TITLE: An infinite $p$-group may not be nilpotent
QUESTION [7 upvotes]: 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.

REPLY [9 votes]: Let $G_c$ be a finite $p$-group of class $c$. Consider the direct sum $G = G_1 \oplus G_2 \oplus G_3 \oplus \ldots$