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.
2026-03-27 18:34:47.1774636487
An infinite $p$-group may not be nilpotent
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Let $G_c$ be a finite $p$-group of class $c$. Consider the direct sum $G = G_1 \oplus G_2 \oplus G_3 \oplus \ldots$