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Let G be a connected linear algebraic group. is there a unique closed normal maximal subgroup in G ?

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of course not: in $PSL_2\times PSL_2$ (in the place of $PSL_2$ you may insert any simple group) for example you have at least two closed maximal normal subgroups –  zroslav Jun 15 '11 at 19:11
No. Try $G = \mathbf{G}_a^2$ over $\mathbf{C}$, say. –  Konstantin Ardakov Jun 15 '11 at 19:11
Or $C^*$ which has no maximal subgroups closed subgroups! –  Bugs Bunny Jun 15 '11 at 21:25
Please see the "how to ask" page before asking another question. –  S. Carnahan Jun 16 '11 at 5:26

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