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.
An infinite $p$-group may not be nilpotent
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$\begingroup$
abstract-algebra
group-theory
p-groups
infinite-groups
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2There is an [example](http://groupprops.subwiki.org/wiki/P-group_not_implies_nilpotent) on grouprops. – 2012-12-10
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2Tarski 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.) – 2012-12-10
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0To see that these groups are simple, see [this old answer](http://math.stackexchange.com/questions/139668/is-there-an-infinite-simple-group-with-no-element-of-order-2/139809#139809) of mine (about infinite simple groups not having elements of order $2$). – 2012-12-10
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0@user1729: I am to fresh my knowledge about nil-potency and nilpotent of groups again so sorry for asking, but Did I ask correctly that "...**may not be** nilpotent?" Shouldn't be "...**is not** nilpotent?" Thanks. – 2012-12-10
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1Hmm...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. – 2012-12-10
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0@user1729: Thanks so much for the link and advices. – 2012-12-10
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2A finitely generated nilpotent group has a finite torsion subgroup, so cannot be an infinite $p$-group. – 2012-12-10
1 Answers
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Let $G_c$ be a finite $p$-group of class $c$. Consider the direct sum $G = G_1 \oplus G_2 \oplus G_3 \oplus \ldots$