Rational points do not detect the group-scheme structure of alpha_p and mu_p

For every commutative $k$-algebra $R$, ${\alpha }_p(R)=\{a\in R:a^p=0\}$ is an additive subgroup of $R$: $(a+b{)}^p=a^p+b^p$ and $(-a{)}^p=(-1{)}^p{a}^p$. Also ${\mu }_p(R)=\{u\in R^{\times }:u^p=1\}$ is a multiplicative subgroup of $R^{\times }$. The closed immersions into ${\mathbf{G}}_a$ and ${\mathbf{G}}_m$ therefore give the induced group laws by [F1]–[F2]. AC is carried through the criterion in [F1], as recorded in [F4]. Each coordinate algebra has dimension $p$ over $k$, so is finite type. The formula $t=1+x$ identifies their underlying rings because $(1+x{)}^p-1=x^p$. In a field $a^p=0$ forces $a=0$, and $t^p=1$ forces $(t-1{)}^p=0$, hence $t=1$. Thus both rational-point groups are singleton while both schemes retain a nonzero nilpotent coordinate.

Every group-scheme homomorphism $f:{\alpha }_p\to {\mathbf{G}}_m$ corresponds by [F3] to a unit $g(x)={\sum }_{j=0}^{p-1}{c}_j{x}^j$ in $k[x]\mathrm{/}(x^p)$ satisfying $g(0)=1$ and $g(x+y)=g(x)g(y)$ in $k[x,y]\mathrm{/}(x^p,y^p)$. Compare coefficients of $x^{r-1}y$ for $1\le r<p$: the left side has coefficient $r{c}_r$, the right side $c_{r-1}{c}_1$. Since $c_0=1$ and $1,\dots ,p-1$ are invertible in $k$, induction gives $c_r=c_1^r\mathrm{/}r!$. Now compare the coefficient of $x^{p-1}y$: the left side is zero because every term of $g(x+y)$ has total degree less than $p$, while the right side is $c_{p-1}{c}_1=c_1^p\mathrm{/}(p-1)!$. Thus $c_1=0$ and all $c_r=0$ for $r>0$. This also covers $p=2$. Hence every such homomorphism is the trivial one, $g=1$.

If ${\alpha }_p\cong {\mu }_p$ as group schemes, compose that isomorphism with the closed subgroup inclusion ${\mu }_p\hookrightarrow {\mathbf{G}}_m$. The result would be nontrivial: on coordinate rings the inclusion pulls $t$ back to its nonconstant class in $k[t]\mathrm{/}(t^p-1)$, and an isomorphism cannot send $t-1\mathrm{\ne }0$ to zero. This contradicts step 2.1. Thus the two group schemes are not isomorphic despite their isomorphic underlying schemes and rational-point groups. Nilpotent test algebras distinguish their laws; for example their common coordinate $x=t-1$ has the additional product term $x\otimes x$ for ${\mu }_p$ written in the counterexample.