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 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.
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$\begingroup$ The language here is certainly subtle. Wikipedia has some useful information but is rarely a complete source and sometimes gets things wrong. SGA3 (now in an edited online version math.jussieu.fr/~polo/SGA3 ) should be authoritative, though the books by Demazure-Gabriel and more recently Conrad-Gabber-Prasad are quite useful. $\endgroup$– Jim HumphreysCommented Mar 27, 2014 at 15:08
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$\begingroup$ Sorry, I think I am responsible for that claim appearing some years ago. I think the argument in my head at the time was that the reduced subscheme is uniquely maximal with respect to the reduced property, and that this uniqueness makes it invariant under automorphisms, hence a characteristic subgroup. $\endgroup$– S. Carnahan ♦Commented Mar 28, 2014 at 9:23
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$\begingroup$ @Carnahan Yes that argument is true. The problem is that a characteristic subgroup in this weak sense that it is invariant under automorphisms does not imply normality. The maybe more useful definition of characteristic subgroup is to require that it remains a characteristic subgroup for any base extension. $\endgroup$– anonymousCommented Mar 28, 2014 at 10:06
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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 $G$.
answered Mar 27, 2014 at 14:00
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$\begingroup$ A variant of this example that arises "in nature" is to consider the relative Frobenius isogeny $F:H \rightarrow H^{(p)}$ for a nontrivial connected semisimple group $H$ over a perfect field $k$ of characteristic $p > 0$ and let $G$ be the schematic preimage $F^{-1}(T)$ for a maximal $k$-torus $T \subset H^{(p)}$. Then $G_{\rm{red}}$ is a maximal $k$-torus of $H$ but it is not normal in $G$. For $H = {\rm{PGL}}_2$ this nearly recovers Moret-Bailly's answer, but using two $\alpha_p$'s (on which $\mathbf{G}_m$ respectively acts by usual scaling and its composition with inversion). $\endgroup$ Commented Mar 27, 2014 at 18:35
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$\begingroup$ @user76758: in fact, you get exactly my example "in nature" with your construction by taking for $H$ the group of affine automorphisms of the line (except this $H$ is not semisimple). $\endgroup$ Commented Mar 27, 2014 at 22:04
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$\begingroup$ @LaurentMoret-Bailly: Good point! $\endgroup$ Commented Mar 27, 2014 at 23:27