Let $G$ be a group. For $g,h \in G$, let $[g,h]=g^{-1}h^{-1}gh$ be a commutator. The normal subgroup $G' = \langle [g,h] \ | \ g,h \in G \rangle$ is called the commutator subgroup or derived subgroup.
An integral of $G$ is a group $H$ such that $H'\simeq G$. The problem of the existence of an integral was first mention by B.H. Neumann is this paper (1956). A group without integral is called non-integrable. The smallest non-integrable finite group is the symmetric group $S_3$; moreover $S_n$ is non-integrable $\forall n \ge 3$.
Here are two recent references about integrals of groups: Filom-Miraftab (2017) and Araújo-Cameron-Casolo-Matucci (2019).
The commutator subgroup is the smallest normal subgroup for which the quotient is commutative. This notion was extended to semisimple Hopf algebra by Burciu (here, 2012) and studied by Cohen-Westreich (see for ex. here). It is called commutator subalgebra. It is the smallest normal left coideal subalgebra for which the quotient is commutative. Then let call a semisimple Hopf algebra integrable if it is isomorphic to the commutator subalgebra of a semisimple Hopf algebra.
Question: Are the Hopf algebras $\mathbb{C}S_n$ integrable? What if $n=3$?
More generally we can ask whether there exist a non-integrable finite group which is integrable as Hopf algebra, and if so, whether there is one which is not, and if no, whether every finite dimensional semisimple Hopf algebra is integrable.
