# An infinite $p$-group may not be nilpotent

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.

• There is an example on grouprops. – JSchlather Dec 10 '12 at 10:02
• 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.) – user1729 Dec 10 '12 at 10:17
• To see that these groups are simple, see this old answer of mine (about infinite simple groups not having elements of order $2$). – user1729 Dec 10 '12 at 10:22
• 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. – user1729 Dec 10 '12 at 12:25
• A finitely generated nilpotent group has a finite torsion subgroup, so cannot be an infinite $p$-group. – Derek Holt Dec 10 '12 at 16:55

## 1 Answer

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