On semisimple Hopf algebras of dimension $pq$

Abstract:

We consider the problem of the classification of semisimple Hopf algebras $A$ of dimension $pq$ where $p<q$ are two prime numbers. First we prove that the order of the group of grouplike elements of $A$ is not $q$, and that if it is $p$, then $q=1$ $(\operatorname {mod} p)$. We use it to prove that if $A$ and its dual Hopf algebra $A^*$ are of Frobenius type, then $A$ is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension $3p$, and a partial classification in dimensions $5p$ and $7p$.