Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G\_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us assume that $k$ is perfect. In general, $G\_{\operatorname{red}}$ is not normal in $G$ (See SGA 3, VI$\_A$, 0.2). The Wikipedia page on [groupschemes](http://en.wikipedia.org/wiki/Group_scheme) claimes that $G\_{\operatorname{red}}$ is normal in $G$ if $G$ is connected. I would be very thankful if somebody could point me to a proof of this.



Is $G_{\operatorname{red}}$ normal in $G$?
The claim is false. Over a field of characteristic $p>0$, take for $G$ the semidirect product $\alpha\_p\rtimes\mathbb{G}\_\mathrm{m}$ where $\mathbb{G}\_\mathrm{m}$ acts on $\alpha\_p$ by scaling. Then $G$ is connected but $\mathbb{G}\_\mathrm{red}=\{0\}\times \mathbb{G}\_\mathrm{m}$ is not normal in $G$.


