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This is the problem:

Exhibit three reductive groups of dimension $4$ and semisimple rank $1$ which are pairwise nonisomorphic (as algebraic groups).

I know that for any reductive group $G$ of semisimple rank $1$, $G = (G,G)Z$, where $(G, G)$, the derived subgroup of $G$ is semisimple, of dimension $3$, and $Z$ is the identity component of the centre of $G$.

As the dimension of $G$ is $4$, it is clear that the dimension of its cetre is $1$.

I think $GL(2,\mathbb{C})$ is an easy example. Other such groups might also appear as matrix groups. But I have no idea as to how to construct them.

Would you please tell me a method or give me some advice? Thanks in advance~

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    The center of $GL_2(k)$ is $k^\times$, the center of $SL_2(k)\times GL_1(k)$ is (isomorphic to) $(\mathbb Z/2\mathbb Z)\times k^\times$ (at least when the characteristic of $k$ is not $2$). The two are not isomorphic (since there is a surjective morphism from one to the other with nontrivial kernel).2012-02-15

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