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What is some normal subgroups of SL(2, R)? I tried to check SO(2, R), UT(2, 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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up vote 1 down vote accepted

${\rm{SL}}_2(\mathbb{R})$ 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) –  Akaichan Oct 30 '12 at 23:37
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