Let $A$ be a (semi)abelian variety over an algebraically closed field $K$, and $X$ be a closed irreducible subvariety. Can $X$ have a nontrivial finite stabilizer ? By stabilizer, I mean the closed subgroup $S\subset A$ such that $X$ is stable by translation by any point in $S$. This question is equivalent to the one in the title since you can always take the quotient by the neutral component of $S$ to reduce to a finite group.
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No. Take a curve in its Jacobian and pull it back by multiplication by some n. The resulting pullback is a curve invariant by the n torsion. 

