Let $S$ be some base scheme, $H$ a finite flat group scheme over $S$, and $\alpha: \mu\_p \to H$ a homomorphism of group schemes ($p$ a prime). Is the kernel of $\alpha$ necessarily flat over $S$?


(I know that kernels of general homomorphisms of FFGS $G \to H$ need not be flat, but I don't know of a counterexample when $G = \mu\_p$.)



Kernels of homomorphisms of group schemes
This holds for any homomorphism $f: G\to H$ with $G$ of multiplicative type and of finite type, and $H$ separated and finitely presented. Here I assume that by "finite flat" you mean "finite locally free". 


Reference: SGA3, IX, Thm 6.8. 


