Let $G$ be any algebraic subgroup of $\mathrm{GL}\_n$ over an algebraically closed field of any characteristic.


If $s$ is a semisimple element of $G$, can the $G$-conjugacy class of $s$ fail to be closed in $G$?



Is a semisimple conjugacy class closed?
No---it's always closed. See 18.2 in Humphreys, *Linear Algebraic Groups*.


