If $G/H$ and $G$ are connected linear algebraic groups must $H$ also be connected?

Question

Let $k$ be a perfect field (e.g of characteristic zero) and let $G$ and $H$ be linear algebraic groups over $k$, with $H$ a normal subgroup of $G$.

If both $G$ and $G/H$ are connected, must $H$ also be?

Answer 1

First, if $G$ is connected, then $G/H$ is connected (the map $G\to G/H$ defining the ``quotient" is faithfully flat, in particular surjective). Take a connected linear algebraic group with disconnected center, e.g., $\mathrm{SL}_2$ in characteristic zero. The quotient by the center is connected, but the center isn't connected (it's étale and non-trivial). In general connected semisimple groups have finite center, and in characteristic zero, the center will be étale, hence not connected if non-trivial.