Groups and geometries

Verify that the group $A_{1}\left(\mathbb {F}_{p}\right)$ of automorphisms of the affine line over a field of $p$ elements ($p$ prime) has order $p(p-1)$. To which group is $A_{1}\left(\mathbb {F}_{3}\right)$ isomorphic?

Give a geometric description of the proper motion $f$ of the Euclidean plane, if

Carry out a classification of the proper motions of four-dimensional Euclidean point space.