# Analog between groups and Hopf algebras

"Subgroups" of a group correspond to "left coideal subalgebras" of a Hopf algebra. Why "subgroups" do not corresponds to "Hopf subalgebras" but "left coideal subalgebras"?

The downsides are notable. It becomes rather non-trivial to define a notion of short exact sequence, for example. While you can map a normal Hopf subalgebra of $H$ to a normal coideal subalgebra via $K\mapsto HK^+$, this is in general not an injective map. So naive attempts at defining short exact sequences run into the issue that they're not particularly well-defined, as the kernels aren't solely dependent on the Hopf subalgebra you'd like to use. A rather less naive approach becomes necessary.