1
$\begingroup$

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.

  • 0
    @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

1 Answers 1

2

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...)