Would you please construct a locally nilpotent group which is not nilpotent? Would you please construct a locally nilpotent group which is not nilpotent?
An example of a locally nilpotent group which is not nilpotent?

I'll give another example.  First, two definitions.


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*The Prüfer $p$-group or $p$-quasicyclic group, denoted $\mathbb{Z}_{p^{\infty}}$ is the group defined by all complex $p^n$th roots of unity for any $n\in \mathbb{N}$.  (Note that this is the Pontryagin dual group to the $p$-adic integers.)


*The generalized dihedral group of an abelian group $H$, denoted $\text{Dih}(H)$, is the semidirect product of $H$ by $\mathbb{Z}_2$, where $\mathbb{Z}_2$ acts by inversion on elements of $H$.  In other words, in $\text{Dih}(H)=H\rtimes \mathbb{Z}_2$, $(h_1,0)(h_2,a)=(h_1h_2,a)$ and $(h_1,1)(h_2,a)=(h_1h_2^{-1},a\oplus1)$.

The example: $\text{Dih}(Z_{2^\infty})$ is locally nilpotent but not nilpotent.
