When is Cartier dual of a finite group etale?

Question

I am trying to solve the following exercise from Waterhouse: Introduction to affine group schemes (Chapter 6, Ex. 12 on page 53) without any success.

Let $char(k)=p >0$ and let $G$ be an abelian etale finite group scheme. Show that $G^D$ (Cartier dual of $G$) is etale iff $\dim_k(k[G])$ is prime to $p$.

I can see this if $G=\mathbb Z/n$, but not in general. Can someone help?

Answer 1

You can reduce to the case when $G=\underline{\mathbb{Z}/n\mathbb{Z}}$ easily by noting that

a) Cartier duality commutes with with base change.

b) Over a separably closed field every etale group scheme is constant.

c) Cartier duality commutes with products.