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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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    Dear ShinyaSakai, You should give some more context. E.g. what field are you working over? (Your example is over $\mathbb C$; is this the general context in which you are working?) Regards,2012-02-13
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    @Matt E: Thanks for the comment. The examples could be based on any algebraically closed fields. Regards,2012-02-13
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    I would guess $GL_2(k)$, $SL_2(k)\times k^\times$, and $GL_2(k)/\pm I$ (the centers are all different). This doesn't work for characteristic $2$, but it can probably be saved by messing around with other finite subgroups of the center...2012-02-15
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    @B R: Thanks for the comment. Is $k^\times$ the multiplication group of nonzero elements of $k$? If it is, I think $SL_2(k) \times k^\times$ is the same as $GL_2(k)$, since $k$ is algebraically closed.2012-02-15
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    Yes, I should have actually written $GL_1(k)$. The center of $SL_2(k)$ is $\pm I$, so the center of $SL_2(k)\times k^\times$ is $\pm I\times k^\times$.2012-02-15
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    @B R: Thanks for the comment. Arn't the two all the same? The center of $GL(2,k)$ is also $\pm I\times k^\times$. ($I\times k^\times$)2012-02-15
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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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