Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see [KSZ06].
We are interesting in an alternative generalization.
First of all note that the following direct generalization is false:
Statement 1: If a prime number $p$ divides $\dim(\mathbb{A})$ then there is a Hopf $\star$-subalgebra $\mathbb{B} \subseteq \mathbb{A}$ of dimension $p$.
It is false because for $\mathbb{A}$ the dual of $\mathbb{C}A_5$, $2$ divides $\dim(\mathbb{A})$ and any Hopf $\star$-subalgebra of dimension $2$ should correspond to an index $2$ subgroup of $A_5$, but any index $2$ subgroup is normal, which is in contradiction with $A_5$ simple (thanks to M. Izumi and R. Ng for showing me this example).
Definition: a left coideal $\star$-subalgebra $\mathbb{I} \subset \mathbb{A}$ is a subalgebra $\mathbb{I}$ with $\mathbb{I}^{\star} = \mathbb{I}$ and $\Delta(\mathbb{I}) \subseteq \mathbb{A} \otimes \mathbb{I}$.
Statement 2: If a prime number $p$ divides $\dim(\mathbb{A})$ then there are left coideal $\star$-subalgebras $\mathbb{I} \subseteq \mathbb{J} \subseteq \mathbb{A}$ such that $\dim(\mathbb{J})/ \dim(\mathbb{I}) = p$.
Question: Is Statement 2 true?
It was checked on every left coideal $\star$-subalgebra lattices mentioned in [DT11].
Remark: the use of left coideal $\star$-subalgebra is natural for the following reason coming from subfactor theory. Let $R$ be the hyperfinite ${\rm II}_1$ factor, then $\mathbb{A}$ acts outerly on $R$ and $(R^{\mathbb{A}} \subseteq R)$ is an irreducible finite index depth $2$ subfactor (moreover every irreducible finite index depth $2$ subfactor is of this form). Now there is a one-to-one Galois correspondence [ILP98] between the intermediate subfactors $R^{\mathbb{A}} \subseteq P \subseteq R$ and the left coideal $\star$-subalgebras $\mathbb{I} \subseteq \mathbb{A}$ (by $P = R^{\mathbb{I}}$).
Whatever the answer (positive or negative) it would be surprising in any case, because if the answer is positive then this unexpected Cauchy theorem would be true, and if the answer is negative, then we could expect the existence of non-trivial maximal Hopf ${\rm C}^{\star}$-algebra! (see Is there a non-trivial Hopf algebra without left coideal subalgebra? and Non weakly-group-theoretical integral fusion category).
References
[DT11] David, Marie-Claude; Thiéry, Nicolas M. Exploration of finite-dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions. J. Algebra Appl. 10 (2011), no. 5, 995--1106
[KSZ06] Kashina, Yevgenia; Sommerhäuser, Yorck; Zhu, Yongchang. On higher Frobenius-Schur indicators. Mem. Amer. Math. Soc. 181 (2006), no. 855, viii+65 pp.
[ILP98] Izumi, Masaki; Longo, Roberto; Popa, Sorin. A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155 (1998), no. 1, 25--63.
