A group scheme is called *multiplicative group scheme* if the following equivalent conditions are satisfied: 1. $G\otimes_k k_s$ is diagonalizable. 1. $G\otimes_k K$ is diagonalizable for a field $K\in M_k$. 1. $G$ is the Cartier dual of an étale $k$-group. 1. $\hat D(G)$ is an étale $k$-formal group. 1. $Gr_k(G,\alpha_k)=0$ 1. (If $p\not =0)$, $V_G$ is an epimorphism 1. (If $p\not =0)$, $V_G$ is an isomorphism