Finite quantum groups and quantum permutation groups

TL;DR

This paper explores finite quantum permutation groups, providing examples from twisting constructions and bicrossed products, and identifies the smallest dimension for non-quantum permutation finite quantum groups.

Contribution

It introduces new examples of finite quantum permutation groups and demonstrates the minimal dimension for non-quantum permutation finite quantum groups.

Findings

Finite quantum permutation groups from twisting and bicrossed products

Existence of finite quantum groups not being permutation groups

Smallest dimension for non-quantum permutation quantum groups is 24

Abstract

We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not quantum permutation groups: one such example occurs as a split abelian extension associated to the exact factorization $\mathbb S_4 = \mathbb Z_4 \mathbb S_3$ and has dimension 24. We show that, in fact, this is the smallest possible dimension that a non quantum permutation group can have.