Given an algebraic group $G$ over an algebraically closed field $K$, if $H$ is a subvariety of $G$, then is $H$ a subgroup of $G$? This seems rather strong. If it is indeed false, is there a geometric characterization of when $H$ is a subgroup of $G$?
2026-03-30 05:12:24.1774847544
Subvariety of an Algebraic Group.
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Just to get this off the unanswered list.
No, it is not true that a subvariety of an algebraic group is an algebraic group. Moreover, there is no real geometric quality that guarantees this. Namely, note that $\mathbb{G}_a^n$ is an algebraic group, and the subvarieties of this (as $n$ varies) are precisely the affine varieties. There is no geometric condition that guarantees that an arbitrary affine variety is even a group scheme.