On Nichols algebras over PGL(2,q) and PSL(2,q)

We compute necessary conditions on Yetter-Drinfeld modules over the groups $\mathbf{P}\mathbf{G}\mathbf{L}(2,q)=\mathbf{P}\mathbf{G}\mathbf{L}(2,{\text{FF}}_q)$ and $\mathbf{P}\mathbf{S}\mathbf{L}(2,q)=\mathbf{P}\mathbf{S}\mathbf{L}(2,{\text{FF}}_q)$ to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that there is no non-trivial finite-dimensional pointed Hopf algebra over the Mathieu groups $M_{20}$ and $M_{21}=\mathbf{P}\mathbf{S}\mathbf{L}(3,4)$.