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A connected group variety is simply connected if every multiplicative isogeny from a connected group variety to it is an isomorphism. (Taken from p.388 of Milne's book on algebraic groups) I would imagine that the answer is yes, but don't know how to prove it.

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For semisimple groups the answer is yes. This can be seen from the fact that they are the semisimple groups whose character group is the weight lattice.

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