Is there any example of a linear algebraic group (over an algebraically closed field) $G$, a finite group $H$ (with discrete topology) and a group homomorphism from $G$ to $H$ which is not continuous, or at least not a morphism of (affine) varieties? If there exist such examples, is there any criterion for checking whether a given group homomorphism is also a morphism of algebraic groups?

  • $\begingroup$ Not continuous in which topology? $\endgroup$ Feb 16, 2022 at 3:51
  • $\begingroup$ @MoisheKohan Zariski topology on $G$ and $H$, where we view $H$ as a linear algebraic group via a faithful representation $H\rightarrow GL_n$. I believe that makes $H$ a discrete topological space. $\endgroup$
    – Absol
    Feb 16, 2022 at 5:36
  • $\begingroup$ You should edit your question to make this clear and also include your thoughts about the problem. $\endgroup$ Feb 16, 2022 at 10:14


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