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.
Following Jug's suggestion: let $\,\,A:=C_3=\langle a\rangle\,,\,B:=C_2=\langle c\rangle\,$ , with the regular action of $\,B\,$ on itself, and form the (regular) wreath product $$A\wr B\cong \left(C_3\times C_3\right)\rtimes_R C_2$$ Take the elements $$\pi=((1,1),c))\,\,,\,\,\sigma=((a,a^2),1)$$It's now easy to check that$$\pi^2=\sigma^3=1\,\,,\,\,\pi\sigma\pi=\sigma^2$$ so we get that $\,\,\langle \pi\,,\,\sigma\rangle\cong S_3\,\,$ and thus $\,\,A\wr B\,\,$ can't be nilpotent, though both $\,\,A,B\,\,$ are (they're even abelian...)