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What are some normal subgroups of SL$(2, \mathbb{R})$? I tried to check SO$(2, \mathbb{R})$, UT$(2, \mathbb{R})$, linear algebraic group and some scalar and diagonal matrices, but still couldn't come up with any. So can anyone give me an idea to continue on, please?

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${\rm{SL}}_2(\mathbb{R})$ is a simple Lie group, so there are no connected normal subgroups.
It's only proper normal subgroup is $\{I,-I\}$

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    Thank you, that makes alot of sense, consider I just got done proving that {I, -I} is al so the center of SL(2, R)2012-10-30