abelian $p$-groups are necessarily infinitely generated. So, the question is now "does there exist an infinite, finitely-generated nilpotent $p$-group". I do not know. This may, in fact, be open...our knowledge of finitely-generated torsion groups is very patchy. You might want to look up the work of Mark Sapir, who was a student of Ol'shanskii, or you could even look up Ol'shanskii himself. Or maybe the pro-$p$ people know about this. $\endgroup$ – user1729 Dec 10 '12 at 12:25
  • 2
    $\begingroup$ A finitely generated nilpotent group has a finite torsion subgroup, so cannot be an infinite $p$-group. $\endgroup$ – Derek Holt Dec 10 '12 at 16:55
7
$\begingroup$

Let $G_c$ be a finite $p$-group of class $c$. Consider the direct sum $G = G_1 \oplus G_2 \oplus G_3 \oplus \ldots$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.