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Is the wreath product of two nilpotent groups always nilpotent?

I know the answer is no due to a condition "The regular wreath product A wr B of a group A by a group B is nilpotent if and only if A is a nilpotent p-group of finite exponent and B is a finite p-group for the same prime p ", but I can't easily construct a counter example to show it.

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    Take any p-group $P$ and q-group $Q$, and then $P\wr Q$ is not nilpotent, as it has no normal Sylow q-subgroup.2012-05-31
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    @SteveD: You should mention your implicit assumption $p\ne q$ ;-). To user31899: To be even more concrete than Steve, take $A$ and $B$ as simple as possible, but violating the condition after "if and only if", e.g., take something like $A=C_3$, $B=C_2$.2012-05-31

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