Indeed five months later a Frucht Theorem for finite quantum groups (using quantum graphs) was established:

arxiv.org/abs/2503.111....

It has various names; some say West Kerry, some say Corcha Dhuibhne, some say the Dingle Peninsula.

But Annascaul, Inch, etc are not "Dingle's" or in Dingle.

You can detect this is a pet peeve!😂

"Dingle's"?

Annascaul Lake is Annascaul's; not Dingle's 👍

Holy Crow.

We also had a counting argument.

Every quotient of Z2*Z3 (free product) in duality is a quantum subgroup of q perm group on five symbols. There are uncountably many of them but countably many graphs.

We showed that when Kac-Paljutkin acts on a graph the graph must have more classical symmetries than the classical version of KP.

And if a q perm group is the q automorphism group of a graph its classical version must be the classical automorphism group.

So KP is not the q autos of a graph.

They are the ones.

Banica and I showed some explicit q permutation groups could not be the quantum automorphism group of a finite graph.

Every compact quantum group has a classical version which is a q subgroup of the compact q group.

When a q group acts on a graph so do subgroups.

(Of course this is in the framework of Woronowicz not Drinfeld and Jimbo).

I didn't know were two Hopfs!

Do enough Hopf algebra stuff and "count" looks like a typo.

The question is what class of objects are e.g. quantum permutation groups the quantum automorphism groups of.

The obvious suggestion is quantum graphs. People are probably currently trying to prove such a theorem.

Every finite permutation group is isomorphic to the automorphism group of a finite graph. This is Frucht's Theorem.

Finite graphs also have quantum automorphism groups but not every quantum permutation group is iso to a quantum automorphism group.