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Does a local commutative algebraic group G be commutative ?

Here local commutative means for any point g, there is an open set U containing g in the algebraic group G and for any x and y belong to U, we have xy=yx.

If G is connected group, it is clear. How about the common situation?

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Any finite group is rather vacuously an 0-dimensional algebraic group and such groups are locally commutative, so the answer is "no". –  David Loeffler Dec 9 '12 at 14:20
Oh, yes, maybe I should suppose the algebraic group has positive dimension. –  Strongart Dec 11 '12 at 11:48
Right, but then you could take the direct product of an arbitrary finite group by a bunch of copies of $\mathbb{G}_a$ and the same would happen. –  David Loeffler Dec 11 '12 at 16:17
Oh, it happens, but I do not like this type of examples. –  Strongart Dec 15 '12 at 15:29
Any algebraic group is an extension of a finite group by a connected group, so there really aren't any other examples to consider. –  David Speyer Feb 11 '13 at 21:03

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