Would you please construct a locally nilpotent group which is not nilpotent? An example of a locally nilpotent group which is not nilpotent?
Would you please construct a locally nilpotent group which is not nilpotent?
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0@Derek: you mean "restricted direct product" (aka direct sum). (or you have a nonstandard definition of "direct product") – 2012-10-31
2 Answers
The Fitting subgroup $\mathrm{Fit}(G)$ of a group $G$ is the product of all nilpotent normal subgroups of $G$. Since the product of finitely many nilpotent normal subgroups is again a nilpotent normal subgroup, $\mathrm{Fit}(G)$ is locally nilpotent. To find your example it therefore suffices to find a nonnilpotent group with $G=\mathrm{Fit}(G)$.
In fact, if $p$ is a prime and $E$ is an infinite elementary abelian $p$-group (i.e. $E$ is abelian of exponent $p$), then the wreath product $\mathbb{Z}_p\wr E$ satisfies this property, but the proof (and even the construction of the wreath product, if you haven't seen it before) is somewhat involved.
I'll give another example. First, two definitions.
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.
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0Thanks @YvesCornulier, I forgot to mention that. – 2012-11-01